Chapter Three. Visualization and Sketching


After studying the material in this chapter, you should be able to:

1. Define the terms vertex, edge, plane, surface, and solid.

2. Identify four types of surfaces.

3. Identify five regular solids.

4. Draw points, lines, angled lines, arcs, circles, and ellipses.

5. Apply techniques that aid in creating legible well-proportioned freehand sketches.

6. Apply techniques to draw irregular curves.

7. Create a single-view sketch.

8. Create an oblique sketch.

9. Create perspective sketches.

10. Create an isometric sketch of an object.

Refer to the following standard:

• ANSI/ASME Y14.3 Orthographic and Pictorial Views

Shaded sketch displaying Details of Wire Placement of a toilet flush.

Shaded Sketch Showing Details of Wire Placement (Courtesy of Quantum Design.)


The ability to envision objects in three dimensions is one of the most important skills for scientists, designers, engineers, and technicians. Learning to visualize objects in space, to imagine constructively, is something you can learn by studying technical drawing. People who are extraordinarily creative often possess outstanding ability to visualize, but with practice anyone can improve his or her ability.

In addition to its role in developing spatial thinking skills, sketching is a valuable tool that allows you to communicate your ideas quickly and accurately. During the development stage of an idea, a picture is often worth a thousand words.

Sketching is also an efficient way to plan your drawing and record notes needed to create a complex object. When you sketch basic ideas ahead of time, you can often complete a final CAD drawing sooner and with fewer errors. Using good technique makes sketching faster, easier, and more legible.

Image of complex models created using 3D CAD.

Complex Surface Models Created Using 3D CAD (Courtesy of Professor Richard Palais, University of California, Irvine, and Luc Benard.)

Understanding Solid Objects

Sketches and drawings are used to communicate or record ideas about the shape of 3D objects. Before starting to sketch, it helps to develop a vocabulary for understanding and discussing 3D shapes.

Three-dimensional figures are referred to as solids. Solids are bounded by the surfaces that contain them. These surfaces can be one of the following four types:


Single curved

Double curved


Regardless of how complex a solid may be, it is composed of combinations of these basic types of surfaces. Figure 3.1 shows examples of the four basic types of surfaces.

Figures display the four basic types of surfaces.

3.1 Types of Surfaces

Types of Solids


Solids that are bounded by plane surfaces are called polyhedra (Figures 3.23.4). These planar surfaces are also referred to as faces of the object. A polygon is a planar area that is enclosed by straight lines.

Five regular polyhedras are displayed.

3.2 Regular Polyhedra

Type of prisms is displayed.

3.3 Right Prisms and Oblique Prisms

Type of pyramids is displayed.

3.4 Pyramids

Regular Polyhedra

If the faces of a solid are equal regular polygons, it is called a regular polyhedron. There are five regular polyhedra: the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron (Figure 3.2).


A prism has two bases, which are parallel equal polygons, and three or more additional faces, which are parallelograms (Figure 3.3). A triangular prism has triangular bases, a rectangular prism has rectangular bases, and so on. (If a prism’s bases happen to be parallelograms, the prism is a called a parallelepiped, a word rarely heard in everyday conversation.)

A right prism has faces and lateral (side) edges that are perpendicular to the bases; an oblique prism has faces and lateral edges that are angled to the bases. If one end is cut off to form an end that is not parallel to the bases, the prism is said to be truncated (a word that simply means “shortened by having a part cut off”).


A pyramid has a polygon for a base and triangular lateral faces that intersect at a common point called the vertex (Figure 3.4). The line from the center of the base to the vertex is called the axis. If the axis is perpendicular to the base, the pyramid is called a right pyramid; otherwise, it is an oblique pyramid. A triangular pyramid has a triangular base, a square pyramid has a square base, and so on. If a portion near the vertex has been cut off, the pyramid is truncated, or it is referred to as a frustum.


A cylinder has a single-curved exterior surface (Figure 3.5). You can think of a cylinder as being formed by taking a straight line and moving it in a circular path to enclose a volume. Each position of this imaginary straight line in its path around the axis is called an element of the cylinder.

Figure shows a cylinder labeled, right circular and a cylinder placed in an angle labeled, oblique circular.

3.5 Cylinder and Oblique Cylinder


A cone has a single-curved exterior surface (Figure 3.6). You can think of it as being formed by moving one end of a straight line around a circle while keeping the other end fixed at a point, the vertex of the cone. An element of the cone is any position of this imaginary straight line.

Type of cones is displayed.

3.6 Cones


A sphere has a double-curved exterior surface (Figure 3.7). You can think of it as being formed by revolving a circle about one of its diameters, somewhat like spinning a coin. The poles of the sphere are the points at the top and bottom of the sphere that would not move while it was spinning. The axis of the sphere is the term for the line between its poles.

Figure shows a sphere.

3.7 Sphere


A torus is shaped like a doughnut (Figure 3.8). Its boundary surface is double curved. You can think of it as being formed by revolving a circle (or other curve) around an axis that is positioned away from (outside) the curve.

Figure shows two concentric ovals labeled, Torus.

3.8 Torus


An oblate or prolate ellipsoid is shaped like an egg (Figure 3.9). You can think of it as being formed by revolving an ellipse about its minor or major axis, respectively.

Figure displays an ellipsoid placed horizontally labeled, Oblate ellipsoid. And an ellipsoid labeled, prolate ellipsoid.

3.9 Ellipsoids

Understanding Sketching Techniques

Analyzing Complex Objects

The ability to break down complex shapes into simpler geometric primitives is an essential skill for sketching and modeling objects.

Before you begin to draw the outline of an object, consider its overall shape and the relationships between its parts. Construction lines can help you preserve the overall dimensions of the object as you sketch.

Bear in mind that you should be thinking in terms of basic shapes whether you are sketching by hand or using a CAD program. Because basic curves and straight lines are the basis of many of the objects that people create, practice in creating the basic elements of a drawing will help you sketch with ease.

Essential Shapes

Look for the essential shapes of objects. If you were to make a clay model of an object, what basic shape would you start with? A ball? A box?

Try squinting your eyes and looking at familiar objects. Do you see their shape as a rectangle? A circle? What other basic shapes do you notice when you look at objects this way?

Think about breaking down more complex objects into their simpler geometric shapes as shown in Figure 3.10. You can block in these shapes using construction lines to show their relationships to one another. Then, add details, continuing to pay attention to the spatial relationships between them.

Photograph of a table lamp is placed on the left. To its right, shapes such as a circle, triangle, and cylinder are joined to create the shape of the table lamp.

3.10 Identifying Essential Shapes

Construction Lines

Artists often begin a sketch by blocking in light guidelines to help them preserve basic shapes and proportions. In technical drawing these are called construction lines (Figure 3.12).

Photograph of a corkscrew is placed on the left. To its right, the essential shapes of the corkscrew are sketched.

3.11 The essential shapes of the corkscrew in the photo are sketched at right.

Construction lines are depicted.

3.12 Using Construction Lines

It is often helpful to begin a sketch by describing the object’s main shapes with construction lines, taking some care to accurately represent the relative size and placement of features.

Use the basic shapes as a guide to place key features, then use those main features as a “reference map” to place smaller details. For example, the sixth fret line is about halfway up the rectangular guitar neck.

Throughout this chapter you will use light construction lines to draw circles, arcs, and ellipses. Section 3.4 discusses the process of estimating and maintaining the proportions of an object in further detail.

Contours and Negative Space

The contours of an object are the main outlines that separate it from the surrounding space. One way to think about the contours of objects is to look at the contrast between the positive and negative space. Positive space is the space occupied by the object. Negative space is the unoccupied space around it.

In Figure 3.13 the space occupied by the contour of a pair of scissors is shown. Note how you can identify specific shapes by looking at the negative space. The individual shapes that make up the negative space are shown in different colors to make them easier for you to see. Some people sketch more accurately when they try to draw the negative space that surrounds the object.

Image explains Contours and Negative Space.

3.13 Negative Space

Tip: Practice Drawing Contours

Photograph of a wooden chair.

Try sketching the negative spaces that define the shape of a chair. Look at each space as an individual shape. What is the shape of the space between the legs? What is the shape of the space between the rungs and the seat?

Make a sketch of a chair, paying careful attention to sketching the negative spaces of the chair as they really appear. The positive and negative spaces should add up to define the chair.

If you have difficulty, make corrections to your sketch by defining the positive shapes and then check to see if the negative shapes match.

An 8.5″ × 11″ sheet of Plexiglas® (available at most glass stores) is an excellent tool for developing sketching ability. Using a dry-erase marker, hold the Plexiglas up in front of an object and trace the object’s contours on the Plexiglas. If you don’t move, the outline should match the object’s outline exactly. Lower the Plexiglas and look at the orientation of the lines. Are they what you expected?

Try looking at the object and drawing the sketch with the Plexiglas lying on your desktop or knees. Then, raise it and see if your drawing matches the object.

To develop sketching ability, try drawing everyday objects like your toaster, printer, or lamp, as well as exterior and interior views of buildings and equipment.

Making a sketch of a chair is explained.


As you sketch objects, keep in mind that you want to maintain a consistent viewpoint, as a camera does. This is easier when you are sketching a picture from a book, because you can’t move around the object. When you move, you see a different view of the object depending on where you stand.

Sometimes people have difficulty sketching because they want to show parts of the object that cannot really be seen from a single viewpoint. For example, knowing that the handle of the rubber stamp in Figure 3.14 appears circular from the top, you may be tempted to show it as round, even though it may appear elliptical from your viewpoint.

Photograph of a rubber stamp.

3.14 Rubber Stamp

When you are sketching an object pictorially, temporarily set aside your knowledge of the shapes the object is actually made of and carefully examine the shapes you see from a single, static viewpoint. In this type of sketching, instead of trying to envision the object as it is, try only to see it as it looks.


Adding shading to your sketch can give it a more realistic appearance because it represents the way the actual object would reflect light. Shading doesn’t mean “coloring in.” You may want to shade only the most prominently shadowed areas. First, identify the darkest and lightest areas on an object. If you want, you can shade various middle tones, placed exactly as they look on the object.

In some ways, shading is like doing a drawing within a drawing, because it is a matter of identifying shapes. When you are shading, instead of identifying the shapes of the object’s contours, you are identifying the shape and relative darkness of the shadows.

Hatching lines, shown in Figure 3.15, and stippling, shown in Figure 3.16, are commonly used methods to add shading because they are easier to reproduce with a photocopier than continuous-tone pencil shading. In the illustration you can see that shadowed areas are darkened simply by adding more hatching lines or stippling dots.

Drawing of a rubber stamp labeled, Hatching.

3.15 Hatching

Drawing of a rubber stamp labeled, Stipping.

3.16 Stippling

It is not uncommon for people to draw outlines by hand and add digital shaded fills to a scan of the outline. Another way to make a subject the clear focal point of a drawing is to stylize the shadows. Industrial designers often use markers to add a stylized shadow to their sketches. You can use a straight edge to protect the original sketch from the marker and quickly sketch the shadow (Figure 3.17).

The concept of Marker Shading is depicted.

3.17 Marker Shading in a Concept Sketch (Courtesy of Douglas Wintin.)

Regardless of how you apply shading, darken the outline to define the shape clearly and boldly. Remember that when you are communicating by using a sketch, its subject should be clear. To make the subject—in this case, a rubber stamp—clear, make it stand out with thick bold contour lines.

Edges and Vertices


An edge of a solid is formed where two surfaces intersect. Edges are represented in drawings by visible or hidden lines (Figure 3.18).

Figure shows a three dimensional inverted L shape. The sides are marked face, corners are marked vertex, and edges are marked.

3.18 Edges and Vertices of a Solid


A vertex (plural, vertices) of a solid is formed where three or more surfaces intersect. The end of an edge is a vertex. These vertices or “points” are very useful in defining the locations of the solid object feature that you will sketch (Figure 3.18).

Points and Lines

A point is used to represent a location in space but has no width, height, or depth (Figure 3.19). A point in a drawing is represented by the intersection of two lines (Figure 3.19a), by a short crossbar on a line (Figure 3.19b), or by a small cross (Figure 3.19c). Do not represent points by simple dots on the paper. This makes the drawing look “blobby” and is not as accurate.

Three figures show a line intersecting with "another line, a short crossbar of line, and a small perpendicular line." The intersections are labeled, Point.

3.19 Showing Points

A line is used in drawings to represent the edge of a solid object. A straight line is the shortest distance between two points and is commonly referred to simply as a “line.” If the line is indefinite in extent, in a drawing the length is a matter of convenience, and the endpoints are not marked (Figure 3.20a). If the endpoints of the line are significant, they are marked by small drawn crossbars (Figure 3.20b). Other common terms are illustrated in Figures 3.20c to i. Either straight lines or curved lines are parallel if the shortest distance between them remains constant. The common symbol for parallel lines is | |, and for perpendicular lines it is ⊥. Two perpendicular lines may be marked with a “box” as shown in Figure 3.20g. Such symbols may be used on sketches, but not on production drawings.

Types of line are displayed.

3.20 Showing Lines


An angle is formed by two intersecting lines. A common symbol for angle is ∠.

There are 360 degrees (360°) in a full circle, as shown in Figure 3.21a. A degree is divided into 60 minutes (60′), and a minute is divided into 60 seconds (60″). The angle value 37°26′10″ is read 37 degrees, 26 minutes, and 10 seconds. When minutes alone are indicated, the number of minutes should be preceded by 0°, as in 0°20′. If the minutes value were listed alone without showing the zero value, it might be mistaken for a distance measurement. For example, 20 minutes, 10 seconds written as 20′10″ might be mistaken for 20 feet 10 inches.

Figures depicts Angles.

3.21 Showing Angles

The different kinds of angles are illustrated in Figure 3.21. Two angles are complementary if they total 90° (Figure 3.21f) and are supplementary if they total 180° (Figure 3.21g).

In sketching, most angles can be estimated. Use a protractor if necessary when drawing odd angles.

Drawings and Sketches

The following are important skills to keep in mind for sketches and drawings:

1. Accuracy. No drawing is useful unless it shows the information correctly.

2. Speed. Time is money in industry. Work smarter and learn to use techniques to speed up your sketching and CAD drawings while still producing neat accurate results.

3. Legibility. A drawing is a means of communicating with others, so it must be clear and legible. Give attention to details. Things that may seem picky and small as you are drawing may be significant and save money or even lives when the product is built.

4. Neatness. If a drawing is to be accurate and legible, it must also be clean.

Freehand Sketching

Freehand sketches are a helpful way to organize your thoughts and record ideas. They provide a quick, low-cost way to explore various solutions to design problems so that the best choices can be made. Investing too much time in creating a detailed layout before exploring your options through sketches can be costly.

The degree of precision needed in a given sketch depends on its use. Quick sketches to supplement verbal descriptions may be rough and incomplete. Sketches can be used to convey important and precise information when they are clearly drawn and annotated.

Freehand sketching requires only pencil, paper, and eraser. Mastering the techniques in this chapter for showing quick single-view, oblique, perspective, and isometric drawings using good freehand line technique will give you a valuable tool for communicating your ideas.

The term freehand sketch does not mean a sloppy drawing. As shown in Figure 3.22, a freehand sketch shows attention to proportion, clarity, and correct line widths. Figure 3.23 shows an as-built drawing with corrected items sketched on the printed CAD drawing.

A hand drawn sketch on a graph paper.

3.22 Sketch on Graph Paper. Sketches are also used to clarify information about changes in design or to provide information on repairing existing equipment.

An as-built drawing with corrected items sketched on the printed CAD drawing.

3.23 An As-Built Drawing with Corrected Items Sketched on the Printed CAD Drawing

3.1 Technique of Lines

The chief difference between a drawing and a freehand sketch lies in the character or technique of the lines. A good freehand line is not expected to be as rigidly straight or exactly uniform. A good freehand line shows freedom and variety, whereas a line drawn using CAD or instruments should be exact. Still, it is important to distinguish between line patterns to make your drawing legible.

The line patterns in Figure 3.24 are examples of good freehand quality. Figure 3.25 shows examples of good and poor technique.

Line patterns are displayed.

3.24 Freehand Alphabet of Lines (Full Size)

Technique of lines is depicted.

3.25 Technique of Lines (Enlarged)


• Make dimension, extension, and centerlines thin, sharp, and black.

• Make hidden lines medium and black.

• Make visible and cutting-plane lines thick and black.

• Make construction lines thick and light.


Even in freehand drawings, thick lines should be twice the width of thin lines.

Thicknesses do not have to be exact, but there should be an obvious difference between thick and thin lines.

Because visible lines and cutting-plane lines are the two thick line patterns, other lines should be distinctly thinner in comparison.

To draw thick and thin lines freehand, you might like to keep two pencils handy, one that is razor sharp for thin lines and another that is dulled, to create thicker lines.

As the sharp point becomes dulled, switch it with the dull pencil, and sharpen the other, so that there is always one sharp and one dulled point ready to use.

Type of lines is explained.

3.2 Sketching Straight Lines

Most of the lines in an average sketch are straight lines. With practice, your straight lines will naturally improve, but these basics may help you improve quickly.

• Hold your pencil naturally, about 1″ back from the point, and approximately at a right angle to the line to be drawn.

• Draw horizontal lines from left to right with a free and easy wrist and arm movement.

• Draw vertical lines downward with a wrist and arm movement.

• Draw curved lines using finger and wrist movements.

Blocking in a Freehand Drawing

Over the years, freehand sketchers have developed all sorts of tricks to improve speed and accuracy. Methods for finding midpoints or quickly blocking in straight vertical and horizontal lines are just a few secrets of the technical sketching craft that can come in handy, even today. When a great idea hits, or you need to sketch quickly at a meeting or on a job site, you might not have access to a CAD system, or even a ruler.

Tips: Drawing Long Freehand Lines

For long freehand lines, make light end marks and lightly sweep your pencil between them, keeping your eye on the mark toward which you are moving. When you are satisfied with the accuracy of your strokes, apply more pressure to make a dark line.

Two figures depict drawing freehand lines.

Blocking in a Border Freehand

Hold your hand and pencil rigidly and glide your fingertips along the edge of the paper to maintain a uniform border.

A rectangle is drawn on a sheet of paper with a pencil. A bi-directional arrow labeled, Keep this distance from the edge is placed between the border of the rectangle and the page.

Blocking in a Border Using a Strip of Paper

Mark the distance on the edge of a card or a strip of paper and use it like a ruler to mark at intervals, then draw a final line through the points.

Two sides of a rectangle are drawn on a sheet of paper. A strip of paper is placed on the other side with outlines marked.

If your line looks like this, you may be gripping your pencil too tightly or trying too hard to imitate mechanical lines.

Figure shows a line that bends downward.

Slight wiggles are OK as long as the line continues on a straight path.

Figure shows a slightly wiggly line.

Occasional very slight gaps are fine and make it easier to draw straight.

Figure shows a line with slight occasional gaps.

Finding a Midpoint Freehand

Use your thumb on your pencil to guess half the distance. Try this distance on the other half. Continue adjusting until you locate the center, then mark it.

Sketch shows a sheet of paper with a line drawn on the top. The sides of the line are marked A and B, and the center is marked C. A hand holding a pencil is placed parallel to the line.

Folding a Paper to Find a Midpoint

Mark the total distance on the edge of a strip of paper, then fold the paper to locate its center at the crease. You can fold one half to find quarter points, and so on.

Sketch shows a sheet of paper with a line drawn at the top. The sides of the line are marked A and B, and the center is marked C. The strip of paper placed on the sheet is seen folded.

Step by Step: Dividing Lines into Equal or Proportional Parts

Proportional Parts

To divide the given line shown into proportions of (for example) 2, 3, and 4 units:

Figure explains Proportional parts.

Image Draw a vertical construction line at one end of the line you are dividing.

Figure shows a horizontal line. A vertical downward arrow is drawn perpendicularly from the end of the line. The angle of intersection is marked, Sketch vertical line from the end.

Image Set the zero point of your scale at the other end of the line.

Image Swing the scale so the desired unit falls on the vertical line. In this case it will be the 9th unit, because 2 + 3 + 4 = 9.

Concept of swinging the scale is depicted.

Image Draw vertical lines upward from the corresponding scale divisions and mark tiny crossbars on the line as shown.

Equal Parts

If you use uniform divisions for the steps above (every third division, for instance) you will get equal parts. Examples of practical applications for dividing lines equally are shown below.

Calculated Proportions

To divide a line into proportions equal to the square of 1, 2, 3, and 4 (1, 4, 9, and 16), find 16 divisions on your scale.

Figure shows a scale with top markings labeled, 1, 4, 9, and 16 respectively. The bottom section of the scale is marked (1 squared), (2 squared), (3 squared), and (4 squared).

Image Set the zero point of your scale at the end of the line and draw a light construction line at any convenient angle from one end of the line you are dividing to the appropriate division on the scale. In this case the 4 mark is 16 equal divisions from the 0.

Image Draw construction lines parallel to the end line through each proportionate scale division.

Figure shows a horizontal line with four points marked 2, 3, and 4. A scale is placed diagonally opposite to the line with markings 0, 2, 5, and 9 on the scale represents the points on the line. A text reading "Swing scale so last desired division lines up with a vertical line" leads to 9.

Tip: Exaggerating Closely Spaced Parallel Lines

Sometimes it is helpful to exaggerate the distance between closely spaced parallel lines so there is no fill-in when the drawing is reproduced.

Usually this is done to a maximum of 3 mm or .120″. When using CAD it is better to draw the features the actual size and include a detail showing the actual spacing.

Practical applications for dividing lines equally are shown.

3.3 Sketching Circles, Arcs, and Ellipses


Small circles can be sketched using one or two strokes, without blocking in any construction lines. Circle templates also make it easy to sketch circles of various sizes.

For better looking freehand sketched circles of larger sizes, try the construction methods shown here. Figure 3.26 shows an object with rounded features to sketch using circles, arcs, and ellipses.

Photograph of a table fan.

3.26 Many objects have rounded features that circles, arcs, and ellipses are used to represent. (Tim Ridley © Dorling Kindersley.)

Tip: The Freehand Compass

Using your hand like a compass, you can create circles and arcs with surprising accuracy after a few minutes of practice.

1. Place the tip of your little finger or the knuckle joint of your little finger at the center.

2. “Feed” the pencil out to the radius you want as you would do with a compass.

3. Hold this position rigidly and rotate the paper with your free hand.

The Freehand Compass method is depicted.

Step by Step: Methods for Sketching Circles

Enclosing Square Method

Sketch of a square with the midpoints of the side marked.

Image Lightly sketch an enclosing square and mark the midpoint of each side.

Sketch of a square with the midpoints of the side marked. Center lines are drawn on arcs connecting the midpoints.

Image Lightly draw in arcs to connect the midpoints.

Sketch of a square with the midpoints of the side marked. Center lines are drawn on arcs connecting the midpoints and are darken to form a circle.

Image Darken the final circle.

Centerline Method

Sketch of center lines.

Image Sketch the two centerlines of the circle.

Arcs are drawn to form a circle on a set of four center lines.

Image Add light 45° radial lines and sketch light arcs across them at an estimated radius distance from the center.

The arcs drawn to form a circle on a set of four center lines are darkened.

Image Darken the final circle.

Paper Method

Sketch shows center lines surrounded by a circle. A scrap of paper is placed along the Radius to determine the circle.

Image Mark the estimated radius on the edge of a card or scrap of paper and set off from the center as many points as desired.

Sketch shows center lines surrounded by a circle. The border of the sector on top-left is dashed lines. A hand with a pencil labeled, sketch circle through points is placed on the circle.

Image Sketch the final circle through these points.

Step by Step: Methods for Sketching Arcs

Radius Method

Image Locate the center of the arc and lightly block in perpendicular lines. Mark off the radius distance along the lines.

Sketch of perpendicular lines. Arcs are drawn at the end of the lines.

Image Draw a 45° line through the center point and mark off the radius distance along it.

Sketch of perpendicular lines. A 45-degree line is drawn from the point of intersection of the lines. Arcs are sketched on the end of the lines.

Image Lightly sketch in the arc as shown. Darken the final arc.

Sketch of perpendicular lines. A 45-degree line is drawn from the point of intersection of the lines. Arcs are sketched on the end of the lines. The arcs are joined to form a sector.

Trammel Method

Image Locate the center of the arc and lightly block in perpendicular lines. Mark off the radius distance along the lines.

Sketch of two perpendicular lines. Arcs are drawn at the end of the lines.

Image Mark the radius distance on a strip of paper and use it as a trammel.

Sketch of two perpendicular lines. Arcs are drawn at the end of the lines. A strip of paper labeled, Trammel is placed at the point of intersection and an arc is drawn.

Image Lightly sketch in the arc, then darken the final arc.

Sketch of two perpendicular lines. Arcs are drawn at the end of the lines. A strip of paper labeled, a pencil is placed at the point of intersection and an arc is drawn.

Tangent Method

Use these steps to draw arcs sketched to points of tangency.

Image Locate the center of the arc and sketch in the lines to which the arc is tangent.

An inclined line and a horizontal line intersects approximately at 30-degree angle. The center of the arc is marked.

Image Draw perpendiculars from the center to the tangent lines.

An inclined line and a horizontal line intersects approximately at 30-degree angle. The center of the arc is marked. Perpendicular lines are drawn from the center of the arc to the tangent.

Image Draw in the arc tangent to the lines ending at the perpendicular lines.

An inclined line and a horizontal line intersects approximately at 30-degree angle. The center of the arc is marked. Perpendicular lines are drawn from the center of the arc to the tangent. An arc is drawn perpendicular to the lines.

Image Darken in the arc and then darken the lines from the points of tangency.

An inclined line and a horizontal line intersects approximately at 30-degree angle. The center of the arc is marked. Perpendicular lines are drawn from the center of the arc to the tangent. The arc drawn perpendicular to the lines is shaded.

Methods for Sketching Ellipses

Freehand Method

Image Rest your weight on your upper forearm and move the pencil rapidly above the paper in an elliptical path.

Image Lower the pencil to draw very light ellipses.

Image Darken the final ellipse.

Sketch of a circle along with three smaller circles above it.
A dark sketch of a circle.

Rectangle Method

Image Lightly sketch an enclosing rectangle.

Image Mark the midpoint of each side and sketch light tangent arcs.

Image Darken in the final ellipse.

A lightly sketched rectangle with center lines. Tangent arcs are drawn at the midpoint of each side.
A lightly sketched rectangle with center lines. Tangent arcs are drawn at the midpoint of each side. The arcs are darkened to form a circle.

Axes Method

Image Lightly sketch in the major and minor axes of the ellipse.

Image Mark the distance along the axes and lightly block in the ellipse.

Image Darken the final ellipse.

Sketch of center lines with tangent arcs drawn across the end of the lines.
Sketch of center lines with tangent arcs drawn across the end of the lines. The arcs are darkened to form a circle.

Trammel Method

Image To sketch accurate ellipses, you can make a trammel.

Sketch of center lines with the top-right section marked Minor axis, and the top-left section marked Major axis.

Image Mark half the desired length of the minor axis on the edge of a strip of paper (A-B). Using the same starting point, mark half the length of the major axis (A-C). (The measurements will overlap.)

Figure explains the second step of the trammel method.

Image Line up the last two trammel points (B and C) on the axes and mark a small dot at the location of the first point (A).

Figure explains the third step of the trammel method.

Image Move the trammel to different positions, keeping B and C on the axes, and mark more points at A. Sketch the final ellipse through the points.

Sketch shows center lines enclosed in a circle.

Sketching Arcs

Sketching arcs is similar to sketching circles. In general, it is easier to hold your pencil on the inside of the curve. Look closely at the actual geometric constructions and carefully approximate points of tangency so that the arc touches a line or other entity at the right point.

Sketching Ellipses

If a circle is tipped away from your view, it appears as an ellipse. Figure 3.27 shows a coin viewed so that it appears as an ellipse. You can learn to sketch small ellipses with a free arm movement similar to the way you sketch circles, or you can use ellipse templates to help you easily sketch ellipses. These templates are usually grouped according to the amount a circular shape would be rotated to form the ellipse. They provide a number of sizes of ellipses on each template but usually include only one or two typical rotations.

The side view, top view, and front view of a coin.

3.27 A Circle Seen as an Ellipse

3.4 Maintaining Proportions

Sketches are not usually made to a specific scale, although it can be handy to do so at times. The size of the sketch depends on its complexity and the size of the paper available. The most important rule in freehand sketching is to keep the sketch in proportion, which means to accurately represent the size and position of each part in relation to the whole. No matter how brilliant the technique or how well drawn the details, if the proportions are off, the sketch will not look right.

To maintain proportions, first determine the relative proportions of height to width and lightly block them in. You can mark a unit on the edge of a strip of paper or use your pencil (as in Figure 3.28) to gauge how many units wide and high the object is. Grid paper can help you maintain proportions by providing a ready-made scale (by counting squares). As you block in the medium-size areas, and later as you add small details, compare each new distance with those already established.

Estimating dimensions is displayed in the figure.

3.28 Estimating Dimensions

Step by Step: Maintaining Proportions in a Sketch

Image If you are working from a given picture, such as this utility cabinet, first establish the relative width compared to the height. One way is to use the pencil as a measuring stick. In this case, the height is about 1-3/4 times the width.

A sketch of a cabinet is displayed. Pencils are placed heightwise and widthwise. Difference between height and width is labeled.

Image Sketch the enclosing rectangle in the correct proportion. This sketch is to be slightly larger than the given picture.

Sketch of the enclosing rectangle of the cabinet is displayed. Width and height of the rectangle are labeled.

Image Divide the available drawer space into three parts with the pencil by trial. Hold your pencil about where you think one third will be and then try that measurement. If it is too short or long, adjust the measurement and try again. Sketch light diagonals to locate centers of the drawers and block in drawer handles. Sketch all remaining details.

Sketch of the cabinet with the available drawer spaces divided into three parts.

Image Darken all final lines, making them clean, thick, and black.

Final sketch of the cabinet.

Step by Step: How to Block in an Irregular Object

Image Capture the main proportions with simple lines.

Sketch shows a vertical rectangle placed on a horizontal rectangular block.

Image Block in the general sizes and direction of flow of curved shapes.

Sketch shows a vertical rectangle placed on a horizontal rectangular block. Direction of the flow of curved shapes is marked.

Image Lightly block in additional details.

Sketch shows a vertical rectangle placed on a horizontal rectangular block. Direction of the flow of curved shapes and additional details are marked.

Image Darken the lines of the completed sketch.

The lines of the completed sketch are darkened.

Step by Step: Geometric Methods for Sketching Plane Figures

Sketching a Polygon by the Triangle Method

Image Divide the polygon into triangles as shown. Use the triangles as a visual aid to sketch the shape.

Sketch of a polygon with a triangle placed inside.

Sketching a Polygon by the Rectangle Method

Image Imagine a rectangle drawn around the polygon as shown.

Image Sketch the rectangle and then locate the vertices of the polygon (points a, b, c, and so on) along the sides of the rectangle.

Image Join the points to complete the shape.

Figure depicts the sketching of Polygon using the rectangle method.

Visual Aids for Sketching Irregular Figures

Image Visualize shapes made up of rectangular and circular forms by enclosing those features in rectangles.

Image Determine where the centers of arcs and circles are located relative to the rectangles as shown.

Image Sketch the features inside the rectangular shapes you have lightly blocked in and darken the final lines.

Sketch shows an irregular shape similar to an arc, enclosed in a rectangle. The features inside the rectangle are shaded.

Creating Irregular Shapes by Offset Measurements

Image Enclose the shape in a rectangle.

Image Use the sides of the rectangle as a reference to make measurements that locate points along the curve.

Sketch shows an irregular shape enclosed in a rectangle.

Enlarging Shapes Using a Grid of Squares

Image Complex curved shapes can be copied, enlarged, or reduced by hand, if necessary.

Image Draw or overlay a grid of squares on the original drawing.

Image To enlarge, draw the containing rectangle and grid of squares at the desired percentage and transfer the lines of the shape through the corresponding points in the new set of squares.

Sketch shows an irregular shape enclosed in a rectangle. A grid of squares is shown on the drawing.

3.5 One-View Drawings

Frequently, a single view supplemented by notes and dimensions is enough information to describe the shape of a relatively simple object.

In Figure 3.29, one view of the shim plus a note indicating the thickness as 0.25 mm is sufficient.

One-view drawing of a shim is displayed.

3.29 One-View Drawing of a Shim

Nearly all shafts, bolts, screws, and similar parts should be represented by single views in this way.

Step by Step: Sketching a Single-View Drawing

Follow the steps to sketch the single-view drawing of the shim shown in Figure 3.29.

Image Lightly sketch the centerlines for the overall width and height of the part. Estimate overall proportions by eye, or if you know the dimensions, use a measuring scale to sketch accurately sized views. Space the enclosing rectangle equally from the margins of the sheet.

A lightly sketched center lines of a shim.

Image Block in all details lightly, keeping the drawing proportions in mind. Use techniques introduced in this chapter to help you.

A lightly sketched drawing of a shim with details blocked in lightly.

Image Locate the centers of circles and arcs. Block in where they will fit using rectangles. Then, sketch all arcs and circles lightly.

A lightly sketched drawing of a shim with the centers of circles and arcs, and rectangles sketched lightly.

Image Darken your final lines.

A lightly sketched drawing of a shim with the final lines darkened.

Image Add annotations to the drawing using neat lettering. Fill in the title block or title strip. Note the scale for the sketch if applicable. If not, letter NONE in the Scale area of the title block.

3.6 Pictorial Sketching

A pictorial sketch represents a 3D object on a 2D sheet of paper by orienting the object so you can see its width, height, and depth in a single view.

Pictorial sketches are used frequently during the ideation phase of engineering design to record ideas quickly and communicate them to others. Their similarity to how the object is viewed in the real world makes them useful for communicating engineering designs to nonengineers. Later in the design process, pictorial drawings are often used to show how parts fit together in an assembly and, in part catalogs and manuals, to make it easy to identify the objects.

This chapter examines three common methods used to sketch pictorials: isometric sketching, which is a subtype of the general category of axonometric projection, oblique sketching, and perspective sketching. Figure 3.30 shows perspective, isometric, and oblique sketches of a stapler. Figure 3.31 shows pictorial sketches for backpack concepts.

Three types of pictorial sketches are shown. They are the pictorial sketches of a stapler in Perspective, Isometric, and Oblique.

3.30 Three Types of Pictorial Sketches

Sketches of multiple backpacks.

3.31 Pictorial sketching is used frequently to convey preliminary design ideas, as in these backpack concept sketches. (Courtesy of André Cotan.)

Each of the pictorial methods differs in the way points on the object are located on the 2D viewing plane (the piece of paper).

A perspective sketch presents the most realistic looking view. It shows the object much as it would appear in a photograph—portions of the object that are farther from the viewer appear smaller, and lines recede into the distance.

An axonometric sketch is drawn so that lines do not recede into the distance but remain parallel. This makes isometric views easy to sketch but takes away somewhat from the realistic appearance.

An oblique sketch shows the front surface of the object looking straight on and is easy to create, but it presents the least realistic representation because the depth of the object appears to be out of proportion.

Various types of pictorial drawings are used extensively in catalogs, sales literature, and technical work. They are often used in patent drawings; in piping diagrams; in machine, structural, architectural design, and in furniture design; and for ideation sketching. The sketches for a wooden shelf in Figure 3.32 are examples of axonometric, orthographic, and perspective sketches.

Sketches of a wooden shelf shown in Axonometric, Orthographic, and Perspective drawing techniques.

3.32 Sketches for a Wooden Shelf Using Axonometric, Orthographic, and Perspective Drawing Techniques. The axonometric projections in this sketch are drawn in isometric. (Courtesy of Douglas Wintin.)

In axonometric and oblique drawings, distant features are not shown proportionately smaller, the way they appear in a photograph or to our vision. Edges that are parallel on the object always appear parallel in an axonometric view. The most common axonometric projection is isometric, which means “equal measure.” When a cube is drawn in isometric, the axes are equally spaced (120° apart). Though not as realistic as perspective drawings, isometric drawings are much easier to draw. CAD software often displays the results of 3D models on the screen as isometric projections. Some CAD software allows you to choose between isometric, axonometric views (see Figure 3.35 for examples), or perspective representation of your 3D models on the 2D computer screen. In sketching, dimetric and trimetric sometimes produce a better view than isometric but take longer to draw and are therefore used less frequently.

3.7 Projection Methods

The four principal types of projection are shown in Figure 3.33. All except the regular multiview projection (Figure 3.33a) are pictorial types, because they show several sides of the object in a single view. In both multiview projection and axonometric projection, the visual rays are parallel to each other and perpendicular to the plane of projection. Both are types of orthographic projection (Figures 3.33a and 3.33b).

The four types of projection are explained.

3.33 Four Types of Projection

In oblique projection (Figure 3.33c), the visual rays are parallel to each other but at an angle other than 90° to the plane of projection.

In perspective (Figure 3.33d), the visual rays extend from the observer’s eye, or station point (SP), to all points of the object to form a “cone of rays,” so that the portions of the object that are farther away from the observer appear smaller than the closer portions of the object.

3.8 Axonometric Projection

The feature that distinguishes axonometric projection from multiview projection is the inclined position of the object with respect to the planes of projection. When a surface or edge of the object is not parallel to the plane of projection, it appears foreshortened. When an angle is not parallel to the plane of projection, it appears either smaller or larger than the true angle.

To create an axonometric view, the object is tipped to the planes of projection so that principal faces, such as the top, side, and front, show in a single view. This produces a pictorial drawing that is easy to visualize, but because the principal edges and surfaces of the object are inclined to the plane of projection, the lengths of the lines are foreshortened. The angles between surfaces and edges appear either larger or smaller than the true angle. There are an infinite variety of ways that the object may be oriented with respect to the plane of projection.

The degree of foreshortening of any line depends on its angle to the plane of projection. The greater the angle, the greater the foreshortening. Once the degree of foreshortening is determined for each of the three edges of the cube that meet at one corner, scales can be easily constructed for measuring along these edges or any other edges parallel to them (Figure 3.34).

The degree of foreshortening is illustrated.

3.34 Measurements are foreshortened proportionately based on the amount of incline.

Use the three edges of the cube that meet at the corner nearest your view as the axonometric axes. Figure 3.35 shows three axonometric projections.

Axonometric projections are displayed.

3.35 Axonometric Projections

Isometric projection (Figure 3.35a) has equal foreshortening along each of the three axis directions.

Dimetric projection (Figure 3.35b) has equal foreshortening along two axis directions and a different amount of foreshortening along the third axis. This is because it is not tipped an equal amount to all the principal planes of projection.

Trimetric projection (Figure 3.35c) has different foreshortening along all three axis directions. This view is produced by an object that is unequally tipped to all the planes of projection.

Axonometric Projections and 3D Models

When you create a 3D CAD model, the object is stored so that vertices, surfaces, and solids are all defined relative to a 3D coordinate system. You can rotate your view of the object to produce a view from any direction. However, your computer screen is a flat surface, like a sheet of paper. The CAD software uses similar projection to produce the view transformations, creating the 2D view of the object on your computer screen. Most 3D CAD software provides a variety of preset isometric viewing directions to make it easy for you to manipulate the view. Some CAD software also allows for easy perspective viewing on screen.

After rotating the object you may want to return to a preset typical axonometric view like one of the examples shown in Figure 3.36.

Solidworks screenshots that display Isometric view, Dimetric view, and Trimetric view. The inset of each of the screenshots displays the axis of the cubes.

3.36 (a) Isometric View of a 1″ Cube Shown in SolidWorks; (b) Dimetric View; (c) Trimetric View (Images courtesy of ©2016 Dassault Systèmes SolidWorks Corporation.)

3.9 Isometric Projection

In an isometric projection, all angles between the axonometric axes are equal. To produce an isometric projection, you orient the object so that its principal edges (or axes) make equal angles with the plane of projection and are therefore foreshortened equally. Oriented this way, the edges of a cube are projected so that they all measure the same and make equal angles (of 120°) with each other, as shown in Figure 3.37.

A cube titled Isometric projection is displayed. The angles between the inner axonometric axes are marked 120 degrees.

3.37 Isometric Projection

Isometric Axes

The projections of the edges of a cube make angles of 120° with each other. You can use these as the isometric axes from which to make measurements. Any line parallel to one of these is called an isometric line. The angles in the isometric projection of the cube are either 120° or 60°, and all are projections of 90° angles. In an isometric projection of a cube, the faces of the cube, and any planes parallel to them, are called isometric planes. See Figure 3.38.

A cube titled Isometric axes is displayed.

3.38 Isometric Axes

Nonisometric Lines

Lines of an isometric drawing that are not parallel to the isometric axes are called nonisometric lines (Figure 3.39). Only lines of an object that are drawn parallel to the isometric axes are equally foreshortened. Nonisometric lines are drawn at other angles and are not equally foreshortened. Therefore the lengths of features along nonisometric lines cannot be measured directly with a scale.

Nonisometric edges of a shape is labeled.

3.39 Nonisometric Edges

Isometric Scales

An isometric scale can be used to draw correct isometric projections. All distances in this scale are 2/3 × true size, or approximately 80% of true size. Figure 3.40a shows an isometric scale. More commonly, an isometric sketch or drawing is created using a standard scale, as in Figure 3.40b, disregarding the foreshortening that the tipped surfaces would produce in a true projection. Figure 3.40c shows an isometric view of a 10″ cube modeled in 3D. When measured parallel to the view, the length appears to be 8.1650 due to the foreshortening (near 80% of the actual size).

Three figures display Isometric and Ordinary Scales.

3.40 Isometric and Ordinary Scales

Tip: Making an Isometric Scale

You can make an isometric scale from a strip of paper or cardboard by placing an ordinary scale at 45° to a horizontal line and the paper (isometric) scale at 30° to the horizontal line. To mark the increments on the isometric scale, draw straight lines (perpendicular to the horizontal line) from the division lines on the ordinary scale.

Alternatively, you can approximate an isometric scale. Scaled measurements of 9″ = 1′–0, or three-quarter-size scale (or metric equivalent) can be used as an approximation.

Figure depicts the making of an isometric scale.

3.10 Isometric Drawings

When you make a drawing using foreshortened measurements, or when the object is actually projected on a plane of projection, it is called an isometric projection (Figure 3.40a). When you make a drawing using the full-length measurements of the actual object, it is an isometric sketch or isometric drawing (Figure 3.40b) to indicate that it lacks foreshortening.

The isometric drawing is about 25% larger than the isometric projection, but the pictorial value is obviously the same in both. Because isometric sketches are quicker (because you can use the actual measurements), they are much more commonly drawn.


Some CAD software will notify you about the lack of foreshortening in isometric drawings (or allow you to select for it) when you print or save them.

Positions of the Isometric Axes

The first step in making an isometric drawing is to decide along which axis direction to show the height, width, and depth, respectively. Figure 3.41 shows four different orientations that you might start with to create an isometric drawing of the block shown. Each is an isometric drawing of the same block, but with a different corner facing your view. These are only a few of many possible orientations.

Four cuboids placed in different positions explain positions of Isometric Axes. The edges of the cuboids are marked 120 degrees.

3.41 Positions of Isometric Axes

A 3D model of a house placed in seven different angles.

3.42 Isometric Options. These views of a simple model of a house are all isometric, but some show the features best.

You may orient the axes in any desired position, but the angle between them must remain 120°. In selecting an orientation for the axes, choose the position from which the object is usually viewed, or determine the position that best describes the shape of the object, or better yet, both.

If the object is a long part, it will look best with the long axis oriented horizontally.

3.11 Making an Isometric Drawing

Rectangular objects are easy to draw using box construction, which consists of imagining the object enclosed in a rectangular box whose sides coincide with the main faces of the object. For example, imagine the object shown in the two views in the Step by Step feature at right enclosed in a construction box, then locate the features along the edges of the box as shown.

The Step by Step feature on the next page shows how to construct an isometric drawing of an object composed of all “normal” surfaces. Normal is used technically to mean “at right angles.” A normal surface is any surface that is parallel to the sides of the box. Notice that all measurements are made parallel to the main edges of the enclosing box—that is, parallel to the isometric axes. No measurement along a nonisometric line can be measured directly with the scale, as these lines are not foreshortened equally to the normal lines. Start at any one of the corners of the bounding box and draw along the isometric axis directions.

Figure shows an Isometric drawing of a table placed on a plane. The Length and width of the plane are marked 48.00. The Height of the table is marked 52.00 and the width of the table is marked 32.50.

3.43 Isometric drawings are not typically dimensioned. When they are, the dimensions should appear in the same plane as the isometric axes and read from the bottom of the sheet or aligned in the isometric planes. (Courtesy of Brandon Wold.)

Step by Step: Box Construction

Follow these steps to create an isometric sketch of a rectangular object.

Figure depicts a step to construct a box.

Image Lightly draw the overall dimensions of the box.

In the drawing of the rectangular box, the dimension of the box is measured and joined to form a rectangular object.

Image Draw the irregular features relative to the sides of the box.

Isometric drawing of a box with its dimensions marked: a, b, c, d, e, and f.

Image Darken the final lines.

Isometric drawing of a box with final lines darkened.

Step by Step: Drawing Normal Surfaces in Isometric

Follow these steps to create an isometric drawing of the object shown below, which has normal surfaces only.

Drawing of a part with orthographic views given.

Image Select axes along which to block in height, width, and depth dimentions.

Sketch shows a box with sides marked A, E, and M. A text reading, "All measurement must be parallel to mainedges of box" points to the box.

Image Locate main areas to be removed from the overall block. Lightly sketch along isometric axes to define portion to be removed.

Sketch shows a box with its isometric axis labeled.

Image Lightly block in any remaining portions to be removed through the whole block.

Sketch shows a box with isometric axis labeled.

Image Lightly block in features to be removed fom the remaining shape along isometric axes.

Sketch shows a box with isometric axis labeled. Lightly block in any remaining portions to be removed through the whole block.

Image Darken final lines.

The final sketch of the box with final lines darkened.

3.12 Offset Location Measurements

Use the method shown in Figures 3.44a and b to locate points with respect to each other. First, draw the main enclosing block, then draw the offset lines (CA and BA) full size in the isometric drawing to locate corner A of the small block or rectangular recess. These measurements are called offset measurements. Because they are parallel to edges of the main block in the multiview drawings, they will be parallel to the same edges in the isometric drawings (using the rule of parallelism).

Offset location measurements is explained.

3.44 Offset Location Measurements

Step by Step: Drawing Nonisometric Lines

How to Draw Nonisometric Lines

The inclined lines BA and CA are shown true length in the top view (54 mm), but they are not true length in the isometric view. To draw these lines in the isometric drawing, use a construction box and offset measurements.

Two drawings placed at the top and bottom depict drawing nonisometric lines.
First step of drawing nonisometric lines is depicted.

Image Directly measure dimensions that are along isometric lines (in this case, 44 mm, 18 mm, and 22 mm).

Second step of drawing nonisometric lines is depicted.

Image Because the 54 mm dimension is not along an isometric axis, it cannot be used to locate point A.

Use trigonometry or draw a line parallel to the isometric axis to determine the distance to point A.

Because this dimension is parallel to an isometric axis, it can be transferred to the isometric.

Third step of drawing nonisometric lines is depicted.

Image The dimensions 24 mm and 9 mm are parallel to isometric lines and can be measured directly.


To convince yourself that nonisometric lines will not be true length in the isometric drawing, use a scrap of paper and mark the distance BA (Step 2) and then compare it with BA on the given top view. Do the same for line CA. You will see that BA is shorter and CA is longer in the isometric than the corresponding lines in the given views.

Step by Step: Drawing Oblique Surfaces in Isometric

Image Find the intersections of the oblique surfaces with the isometric planes. Note that for this example, the oblique plane contains points A, B, and C.

First step of drawing oblique surfaces in isometric is depicted.

Image To draw the plane, extend line AB to X and Y, in the same isometric plane as C. Use lines XC and YC to locate points E and F.

Second step of drawing oblique surfaces in isometric is depicted.

Image Draw AD and ED using the rule, “Parallel edges on a part will appear parallel in any axonometric view.” Because the inclined surface is planar and the surfaces it intersects are parallel to each other, the edges that lie in those planes will also be parallel.

Third step of drawing oblique surfaces in isometric is depicted.

Isometric Drawings of Inclined Surfaces

Figure 3.45 shows how to construct an isometric drawing of an object that has some inclined surfaces and oblique edges. Notice that inclined surfaces are located by offset or coordinate measurements along the isometric lines. For example, dimensions E and F are measured to locate the inclined surface M, and dimensions A and B are used to locate surface N.

Inclined surfaces in Isometric projection is depicted.

3.45 Inclined Surfaces in Isometric

3.13 Hidden Lines and Centerlines

Hidden lines in a drawing represent the edges where surfaces meet but are not directly visible. Hidden lines are omitted from pictorial drawings unless they are needed to make the drawing clear. Figure 3.46 shows a case in which hidden lines are needed because a projecting part cannot be clearly shown without them. Sometimes it is better to include an isometric view from another direction than to try to show hidden features with hidden lines. You will learn more about hidden lines in Chapter 6.

Figure shows a case with its hidden lines marked as dashed lines.

3.46 Using Hidden Lines

Draw centerlines locating the center of a hole only if they are needed to indicate symmetry or for dimensioning. In general, use centerlines sparingly in isometric drawings. If in doubt, leave them out, as too many centerlines will look confusing.

3.14 Angles in Isometric

Angles project true size only when the plane containing the angle is parallel to the plane of projection. An angle may project to appear larger or smaller than the true angle depending on its position.

Because the various surfaces of the object are usually inclined to the front plane of projection, they generally will not be projected true size in an isometric drawing.

Step by Step: How to Draw Angles in Isometric

The multiview drawing at left shows three 60° angles. None of the three angles will be 60° in the isometric drawing.

Two drawings placed at the top and bottom depict drawing angles in isometric.

Image Lightly draw an enclosing box using the given dimensions, except for dimension X, which is not given.

First step of drawing angles in isometric is depicted.

Image To find dimension X, draw triangle BDA from the top view full size, as shown.

Second step of drawing angles in isometric is depicted.

Image Transfer dimension X to the isometric to complete the enclosing box. Find dimension Y by a similar method and then transfer it to the isometric.

Third step of drawing angles in isometric is depicted.

Image Complete the isometric by locating point E by using dimension K, as shown. A regular protractor cannot be used to measure angles in isometric drawings. Convert angular measurements to linear measurements along isometric axis lines.

Fourth step of drawing angles in isometric is depicted.

Tip: Checking Isometric Angles

To convince yourself that none of the angles will be 60°, measure each angle in the isometric in the figure above with a protractor or scrap of paper and note the angle compared with the true 60°. None of the angles shown are the same in the isometric drawing. Two are smaller and one is larger than 60°.

Estimating 30° Angles

If you are sketching on graph paper and estimating angles, an angle of 30° is roughly a rise of 1 to a run of 2.

3.15 Irregular Objects

You can use the construction box method to draw objects that are not rectangular (Figure 3.47). Locate the points of the triangular base by offsetting a and b along the edges of the bottom of the construction box. Locate the vertex by offsetting lines OA and OB using the top of the construction box.

Irregular object in Isometric projection is displayed.

3.47 Irregular Object in Isometric


It is not always necessary to draw the complete construction box as shown in Figure 3.47b. If only the bottom of the box is drawn, the triangular base can be constructed as before. The orthographic projection of the vertex O’ on the base can be drawn using offsets O’A and O’B, as shown, and then the vertical line O’O can be drawn, using measurement C.

Figure shows a cube enclosed in a white rectangle, that is enclosed in a black rectangle.

Tennis Ball (Factory Reject) (Excerpted from Droodles – The Classic Collection by Roger Price ©2000 by Tallfellow Press. Used by permission. All rights reserved.)

3.16 Curves in Isometric

You can draw curves in isometric using a series of offset measurements similar to those discussed in Section 3.12. Select any desired number of points at random along the curve in the given top view, such as points A, B, and C in Figure 3.48. Choose enough points to accurately locate the path of the curve (the more points, the greater the accuracy). Draw offset grid lines from each point parallel to the isometric axes and use them to locate each point in the isometric drawing as in the example shown in Figure 3.48.

Figures depict Curves in Isometric projection.

3.48 Curves in Isometric

3.17 True Ellipses in Isometric

If a circle lies in a plane that is not parallel to the plane of projection, the circle projects as an ellipse. The ellipse can be constructed using offset measurements.

Step by Step: Drawing An Isometric Ellipse by Offset Measurements

Random-Line Method

Image Draw parallel lines spaced at random across the circle.

First step of random-line method is depicted.

Image Transfer these lines to the isometric drawing. Where the hole exits the bottom of the block, locate points by measuring down a distance equal to the height d of the block from each of the upper points. Draw the ellipse, part of which will be hidden, through these points. Darken the final drawing lines.

Second step of random-line method is depicted.

Eight-Point Method

Image Enclose the given circle in a square, and draw diagonals. Draw another square through the points of intersection of the diagonals and the circle as shown.

First step of eight-point method is depicted.

Image Draw this same construction in the isometric, transferring distances a and b. (If more points are desired, add random parallel lines, as above.) The centerlines in the isometric are the conjugate diameters of the ellipse. The 45° diagonals coincide with the major and minor axes of the ellipse. The minor axis is equal in length to the sides of the inscribed square.

Second step of eight-point method is depicted.

When more accuracy is required, divide the circle into 12 equal parts, as shown.

Drawing depicts the 12-point method.

Nonisometric Lines

If a curve lies in a nonisometric plane, not all offset measurements can be applied directly. The elliptical face shown in the auxiliary view lies in an inclined nonisometric plane.

Image Draw lines in the orthographic view to locate points.

Drawing depicts the first step of nonisometric lines.

Image Enclose the cylinder in a construction box and draw the box in the isometric drawing. Draw the base using offset measurements, and construct the inclined ellipse by locating points and drawing the final curve through them.

Measure distances parallel to an isometric axis (a, b, etc.) in the isometric drawing on each side of the centerline X–X. Project those not parallel to any isometric axis (e, f, etc.) to the front view and down to the base, then measure along the lower edge of the construction box, as shown.

Drawing depicts the second step of nonisometric lines.

Image Darken final lines.

Drawing of athe elliptical face with final lines darkened.

3.18 Orienting Ellipses in Isometric Drawings

Figure 3.49 shows four-center ellipses constructed on the three visible faces of a cube. Note that all the diagonals are horizontal or at 60° with horizontal. Realizing this makes it easier to draw the shapes.

Figure shows a cube that encloses circles on its sides. The diameter of the circle is marked, R.

3.49 Four-Center Ellipses

Approximate ellipses such as these, constructed from four arcs, are accurate enough for most isometric drawings. The four-center method can be used only for ellipses in isometric planes. Earlier versions of CAD software, such as AutoCAD Release 10, used this method to create the approximate elliptical shapes available in the software. Current releases use an accurate ellipse.

Step by Step: Drawing a Four-Center Ellipse

Image Draw or imagine a square enclosing the circle in the multiview drawing. Draw the isometric view of the square (an equilateral parallelogram with sides equal to the diameter of the circle).

The first step for drawing a four-center ellipse is displayed.

Image Mark the midpoint of each line and draw a perpendicular line from each.

The second step for drawing a four-center ellipse is displayed.


Here is a useful rule. The major axis of the ellipse is always at right angles to the centerline of the cylinder, and the minor axis is at right angles to the major axis and coincides with the centerline.

Sketch shows two circles placed on either end of a cuboid. The major axis is marked 90 degrees. A text that reads, Minor axis coincides with centerline points to the centerline.

Image Draw the two large arcs, with radius R, from the intersections of the perpendiculars in the two closest corners of the parallelogram.

The third step for drawing a four-center ellipse is displayed.

Image Draw the two small arcs, with radius r, from the intersections of the perpendiculars within the parallelogram, to complete the ellipse.

The fourth step for drawing a four-center ellipse is displayed.


As a check on the accurate location of these centers, you can draw a long diagonal of the parallelogram as shown in Step 4. The midpoints of the sides of the parallelogram are points of tangency for the four arcs.

More Accurate Ellipses

The four-center ellipse deviates considerably from a true ellipse. As shown in Figure 3.50a, a four-center ellipse is somewhat shorter and “fatter” than a true ellipse. When the four-center ellipse is not accurate enough, you can use a closer approximation called the Orth four-center ellipse to produce a more accurate drawing.

Inaccuracy of the Four-Center Ellipse is depicted in three figures.

3.50 Inaccuracy of the Four-Center Ellipse

Step by Step: Drawing an Orth Four-Center Ellipse

Sketch of an ellipse drawn using Orth method.

To create a more accurate approximate ellipse using the Orth method, follow the steps for these methods. The centerline method is convenient when starting from a hole or cylinder.

Centerline Method

Image Draw the isometric centerlines. From the center, draw a construction circle equal to the actual diameter of the hole or cylinder. The circle will intersect the centerlines at four points, A, B, C, and D.

The first step for drawing an orth four-center ellipse using centerline method.

Image From the two intersection points on one centerline, draw perpendiculars to the other centerline. Then draw perpendiculars from the two intersection points on the other centerline to the first centerline.

The second step for drawing an orth four-center ellipse using centerline method.

Image With the intersections of the perpendiculars as centers, draw two small arcs and two large arcs.

Drawing depicts the third step of centerline method.

Enclosing-Rectangle Method

Image Locate center of circle and block in enclosing isometric rectangle.

Drawing depicts the first step of enclosing-rectangle method.

Image Use the midpoint of the isometric rectangle (the distance from A to B) to locate the foci on the major axis.

Drawing depicts the second step of enclosing-rectangle method.

Image Draw lines at 60° from horizontal through the foci (points C and D) to locate the center of the large arc R.

Drawing depicts the third step of enclosing-rectangle method.

Image Draw the two large arcs R tangent to the isometric rectangle. Draw two small arcs r, using foci points C and D as centers, to complete the approximate ellipse.

Drawing depicts the fourth step of enclosing-rectangle method.

Note that these steps are exactly the same as for the regular four-center ellipse, except for the use of the isometric centerlines instead of the enclosing parallelogram. (When sketching, it works fine to just draw the enclosing rectangle and sketch the arcs tangent to its sides.)

Tip: Isometric Templates

Special templates like this isometric template with angled lines and ellipses oriented in various isometric planes make it easy to draw isometric sketches. The ellipses are provided with markings to coincide with the isometric centerlines of the holes—a convenient feature in isometric drawing.

You can also draw ellipses using an ellipse template. Select the size and alignment of the major and minor axes of the ellipse to match the orientation of the shape in your drawing.

Drawing shows an isometric template.

3.19 Drawing Isometric Cylinders

A typical drawing with cylindrical shapes is shown in Figure 3.51. Note that the centers of the larger ellipse cannot be used for the smaller ellipse, though the ellipses represent concentric circles. Each ellipse has its own parallelogram and its own centers. Notice that the centers of the lower large ellipse are drawn by projecting the centers of the upper large ellipse down a distance equal to the height of the cylinder.

Isometric drawing of a bearing is displayed.

3.51 Isometric Drawing of a Bearing

3.20 Screw Threads in Isometric

Parallel partial ellipses equally spaced at the symbolic thread pitch are used to represent only the crests of a screw thread in isometric (Figure 3.52). The ellipses may be sketched, drawn by the four-center method, or created using an ellipse template.

Screw thread in isometric view.

3.52 Screw Threads in Isometric

3.21 Arcs in Isometric

The four-center ellipse construction can be used to sketch or draw circular arcs in isometric. Figure 3.53a shows the complete construction. It is not necessary to draw the complete constructions for arcs, as shown in Figures 3.53b and c. Measure the radius R from the construction corner, then at each point, draw perpendiculars to the lines. Their intersection is the center of the arc. Note that the R distances are equal in Figures 3.53b and c, but that the actual radii used are quite different.

Drawing of an arc in isometric view with its radius marked, R and diameter marked, D.

3.53 Arcs in Isometric

3.22 Spheres in Isometric

The isometric drawing of any curved surface can be constructed from all lines that can be drawn on that surface. For spheres, select the great circles (circles cut by any plane through the center) as the lines on the surface. Because all great circles, except those that are perpendicular or parallel to the plane of projection, are shown as ellipses having equal major axes, they enclose a circle whose diameter is the major axis of the ellipse.

Figure 3.54 shows two given views of a sphere enclosed in a construction cube. Next, an isometric of a great circle is drawn in a plane parallel to one face of the cube. There is no need to draw the ellipse, since only the points on the diagonal located by measurements a are needed to establish the ends of the major axis and thus to determine the radius of the sphere.

Isometric of a sphere is shown in four sections.

3.54 Isometric of a Sphere

In the resulting isometric drawing, the diameter of the circle is Image times the actual diameter of the sphere. The isometric projection is simply a circle whose diameter is equal to the true diameter of the sphere.

Step by Step: Isometric Sketching from an Object

Positioning the Object

To make an isometric sketch from an actual object, first hold the object in your hand and tilt it toward you, as shown in the illustration. In this position the front corner will appear vertical. The two receding bottom edges and those edges parallel to them should appear to be at about 30° with horizontal. The steps for sketching the object follow.

A 3D model demonstrates isometric sketching from an object.

Image Sketch the enclosing box lightly, making AB vertical and AC and AD approximately 30° with horizontal. These three lines are the isometric axes. Make AB, AC, and AD approximately proportional in length to the actual corresponding edges on the object. Sketch the remaining lines parallel to these three lines.

Sketch demonstrates the first step of isometric sketching from an object.

Image Block in the recess and the projecting block.

Sketch demonstrates the second step of isometric sketching from an object.

Image Darken all final lines.

Drawing of a couch formed by placing rectangular boxes.

CAD at Work: Isometric Sketches Using AutoCAD Software

Need a quick isometric sketch? AutoCAD software has special drafting settings for creating an isometric style grid.

Figure A shows the Drafting Settings dialog box in AutoCAD. When you check the button for Isometric snap, the software calculates the spacing needed for an isometric grid. You can use it to make quick pictorial sketches like the example shown in Figure B. Piping diagrams are often done this way, although they can also be created using 3D tools.

Screenshot of Drafting Settings dialog box.

(A) Selecting Isometric Snap in the AutoCAD Drafting Settings Dialog Box (Autodesk screen shots reprinted courtesy of Autodesk, Inc.)

A pictorial sketch created from a flat drawing using Isometric Snap.

(B) A Pictorial Sketch Created from a Flat Drawing Using Isometric Snap

Even though the drawing in Figure B looks 3D, it is really drawn in a flat 2D plane. You can observe this if you change the viewpoint so you are no longer looking straight onto the view.

The Ellipse command in AutoCAD has a special Isocircle option that makes drawing isometric ellipses easy. The isocircles are oriented in different directions depending on the angle of the snap cursor. Figure C shows isocircles and snap cursors for the three different orientations. In the software, you press <Ctrl> and E simultaneously to toggle the cursor appearance.

Figure shows three isometric circles placed at different angles. The corresponding snap cursors are placed down as follows: center cursor, right cursor, and left cursor.

(C) Variously Oriented Isometric Circles and the Corresponding Snap Cursors Used to Create Them

3.23 Oblique Sketches

Oblique drawing is an easy method for creating quick pictorials (Figure 3.55). In most oblique sketches, circles and angles parallel to the projection plane are true size and shape and are therefore easy to construct. Although circular shapes are easy to sketch in the front oblique plane, they would appear elliptical in the top or sides. Oblique views are primarily a sketching technique used when the majority of circular shapes appear in the front view or when the object can be rotated to position circles in the front view.

Oblique Projection theory is depicted.

3.55 Oblique Projection Theory

CAD is not typically used to create oblique views because better-appearing isometric or perspective drawings can be created easily from 3D CAD models.

Appearance of Oblique Drawings

Three things affect the appearance of your oblique sketch:

1. The surface of the object you choose to show parallel to the projection plane

2. The angle and orientation you choose for the receding lines that depict the object’s depth

3. The scale you choose for the receding lines depicting the object’s depth (Figure 3.56)

Figure shows two cuboids with a rectangle engraved on one side and a circle on the other. The cuboids are placed in different angles.

3.56 Angle of Receding Axis

Sketch of different cereal boxes.

Meat Cereals. Oblique sketching shows the front surface parallel to the view, making these meat cereal boxes easy to draw. (Courtesy of Randall Munroe.)

Choosing the Front Surface

Think about which surface of the object would be the best one to think of as parallel to the plane of projection. For example, a cube has six different surfaces. As you are creating your sketch, any of those six surfaces could be oriented as the “front” of the part. Of course, with a cube it wouldn’t matter which one you chose. But a cube with a hole through it will make a much better oblique sketch when the round hole is oriented parallel to the projection plane.

Angle of Receding Lines

An angle of 45° is often chosen for the angle of the receding lines because it makes oblique sketches quick and easy. You can use graph paper and draw the angled lines through the diagonals of the grid boxes. An angle of 30° is also a popular choice. It can look more realistic at times. Any angle can be used, but 45° is typical. As shown in Figure 3.57, you can produce different oblique drawings by choosing different directions for the receding lines.

Figure shows 9 cuboids in a set of three. The angles on all the edges are highlighted.

3.57 Variation in Direction of Receding Axis

3.24 Length of Receding Lines

Theoretically, oblique projectors can be at any angle to the plane of projection other than perpendicular or parallel. The difference in the angle you choose causes receding lines of oblique drawings to vary in angle and in length from near zero to near infinity. However, many of those choices would not produce very useful drawings. Figure 3.58 shows a variety of oblique drawings with different lengths for the receding lines.

Foreshortening of receding lines is depicted.

3.58 Foreshortening of Receding Lines

Because we see objects in perspective (where receding parallel lines appear to converge) oblique projections look unnatural to us. The longer the object in the receding direction, the more unnatural the object appears. For example, the object shown in Figure 3.58a is an isometric drawing of a cube in which the receding lines are shown full length. They appear to be too long and they appear to widen toward the rear of the block. Figure 3.59b shows how unnatural the familiar pictorial image of railroad tracks leading off into the distance would look if drawn in an oblique projection. To give a more natural appearance, show long objects with the long axis parallel to the view, as shown in Figure 3.60.

Figure a shows a railroad track with electric poles beside it titled, Perspective. Figure b shows a railroad track titled, Oblique.

3.59 Unnatural Appearance of Oblique Drawing

Figure shows an object placed vertically labeled, Poor and an object p[laced horizontally, labeled good.

3.60 Long Axis Parallel to Plane of Projection

Cavalier Projection

When the receding lines are true length—(the projectors make an angle of 45° with the plane of projection)—the oblique drawing is called a cavalier projection (Figure 3.58a). Cavalier projections originated in the drawing of medieval fortifications and were made on horizontal planes of projection. On these fortifications the central portion was higher than the rest, so it was called cavalier because of its dominating and commanding position.

Cabinet Projection

When the receding lines are drawn to half size (Figure 3.58d), the drawing is known as a cabinet projection. This term is attributed to the early use of this type of oblique drawing in the furniture industries. Figure 3.61 shows a file drawer drawn in cavalier and cabinet projections.

Cavalier and cabinet projections are depicted in two figures.

3.61 Comparison of (a) Cavalier and (b) Cabinet Projections

3.25 Choice of Position in Oblique Drawings

Orient the view so that important shapes are parallel to the viewing plane, as shown in Figure 3.62. In Figures 3.62a and c, the circles and circular arcs are shown in their true shapes and are easy to draw. In Figures 3.62b and d they are not shown in true shape and must be plotted as free curves or ellipses.

Essential contours parallel to plane of projection is displayed.

3.62 Essential Contours Parallel to Plane of Projection

3.26 Ellipses for Oblique Drawings

Circular shapes that are parallel to the front plane of projection appear as circles in oblique view. In isometric views they appear elliptical. Figure 3.63 shows a comparison of a cylinder drawn in isometric and oblique views.

Figure shows two cylinders labeled Isometric and Oblique respectively placed horizontally. The edge of the Isometric cylinder is labeled, Elliptical arc and the edge of Oblique cylinder is labeled, Circular arc.

3.63 Oblique and Isometric Projections for a Cylinder

It is not always possible to orient the view of an object so that all its rounded shapes are parallel to the plane of projection. For example, the object shown in Figure 3.64a has two sets of circular contours in different planes. Both cannot be simultaneously placed parallel to the plane of projection, so in the oblique projection, one of them must be viewed as an ellipse.

Circles and arcs not parallel to plane of projection is displayed in two figures.

3.64 Circles and Arcs Not Parallel to Plane of Projection

If you are sketching, you can just block in the enclosing rectangle and sketch the ellipse tangent to its sides. Using CAD, you can draw the ellipse by specifying its center and major and minor axes. In circumstances where a CAD system is not available, if you need an accurate ellipse, you can draw it by hand using a four-center arc approximation.

3.27 Angles in Oblique Projection

When an angle that is specified in degrees lies in a receding plane, convert the angle into linear measurements to draw the angle in an oblique drawing. Figure 3.65a shows a drawing with an angle of 30° specified.

Three drawings explain Angles In Oblique Projection.

3.65 Angles In Oblique Projection

To draw the angle in the oblique drawing, you will need to know distance X. The distance from point A to point B is given as 32 mm. This can be measured directly in the cavalier drawing (Figure 3.65b). Find distance X by drawing the right triangle ABC (Figure 3.65c) using the dimensions given, which is quick and easy using CAD.

You can also use a mathematical solution to find the length of the side: The length of the opposite side equals the tangent of the angle times the length of the adjacent side. In this case, the length of the opposite side, X, is about 18.5 mm. Draw the angle in the cavalier drawing using the found distance.

Remember that all receding dimensions must be reduced to match the scale of the receding axis. In the cabinet drawing in Figure 3.65c, the distance BC must be half the side BC of the right triangle in Figure 3.65c.

Step by Step: Using Box Construction to Create an Oblique Drawing

A cavalier drawing of the rectangular object shown in the two orthographic views.

Follow these steps to draw a cavalier drawing of the rectangular object shown in the two orthographic views.

Image Lightly block in the overall width (A) and height (B) to form the enclosing rectangle for the front surface. Select an angle for the receding axis (OZ) and draw the depth (C) along it.

Sketch displays the first step of constructing a box to create an oblique drawing.

Image Lightly block in the details of the front surface shape including the two holes, which will appear round. Add the details of the right-side surface shape. Extend lines along the receding axis connecting the edges to form the remaining surface edges.

Sketch displays the second step of constructing a box to create an oblique drawing.

Image Darken the final lines.

Final lines of the box are darkened.

Step by Step: Using Skeleton Construction in Oblique Drawing

Oblique drawings are especially useful for showing objects that have cylindrical shapes built on axes or centerlines. Follow these steps to construct an oblique drawing of the part shown using projected centerlines.

Oblique drawing of a cylindrical shape is displayed.

Image Position the object in the drawing so that the circles shown in the given top view are parallel to the plane of projection. The circles will show true shape in the oblique view. Draw the circular shape in the front plane of the oblique view and extend the center axis along the receding axis of the oblique drawing.

First step of using skeleton construction in oblique drawing is depicted.

Image Add the centerline skeleton as shown.

Second step of using skeleton construction in oblique drawing is depicted.

Image Build the drawing from the location of these centerlines.

Third step of using skeleton construction in oblique drawing is depicted.

Image Construct all important points of tangency.

Fourth step of using skeleton construction in oblique drawing is depicted.

Image Darken the final cavalier drawing.

Fifth step of using skeleton construction in oblique drawing is depicted.

3.28 Sketching Assemblies

Assembly drawings are used to show how parts fit together. Because they do not need to provide all the information to make individual parts, isometric sketches or drawings are often used that show only the exterior view of the assembled parts. You can use assembly sketches as you design to identify areas that must fit together or have common dimensions, or to maintain critical distances to ensure that the device will function as expected. Figure 3.66 shows an isometric sketch of an assembly.

Drawing of an Isometric Assembly Sketch.

3.66 Isometric Assembly Sketch

Sketched assembly drawings are useful in documenting your ideas in a rougher state. An exploded isometric assembly drawing shows the individual parts moved apart from one another along the isometric axis line directions. A centerline pattern is used to show how the parts relate and fit together.

When creating an exploded assembly sketch, keep the parts lined up with the other parts where they assemble. If possible, move any single parts along only one axis direction. In Figure 3.67, all exploded parts are moved along only one axis direction. Notice the centerline pattern showing how the parts align. If you must move a part in two different directions so that it can be seen clearly, add a centerline showing how the part must move to get to its proper location in the assembly. In Figure 3.68, parts have been moved in two axis directions. You can see the jog in the centerline indicating how the part has been moved.

Drawing of an Exploded Isometric Assembly Sketch.

3.67 Exploded Isometric Assembly Sketch. Centerlines have been added to show where parts line up in the assembly.

Isometric assemble sketch with parts been exploded in two directions. Centerlines showing movement along two axes is labeled.

3.68 Isometric Assembly Sketch. Some parts have been exploded in two directions so they do not overlap other parts in the assembly. The centerlines indicate the path the part must follow to assemble correctly. (Courtesy of Albert W. Brown, Jr.)

Finished assembly drawings that will be given to manufacturing workers to show them how to produce a device also contain a parts list that identifies each part in the assembly, the quantity necessary, the material of each part, and a number identifying which item it is on the drawing. A parts list can also be included on a sketched assembly drawing. Sometimes it is helpful to give a quick assembly sketch to manufacturing along with part drawings, especially if they are parts you need to have manufactured quickly from a sketch. Seeing how the parts fit together in assembly will help the manufacturer understand the entire device and identify potential problems.

3.29 Sketching Perspectives

Perspective pictorials most closely approximate the view produced by the human eye. Perspective views are the type of drawing most like a photograph. Examples of a perspective drawing can be seen in Figures 3.69, 3.70a, and 3.72. Although complex perspective views are time consuming to sketch, they are easy to create from 3D CAD models.

Perspective drawing theory is illustrated.

3.69 Perspective Drawing Theory

Figure a shows a railroad track with electric poles beside it titled, Perspective. Figure b shows a railroad track with placed in an angle titled, Oblique.

3.70 Perspective vs. Oblique. Perspective drawings appear more natural than oblique drawings.

Unlike parallel types of projection, perspective projectors converge. The point at which the projectors converge is called the vanishing point. This is clearly seen in Figure 3.70a.

The first rule of perspective is that all parallel lines that are not parallel to the picture plane vanish at a single vanishing point, and if these lines are parallel to the ground, the vanishing point will be on the horizon. Parallel lines that are also parallel to the picture plane remain parallel and do not converge toward a vanishing point (Figures 3.71 and 3.72).

Illustration explains picture plane.

3.71 Looking through the Picture Plane

Figure depicts Perspective.

3.72 A Perspective

When the vanishing point is placed above the view of the object in the picture plane, the result is a bird’s-eye view, looking down onto the object. When the vanishing point is placed below the view of the object, the result is a worm’s-eye view looking up at the object from below (see Figure 3.74 on page 107).

There are three types of perspective: one-point, two-point, and three-point perspective, depending on the number of vanishing points used.

The Three Types of Perspective

Perspective drawings are classified according to the number of vanishing points required, which in turn depends on the position of the object with respect to the picture plane.

If the object sits with one face parallel to the plane of projection, only one vanishing point is required. The result is a one-point perspective, or parallel perspective.

If the object sits at an angle with the picture plane but with vertical edges parallel to the picture plane, two vanishing points are required, and the result is a two-point perspective, or an angular perspective. This is the most common type of perspective drawing.

If the object sits so that no system of parallel edges is parallel to the picture plane, three vanishing points are necessary, and the result is a three-point perspective.

One-Point Perspective

To sketch a one-point perspective view, orient the object so that a principal face is parallel to the picture plane. If desired, this face can be placed in the picture plane. The other principal face is perpendicular to the picture plane, and its lines will converge toward a single vanishing point.

Sketch of Excerpted from Droodles.

8:12 Train as Seen by 8:12 1/2 Commuter (Excerpted from Droodles – The Classic Collection by Roger Price ©2000 by Tallfellow Press. Used by permission. All rights reserved.)

Step by Step: One-Point Perspective

To sketch the bearing in one-point perspective—that is, with one vanishing point—follow the steps illustrated below.

Image Sketch the true front face of the object, just as in oblique sketching. Select the vanishing point for the receding lines. In many cases it is desirable to place the vanishing point above and to the right of the picture, as shown, although it can be placed anywhere in the sketch. However, if the vanishing point is placed too close to the center, the lines will converge too sharply and the picture will be distorted.

First step of one-point perspective is depicted.

Image Sketch the receding lines toward the vanishing point.

Second step of one-point perspective is depicted.

Image Estimate the depth to look good and sketch in the back portion of the object. Note that the back circle and arc will be slightly smaller than the front circle and arc.

Third step of one-point perspective is depicted.

Image Darken all final lines. Note the similarity between the perspective sketch and the oblique sketch earlier in the chapter.

A final sketch of a part drawn using one-point perspective.

Two-Point Perspective

Two-point perspective is more true to life than one-point perspective. To sketch a two-point perspective, orient the object so that principal edges are vertical and therefore have no vanishing point; edges in the other two directions have vanishing points. Two-point perspective is especially good for representing buildings and large civil structures, such as dams or bridges.

Step by Step: Two-Point Perspective

To sketch a desk using two vanishing points, follow these steps.

Image Sketch the front corner of the desk at true height. Locate two vanishing points (VPL and VPR) on a horizon line (at eye level). Distance CA may vary—the greater it is, the higher the eye level will be and the more we will be looking down on top of the object. A rule of thumb is to make C–VPL one third to one fourth of C–VPR.

A line with either side marked VPL and VPR is displayed. True height is marked, AB.

Image Estimate depth and width, and sketch the enclosing box.

A line with either side marked VPL and VPR is displayed. Estimated depth and width are labeled D and W. Text reading, W and D are estimated to look good is placed below the drawing.

Image Block in all details. Note that all parallel lines converge toward the same vanishing point.

A line with either side marked VPL and VPR is displayed. The light sketch of the desk is displayed.

Image Darken all final lines. Make the outlines thicker and the inside lines thinner, especially where they are close together.

A line with either side marked VPL and VPR is displayed. The dark sketch of the desk is displayed.

Three-Point Perspective

In three-point perspective, the object is placed so that none of its principal edges is parallel to the picture plane. Each of the three sets of parallel edges has a separate vanishing point. See Figure 3.73.

Image depicts Three-Point Perspective.

3.73 Three-Point Perspective

Bird’s-Eye View versus Worm’s-Eye View

The appearance of a perspective sketch depends on your view-point in relation to the object. Select some reachable object in the room and move so that you are looking at it from above and really notice its shape. Now gradually move so that you are looking at it from below. Notice how the change of viewpoint changes the appearance of its surfaces—which ones are visible and their relative size.

The horizon line in a perspective sketch is a horizontal line that represents the eye level of the observer. Locating the sketched object below the horizon line produces a view from above (or a bird’s-eye view). Locating the sketched object above the horizon line produces a view from below (or a worm’s-eye view). Figure 3.74 illustrates the horizon line in a drawing and the effect of placing the object above or below the horizon line.

Bird’s-eye view and worm's eye view is depicted.

3.74 (a) Object below the Horizon Line; (b) Object above the Horizon Line

3.30 Curves and Circles in Perspective

If a circle is parallel to the picture plane, its perspective view is a circle. If the circle is inclined to the picture plane, its perspective drawing may be any one of the conic sections in which the base of the cone is the given circle, the vertex is the station point, and the cutting plane is the picture plane (see Figure 3.71). The centerline of the cone of rays is usually approximately perpendicular to the picture plane, so the perspective will usually be an ellipse.

An ellipse may be drawn using a method of blocking in its center and a box tangent to the ellipse as shown in Figure 3.75. Also shown is a convenient method for determining the perspective of any planar curve.

2D and 3D image of a shape represents blocking in curves in two-point perspective.

3.75 Blocking in Curves in Two-Point Perspective

3.31 Shading

Shading can make it easier to visualize pictorial drawings, such as display drawings, patent drawings, and catalog drawings. Ordinary multiview and assembly drawings are not shaded. The shading should be simple, reproduce well, and produce a clear picture. Some of the common types of shading are shown in Figure 3.76. Two methods of shading fillets and rounds are shown in Figures 3.76c and d. Shading produced with dots is shown in Figure 3.76e, and pencil tone shading is shown in Figure 3.76f. Pencil tone shading used in pictorial drawings on tracing paper reproduces well only when making blueprints, not when using a copier.

Six images of a part shaded in different patterns.

3.76 Methods of Shading

3.32 Computer Graphics

Pictorial drawings of all types can be created using 3D CAD (Figures 3.77 and 3.78). To create pictorials using 2D CAD, use projection techniques similar to those presented in this chapter. The advantage of 3D CAD is that once you make a 3D model of a part or assembly, you can change the viewing direction at any time for orthographic, isometric, or perspective views. You can also apply different materials to the drawing objects and shade them to produce a high degree of realism in the pictorial view.

Shaded diametric pictorial view from a 3D model is displayed.

3.77 Shaded Dimetric Pictorial View from a 3D Model (Courtesy of Robert Kincaid.)

Isometric Assembly Drawing of a part.

3.78 Isometric Assembly Drawing (Courtesy of Robert Kincaid.)

3.33 Drawing on Drawing

Because CAD helps people produce accurate drawings that are easy to alter, store, and repurpose, computer automation has made the painstaking aspects of hand-rendering technical drawings nearly obsolete. Still, the ability to sketch with clarity is an immediate and universal way to record and communicate ideas.

The ability to think of complex objects in terms of their basic solid components and to identify relationships between various surfaces, edges, and vertices is basic to creating both hand-drawn (Figure 3.79) and computer-generated technical drawings. By understanding how to represent objects accurately, you can communicate efficiently as part of a team and increase your constructive visualization ability, or your ability to “think around corners.”

Hand Sketch of a workstation. Particle board, frosted acrylic, laminate, natural steel, and rubber tackable are labeled.

3.79 Hand Sketch Conveying Custom Workstation Furniture Concept (Courtesy of Jacob A. Baron-Taltre.)

CAD at Work: Sketching and Parametric Modeling

The Design Process

The parametric modeling process in many ways mirrors the design process. Parametric CAD software lets you specify geometry, dimensions, and other essential parameters that control the model. To get the rough ideas down, the designer starts by making hand sketches. Then as the ideas are refined, more accurate drawings are created either with instruments or using CAD. Necessary analysis is performed and, in response, the design may change. The drawings are revised as needed to meet the new requirements. Eventually the drawings are approved so that the parts may be manufactured.

Rough Sketches

Using parametric modeling software, the designer first roughly sketches the basic shapes on the computer screen as though sketching freehand. These sketches do not need to have perfectly straight lines or accurate corners. The software interprets the sketch much as you would interpret a rough sketch given to you by a colleague. If the lines are nearly horizontal or vertical, the software assumes that you meant them to be so. If the line appears to be perpendicular, the software assumes that it is.

Screenshot shows a rough sketch in pro/ENGINEER sketcher.

A Rough Sketch in Pro/ENGINEER Sketcher

Constraining the Sketch

Using a parametric CAD system, you next refine the two-dimensional sketch by adding geometric constraints, which tell how to interpret the sketch, and by adding parametric dimensions, which control the size of sketch geometry. Once the sketch is refined, you can create it as a 3D feature to which other features can be added. As the design changes, you can change the dimensions and constraints that control the sketch geometry, and the parametric model will be updated to reflect the new design.

When you are creating sketches by hand or for parametric modeling, think about the implications of the geometry you are drawing. Does the sketch imply that lines are perpendicular? Are the arcs that you have drawn intended to be tangent or intersecting? When you create a parametric model, the software applies rules to constrain the geometry, based on your sketch. You can remove, change, or add new constraints, but to use the software effectively you need to accurately depict the geometry you want formed.

Screenshot shows a constrained sketch in pro/ENGINEER sketch.

A Constrained Sketch

CAD at Work: Perspective Views in AutoCAD

AutoCAD software provides options for parallel and perspective viewing. You can use the menu for the ViewCube to quickly select Parallel or Perspective viewing options (Figure A). The defaults for perspective viewing determine the basic appearance when you select the perpective view types. Notice that the grid squares in the figure appear larger closer to the view and smaller when they are farther away. When parallel projection is selected, the grid lines remain parallel.

Screenshot of Autodesk AutoCAD 2016.

(A) A Perspective View Created Using the DView Command in AutoCAD (Autodesk screen shots reprinted courtesy of Autodesk, Inc.)

An interactive command, Dview (dynamic viewing), lets you show 3D models and drawings in perspective (or parallel view) and interactively change the viewing options. Dview uses a camera and target analogy to create perspective views. Using the Camera option, you select a camera position (similar to the station point) with respect to the target point at which the camera is aimed. The Dview Distance option lets you change the distance between the camera and the object and calculate a new view. The Off option of the command turns perspective viewing off to return to a parallel view.

Specifying a distance that is too close can fill up the entire view with the object, in the same way that a close-up shot with a camera does.

The Zoom option of the Dview command acts like a normal zoom command when perspective viewing is off. When perspective viewing is on, you can zoom dynamically by moving the slider bar shown in Figure B to adjust the camera lens to change the field of view. The default is to show the view similar to what you would see through a camera with a 50 mm lens. During the Zoom option, moving the slider is similar to choosing the focal length for a camera lens, which changes the relative perspective appearance (Figure B). Shorter focal lengths exaggerate the perspective effect of closer features.

Screenshot of Autodesk AutoCAD 2016.

(B) Using the Zoom Slider to Adjust Field of View (Autodesk screen shots reprinted courtesy of Autodesk, Inc.)

Industry Case: Sketching for Ideation: A CrossAction Breakthrough

When Oral-B hired LUNAR to create an innovative approach to brushing (and a new flagship product), they had no preconceived notions of what that would mean. But they knew that they wanted a brush that was ergonomically superior to the traditional “flat stick” design, and they wanted the brush to improve brushing effectiveness. The development of the Oral-B CrossAction Toothbrush is an example of the power of ideation in product design.

Jeff Salazar, lead designer for the project, started first not with sketches but with extensive research into the kinesiology of brushing teeth. A research firm had been hired to do a study of how people hold their toothbrushes as they brush. They found that there are five basic grips. Each person might use more than one depending on the part of the mouth to be brushed. Two or three grips were the ones used by most people.

As the kinesiology study was underway the design team at LUNAR undertook their own study of brushing. Given that the design constraints were focused almost totally on ergonomics, they studied how people brush. They engaged team members and others and watched them brush. They used models of teeth to investigate brushing. They bought “flat stick” toothbrushes, heated them so they could be bent at different angles, and had a variety of people report back on their brushing experience with handles of different shapes. People of various sizes and backgrounds, male and female, were asked to brush with the angled handles. This helped them understand the extremes of the angles forward and back that the handle could form.

Photograph of a toothbrush.

They also studied the competition. A competitor had come out with what it promised was a superior toothbrush—one that was “bent like a dental instrument.” After trying the toothbrush, however, the team concluded that the shape was only superior if someone else (e.g., the dental hygienist) was brushing your teeth. This key insight helped them focus the next stage of ideation on shapes with potential.

After discussing their results, the team selected five to six angle configurations for further study. The ergonomic results of the team’s efforts were an instinct that people wanted a toothbrush that would “fill the hand” more than the thin stick. But where should the brush be thicker? The team created a matrix that paired each of the angle configurations with the range of options for the handles’ thickness (Figure 3.80). Then, Jeff began sketching all the various combinations (see Figure 3.81).

Figure explains the matrix used to drive concept generation. The figure shows three sets of toothbrushes labeled, Angles, handles, and heads.

3.80 The Matrix Used to Drive Concept Generation

Various combinations of sketches of toothbrushes.

3.81 Sketches and Notes on the Concept as It Was Developed

After about a week of generating pencil sketches, Jeff decided he needed to “sketch” in 3D. Starting with a full-size sketch (Figure 3.82), he used a band saw to cut the shape of the sketch in foam, then shaped the foam into a model that he and others could hold and evaluate for various grips (Figure 3.83). Those that didn’t work for one grip or another were eliminated. The LUNAR team and their counterparts at Oral-B reviewed the various options and selected four for further development.

Side and rear view sketches of eight toothbrushes.

3.82 Full-Size Sketches Used to Create the Foam “3D Sketches”

Foam models od toothbrushes.

3.83 Foam Models Used as 3D Sketches

At this point, the team created 3D models of the four candidates. Oral-B had prototypes made from the CAD models for the handles (Figure 3.84), added bristles, and embarked on consumer research that would drive the next stage of refinement. Consumers were asked to brush for one month, then attend a focus group to report on their experience. LUNAR Design was present to hear consumer reactions.

Prototypes of toothbrushes generated from the 3D model.

3.84 Prototypes Generated from the 3D Model

What they found was that consumers loved the fatter handle. Its shape in the hand gave them more control when moving the brush to different parts of their mouth. The form provided optimal ergonomics for each of the five common grips. The faceted surface on the handle gave users a tactile clue as to where to place their fingers when gripping. The rubber like material used for the facets was located where users exerted the most finger pressure—and it made it easier to hold the brush when wet. The very positive response indicated that the ergonomic handle could make a significant difference in brushing that consumers would appreciate. People reported that they loved their toothbrush now!

At the same time, Jeff and the LUNAR and Oral-B teams had to listen and interpret what they heard. Focus groups are a wealth of different reactions, and it is up to the team to determine what is relevant and what is peripheral. For example, one reaction was that the fatter handle did not fit in the standard toothbrush holder. The team had to assess whether this was an issue that would hamper acceptance of the toothbrush or whether the improved brushing experience would outweigh its importance to the customer.

The refinement of the design proceeded through two more prototypes (and additional ideation around packaging; see Figure 3.85) before the final Oral-B CrossAction was launched. The intensity of the ideation stage—and the many techniques used to generate and evaluate design ideas—were critical to the team’s success in achieving this breakthrough in brushing.

Ideation sketches for crossaction packaging of a toothbrush is displayed.

3.85 Ideation Sketches for CrossAction Packaging

(All images courtesy of LUNAR.)

Key Words


Angular Perspective

Axonometric Projection

Box Construction

Cabinet Projection

Cavalier Projection


Construction Lines



Dimetric Projection





Freehand Sketch


Horizon Line

Isometric Axes

Isometric Drawing

Isometric Projection

Isometric Scale

Isometric Sketch


Line Patterns

Multiview Projection

Negative Space

Nonisometric Lines


Oblique Drawing

Oblique Projection

Oblique Projectors

Offset Measurements

One-Point Perspective

Orthographic Projections


Pictorial Sketch







Receding Lines

Regular Polyhedron







Three-Point Perspective


Trimetric Projection

Two-Point Perspective

Vanishing Point




Chapter Summary

• Sketching is a quick way of visualizing and solving a drawing problem. It is an effective way of communicating with all members of the design team.

• Three-dimensional figures are bounded by surfaces that are either planar, single-curved, double-curved, or warped.

• Prisms, pyramids, cylinders, cones, tori, and ellipsoids are common shapes in engineering drawings. There are also five regular polyhedra: the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron.

• There are special techniques for sketching lines, circles, and arcs. These techniques should be practiced so they become second nature.

• Using a grid makes sketching in proportion an easy task.

• Circles can be sketched by constructing a square and locating the four tangent points where the circle touches the square.

• A sketched line does not need to look like a CAD or mechanical line. The main distinction between CAD and instrumental drawing and freehand sketching is the character or technique of the line work.

• Freehand sketches are made to proportion, but not necessarily to a particular scale.

• Sketching is one of the most important skills for accurately recording ideas.

• Isometric, oblique, and perspective are three methods used to create pictorial sketches.

Review Questions

1. What are the advantages of using grid paper for sketching?

2. What is the correct technique for sketching a circle or arc?

3. Sketch the alphabet of lines. Which lines are thick? Which are thin? Which are very light and will not reproduce when copied?

4. What type of pictorial drawing can easily be drawn on square grid paper?

5. What is the advantage of sketching an object first before drawing it using CAD?

6. What is the difference between proportion and scale?

Sketching Exercises

Exercise 3.1 Sketch the objects shown using isometric, oblique, and one- or two-point perspective.

3D images of 11 parts.

Exercise 3.2 Quick Sketch.

1. Practice the sketching skills and techniques you have learned for construction lines and ellipses. Set a timer for 10 minutes and make quick sketches of these nine different containers.

2. Select one container and create isometric, oblique, and perspective drawings.

3. Design a new piece of drinkware, using your sketching skills.

4. Select one container. Draw an enclosing box and shade in the negative space so that the contour of the container remains white.

Photograph of a flower pot, teacup, jug, glass, whiskey glass, wine glass, coffee mug, cup, vase, and kettle.

Exercise 3.3 Quick Sketch. Practice the sketching skills and techniques you have learned for construction lines and ellipses. Set a timer for 10 minutes and make quick sketches of these six different chairs.

Photograph of five different types of chair.

Exercise 3.4 Quick Sketch. Practice the sketching skills and techniques you have learned for construction lines and ellipses. Set a timer for 10 minutes and make quick sketches of these tools.

Photograph of a toolbox, tongs, fishing reel, measuring tape, clamp, and hand saw.

Exercise 3.5 Sketching. Practice the sketching skills and techniques you have learned for construction lines and ellipses. Set a timer for 10 minutes and make quick sketches of these items.

Photograph of the bottom view and top view of a skateboard, gratters, and ladder.
Photograph of a parking meter.

Exercise 3.6 Parking Meter. Create an oblique pictorial sketch of the parking meter shown here. Use construction lines to keep the features in proportion.

Three free-body diagram sketches are shown.

Exercise 3.7 Free-Body-Diagram Sketches. Sketch the free body diagrams as shown.

Drawing of a rocker arm with its dimensions marked.

Exercise 3.8 Rocker Arm. Create a freehand sketch. Use the dimensions to help in estimating proportions. Leave the dimensions off your sketch unless directed otherwise by your instructor.

Drawing of a special cam with its dimensions marked.

Exercise 3.9 Special Cam. Create a freehand sketch. Use the dimensions to help in estimating proportions. Leave the dimensions off your sketch unless directed otherwise by your instructor.

Drawing of a Boiler Stay with its dimensions marked.

Exercise 3.10 Boiler Stay. Create a freehand sketch. Use the dimensions to help in estimating proportions. Leave the dimensions off your sketch unless directed otherwise by your instructor.

Drawing of a Outside Caliper with its dimensions marked.

Exercise 3.11 Outside Caliper. Create a freehand sketch. Use the dimensions to help in estimating proportions. Leave the dimensions off your sketch unless directed otherwise by your instructor.

Drawing of a Gear Arm with its dimensions marked.

Exercise 3.12 Gear Arm. Create a freehand sketch. Use the dimensions to help in estimating proportions. Leave the dimensions off your sketch unless directed otherwise by your instructor.

Drawing of a Special S-Wrench with its dimensions marked.

Exercise 3.13 Special S-Wrench. Create a freehand sketch. Use the dimensions to help in estimating proportions. Leave the dimensions off your sketch unless directed otherwise by your instructor.

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