Parabola
Parabola
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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 and had 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 generator.

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 compass.

Since the eccentricity of a parabola is 1, the vertex is equidistant from the directrix and focus (vD = vF).
  1. Using a straightedge, draw the directrix.
  2. 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 focus.
  3. 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 distance.
  4. 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 directrix.
  5. Repeat steps 3 and 4 until enough points are established to draw the parabola.
  6. Connect the dots and you have a parabola.
  7. 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.
  1. A compass turned by a gear opens a parallelogram placed on the directrix.
  2. For convenience, the first setting is equal to vD (= vF).
  3. 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 adjustable pin.
  4. The same pin is 'bumped up' as it crosses the parallelogram's movable green arm, and makes a red dot on a transparent screen.
  5. The yellow link and parallelogram are returned to the starting position.
  6. A new 'compass setting' is chosen, and the process repeats itself.
  7. 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 particular curve.

More Conic Section definitions are included on the Hyperbola page.

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 devices.

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 the sine page.)

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