# Observations on Conics

#### soroban

##### Well-known member

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.