The parabola is probably the most important of the three conic
curves (ellipse, parabola, and hyperbola) in terms of everyday
use. It is the shape of reflectors in flashlights, headlights,
antenna dishes, optical and radio telescopes, and spotlights.
It can direct parallel incoming waves to a point (the focus).
And, It projects parallel waves when the wave source is at the
focus. Also, it is the basic trajectory of a projectile,
without correction for air resistance.
I stated drawing 'Curve Generators' when I wanted to make
animations of gears. The common illustration of two gears
meshing was too mundane. I decided to add a rack to two gears
so I could hang a transparent screen from it and trace a
Cycloid Curve on the screen. Trochoid Curves and Sine and
Cosine Curves lend themselves to this same arrangement, since
they are repeating curves.
When I started working on the conic sections, I already knew
of the pin and string method for drawing an Ellipse
made a wooden toy in my workshop that is, in fact, the Trammel of Archimedes
These were my first Conic Section Animations: one method and one
I needed to add to my collection of generators, so I came up
with the parabola generator, above. After studying the
geometric definition of the conic sections, I realized that
the parabola could be drawn with just a straight edge and a
Since the eccentricity of a parabola is 1, the vertex is
equidistant from the directrix and focus (vD = vF).
- Using a straightedge, draw the directrix.
- Using the straightedge and a compass, construct a line
perpendicular to the directrix. The vertex and focus lie
on this line. Choose a convenient compass setting and
mark, from the directrix, the vertex, and from there, the
- Draw a line parallel to the directrix, again using
only the straight edge and the compass. Use a compass
setting greater than the one chosen for the vertex/focus
- Keeping the same compass setting, draw arcs
intersecting the parallel line, using the focus as the
center of the arcs. Points on the parabola are indicated
where the arc intersects the line that parallels the
- Repeat steps 3 and 4 until enough points are
established to draw the parabola.
- Connect the dots and you have a parabola.
- Time consuming, but it works.
Having a method for drawing a parabola without plotting points
on an X-Y axis, I had to come up with a way to mechanize it.
- A compass turned by a gear opens a parallelogram
placed on the directrix.
- For convenience, the first setting is equal to vD
- Keeping the same setting, the compass is moved to the
focus and rotated 360o, intersecting the parallelogram at
two points. The compass is the yellow link with the
- The same pin is 'bumped up' as it crosses the
parallelogram's movable green arm, and makes a red dot on
a transparent screen.
- The yellow link and parallelogram are returned to the
- A new 'compass setting' is chosen, and the process
- Many points, close together, on either side of the
vertex are needed to see the completed parabola. I have
plotted points and drawn a line, connecting the dots, to
complete the animation.
After completing the Parabola Generator, I realized that the
device works on the polar coordinates definition of the
parabola, whereas my Trochoid and Sine and Cosine Generators
work on the rectangular coordinates definition of the
More Conic Section definitions are included on the Hyperbola
More complete mathematical discussions of the curves
mentioned above can be found at other sites on the web. Xah's
collection of curves is very thorough
and has many animations in other formats. Xah is a professional
programmer with a very extensive knowledge of curves, whereas I am
but a struggling animator with a flair for geometry and mechanical
I would like to thank J Buck, again, for contributing to
Edwin's Animated Images. Mr. Buck helped with the content of
this page. He edited the description of my parabola generator
and its creation. Mr. Buck is also responsible for the sine
and cosine generators being developed. (See acknowledgment on