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Observations on Conics

soroban

Well-known member
Feb 2, 2012
409

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 $x^2 \,=\,4py$ 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.

$\text{Consider the distance }d\text{ between the two foci.}$


$\text{If }d = 0\text{, we have a }circle.$
Code:
               * * *
           *     |     *
         *       |       *
        *        |        *
                 |
       *       F1|         *
       * - - - - o - - - - *
       *         |F2       *
                 |
        *        |        *
         *       |       *
           *     |     *
               * * *


$\text{If }d\text{ is finite and nonzero, we have an }ellipse.$
Code:
               | * * *
           *   |           *
         *     |             *
        *      |              *
               |
       *     F1|      F2       *
       * - - - o - - - o - - - *
       *       |   d           *
               |
        *      |              *
         *     |             *
           *   |           *
               | * * *


$\text{If }d = \infty\text{, we have a }parabola.$
Code:
               |           *
               |   *
             * |
         *     |
        *      |
             F1|
       * - - - o - - - - - -  F2 → →
               |
        *      |
         *     |
             * |
               |   *
               |         *
We have an "infinitely long ellipse".



$\text{If }\color{purple}{d\,>\,\infty}\text{, we have a }hyperbola.$
Code:
                               |
       *                       |   *
           *                   *
              *             *  |
                *         *    |
       → → o - - * - - - * - - o
           F2   *         *    |F1
              *             *  |
           *                   *
       *                       |   *
                               |
We once again have an infinitely long ellipse.
This one has been "stretched around the Universe".
Hence, we can again see both ends.