11
CHAPTER 11
CONSTRUCTING
CONICS
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Conic sections (which we introduced in Chapter 10) turn up in many applications. They come up all the
time in phenomena as different as the path of a ball tossed up in the air, the orbits of the planets, and
the designs of telescopes and satellite dishes. We will explore ways to draw them and suggest places
you can find them in the wild.
The approach in this chapter is a little unconventional. Typically, people start with the same definitions
of the relationships of various features of conic sections, and then (using a lot of algebra) derive equa-
tions from them. This gets messy pretty quickly and walks into algebra that is beyond what we assume
you know how to do at this point.
Instead, we’re going to use a very simple 3D printed model to construct the conics geometrically. The
model is so simple that you could probably figure out alternatives with stuff you have lying about (more
on this later), but the 3D printed model is convenient.
ELLIPSES AND CIRCLES
How are ellipses and circles different from each other? We can define a circle as a constant distance
from one point. You learned that when you drew circles with a compass in Chapter 4. If you sweep the
pencil point all the way around, you get a circle.
What about an ellipse, though? We saw in Chapter
10 that if we take a slice of a cone parallel to the
base of the cone, the cross-section is a circle.
However, if we start to tilt that cut a little, the
circle will stretch out into an ellipse. It turns out
that ellipses have some interesting properties,
and we will start with the classic way to draw
one.
Take a piece of cardboard or foam core, thick
enough so that push pins won’t poke all the way
through. (We used two layers of a shipping card-
board box). Tape a piece of paper over this base.
Take two push pins and use them to pin down the
ends of a piece of string (Figure 11-1). You can tie
a knot in the string to prevent it from unraveling
and make a more solid anchor for the push pins.
The string should be loose. About 25% longer
than the distance between the pins should be
about right. String used to tie up packages works,
or you can use thinner string if you prefer.
3D Printable Models Used in
this Chapter
See Chapter 2 for directions on where and how to
download these models.
centerfinders.scad
A series of nested half-circular pieces used in the
constructions in this chapter.
Other things you will need:
• Piece of foam core board or heavy cardboard
that will hold a pushpin
• Some paper to draw on
• Two push pins or thumbtacks
• String (like package string) that is not
stretchy
• A pencil
• A drawing compass
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Now, put your pencil alongside the string, and stretch it taut, but not so tight
that the push pins want to pop out (Figure 11-2).
Now, keeping the string taut, you can move the pencil around the push pins.
You should be able to draw the top or bottom half of an ellipse(depending on
where you started), and then adjust your pencil to make the other half (Figure
11-3).
Finally, you will have the entire ellipse (Figure 11-4).
Try moving the push pins farther apart. The ellipse will get longer and skin-
nier (Figure 11-05). On the other hand, if you imagined putting them right on
top of one another (or, more likely, using only one push pin), you’d just have a
circle.
The longer dimension of the ellipse (all the way from one end to the other,
like a circle’s diameter) is called the major axis. Half of it, like the radius of
FIGURE 111: Setting up the string to draw an ellipse
FIGURE 113: Drawing the bottom of the ellipse
FIGURE 112: Placing your pencil
FIGURE 114: Finished ellipse
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a circle, is called the semimajor axis, and usually
referred to as “a”. The narrower dimension is called
the minor axis, and half of it is called the semiminor
axis, usually shown as “b”.
The places where our push pins reside are called
the foci (pronounced “foe-sigh” or “foe-kigh”,
depending on where you are from). The singular is
focus. The foci are a distance from the center of the
ellipse that is often called “c” (Figure 11-6).
The dimensions a, b, and c of an ellipse are all related like this:
c=a-b
Where does that come from? Try to figure it out, then check out “Finding the
locations of the Foci” in the Answers at the end of the chapter. (Hint: remem-
ber Pythagoras back in Chapter 6.)
You can also use a loop of string, instead of pinning the string (Figure 11-7).
This has the advantage that it lets you run your pencil all the way around
the ellipse without picking it up, but the math is just a bit more complicated.
In this case, the circumference of your loop would be 2(a + c). If you want to
see how the ellipse changes when you adjust the distance between the foci
without changing the major axis, you would need to adjust the loop so that its
circumference remains 2(a + c).
FOCI OF AN ELLIPSE AND INTERNAL REFLECTION
Our string was a constant length as we drew our ellipse around the circle,
FIGURE 115: Moving the push pins farther apart
FIGURE 117: The loop of string variation
FIGURE 116: Semi-major and semi-minor axes and foci of an
ellipse.
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held down at the foci at each end. That means that for any point on the
ellipse, the sum of the distances from the two foci will be the same. This
gives an ellipse some interesting properties.
Imagine that there was a light at one focus and that the inside of the ellipse
was a mirror. Light that reflects off of a mirror will leave the surface at the
same angle that it came from but in the opposite direction. Think of the string
as a ray of light coming from one focus and reflecting back to another. No
matter where the light from the first focus hits the side of the ellipse, it will
reflect back to the second focus and will have to travel the same total dis-
tance. (A physicist would say that the light waves stay in phase). This is true
for sound waves, too. Note that we are just talking about a two-dimensional
abstraction here, and the behavior can get quite complicated in a 3D space.
WHISPERING GALLERIES
One feature of elliptical (and other curved) spaces is that sound from one
focus can be heard particularly well at the other focus. This has been
exploited in “whispering galleries.” If you stand at certain places in some
buildings with curved walls and domes you can hear sound from the other
focus of the ellipse, even if it is far away and just a whisper. Most famously,
this is said to occur at St. Paul’s Cathedral in London and one place in Grand
Central Terminal in New York City near the Oyster Bar. Whispering galleries
occur in other curved rooms, and their physics is a little tricky. We’ll talk
about parabolic “acoustic mirrors” in the next section.
Sticking to ellipses, the sound from anywhere in this elliptical space (not at
the foci) will arrive at the two foci as well, but will not be, well, focused in the
same way that sound from the other focus will be. A physicist would say that
they would interfere with each other, which just means that different sound
waves bouncing around in the room would cancel each other out. However, a
sound coming from the other focus will be clear even if there is a lot of other
noise bouncing around the room.
It would be pretty tricky to make yourself a whispering gallery. The wall
has to be hard and smoothly curved, which isn’t something one encounters
routinely. But if you are someplace with curved walls, you might see if you
can find a place where this phenomenon will happen. Cloth, unfortunately, is
likely to absorb sound, so hanging a sheet in a curve won’t make you a whis-
pering gallery. However, someplace with a lot of curved concrete (like a skate
park) might have some interesting sound reflections.
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