I made these remarkable "discoveries" while in college.


If you have a table lamp with a cylinderical shade,
turn it on and look at the pattern on the nearby wall.
You will see a hyperbola.


I was in the cafeteria, staring into my coffee mug.
There was particularly bright source of light nearby.
I saw what looked like a cardioid on the surface.
My friend and I spent the next hour proving it.
(We missed the next class.)


After working with [tex]x^2 \,=\,4py[/tex] for weeks, it finally
occured to me that a parabola has one parameter.
Other than orientation, location and scale,
there is exactly one parabola. .How can this be?

Aren't these two different parabolas?
Code:
               Fig. 1                              Fig. 2

                  |                                   |
                  |                            *      |      *
                  |                                   |
      *           |           *                 *     |     *
        *         |         *                    *    |    *
            *     |     *                          *  |  *
    - - - - - - - * - - - - - - -         - - - - - - * - - - - - -
                  |                                   |
Answer: No.

Fig. 1 is an enlargement (close-up view) of Fig. 2.



There is one conic curve.

Consider the distance [tex]d[/tex] between the two foci.


If [tex]d = 0[/tex], we have a circle.
Code:
              * * *
          *     |     *
        *       |       *
       *        |        *
                |
      *       F1|         *
      * - - - - o - - - - *
      *         |F2       *
                |
       *        |        *
        *       |       *
          *     |     *
              * * *

If [tex]d[/tex] is finite and nonzero, we have an ellipse.
Code:
              | * * *
          *   |           *
        *     |             *
       *      |              *
              |
      *     F1|      F2       *
      * - - - o - - - o - - - *
      *       |   d           *
              |
       *      |              *
        *     |             *
          *   |           *
              | * * *

If [tex]d = \infty[/tex], we have a parabola.
Code:
              |           *
              |   *
            * |
        *     |
       *      |
            F1|
      * - - - o - - - - - -  F2 → →
              |
       *      |
        *     |
            * |
              |   *
              |         *

Brace yourself!

If [tex]\color{purple}{d > \infty}[/tex], we have a hyperbola.
Code:
                              |
      *                       |   *
          *                   *
             *             *  |
               *         *    |
      → → o - - * - - - * - - o
          F2   *         *    |F1
             *             *  |
          *                   *
      *                       |   *
                              |