orbit that is a long, skinny ellipse with the sun at
a focus way out near the end of the semimajor
axis. In the 3D print, the foci are pretty much
buried in the wall at each end of the ellipse.
Figure 11-10 shows the orbits and orbital veloc-
ities of Mercury, Venus, and Earth (to scale with
each other, but not to Halley’s Comet).
As you can see from the height of the model,
the comet is moving very fast near the sun, then
slows to a crawl at the other extreme as it moves
along in its 75-year-long orbit. Mercury, too, has
a somewhat elliptical orbit and will go faster in
the part of its orbit nearer the sun than it will in
the other. Looking at the set of three orbits, as
we go farther from the sun, the orbits of Venus
and Earth are each slower than the planets
closer to the sun.
This variation of the speed of a planet, like Earth,
in its trip around the sun is a major component
in the Equation of Time. We learned about the
Equation of Time in Chapter 7 when we had to
look up its value to deduce our location based
on when the sun was highest in the sky. Earth’s
orbit is very nearly circular, but even that small
difference matters. In Chapter 12, we will reprise
that, and go into more depth about how to take
these effects into account when using the sun to
tell time.
PARABOLA
As we saw in Chapter 10, a parabola is created
only when a cutting plane is exactly parallel to
the slant angle of the cone. If the cutting plane is
any shallower (relative to the bottom of the cone) you get an ellipse; any steeper, you get a hyperbola.
A parabola can be thought of as the special case where an ellipse transitions into a hyperbola. It is
shaped somewhere between the letters U and V (never quite coming to an actual point). The sharpest
point of its turnaround is called the vertex. Some people like to think of a hyperbola as an inside-out
ellipse. After we talk about all three, we’ll come back to that.
Directrix and Generatrix
Getting further into the details of conic sections
requires that we make the acquaintance of a new
pair of abstract ideas, the directrix and generatrix
(sometimes called a generator instead). Despite
sounding like pets of superheroes, they are imag-
inary tools that can help us draw a curve. Since
we are all about being hands-on, we are going to
make and use physical versions in the next sec-
tions of this chapter.
A generatrix is a point, curve, or surface that
when moved along a certain path carves out a
desired line, surface, or 3D shape, respectively. A
directrix is that path.
For example, take a straight line that has the
top end pinned at a point and the bottom swept
around a circle to sweep out a cone. The line is
the generatrix, and the circle is the directrix. We
are going to use a directrix to construct conic
sections (specifically, hyperbolas and parabolas)
without (much) algebra, in case you want to make
something physical in one of these shapes. You
know how to make an ellipse now (with a pinned
string or rope). Let’s see how to draw parabolas
and hyperbolas, and why you might want to.
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