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 \(\displaystyle x^2 \,=\,4py\) 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 \(\displaystyle d\) between the two foci.
If \(\displaystyle d = 0\), we have a circle.
Code:
* * *
* | *
* | *
* | *
|
* F1| *
* - - - - o - - - - *
* |F2 *
|
* | *
* | *
* | *
* * *If \(\displaystyle d\) is finite and nonzero, we have an ellipse.
Code:
| * * *
* | *
* | *
* | *
|
* F1| F2 *
* - - - o - - - o - - - *
* | d *
|
* | *
* | *
* | *
| * * *If \(\displaystyle d = \infty\), we have a parabola.
Code:
| *
| *
* |
* |
* |
F1|
* - - - o - - - - - - F2 → →
|
* |
* |
* |
| *
| *Brace yourself!
If \(\displaystyle \color{purple}{d > \infty}\), we have a hyperbola.
Code:
|
* | *
* *
* * |
* * |
→ → o - - * - - - * - - o
F2 * * |F1
* * |
* *
* | *
|