## Some observations on Conics

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

|                                   |
|                            *      |      *
|                                   |
*           |           *                 *     |     *
*         |         *                    *    |    *
*     |     *                          *  |  *
- - - - - - - * - - - - - - -         - - - - - - * - - - - - -
|                                   |

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

There is one conic curve.

Consider the distance $d$ between the two foci.

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

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

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

Brace yourself!

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