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 #1
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
 
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Fig. 1 is an enlargement (closeup 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:
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* F1 *
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$\text{If }d\text{ is finite and nonzero, we have an }ellipse.$
Code:
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* F1 F2 *
*    o    o    *
*  d *

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$\text{If }d = \infty\text{, we have a }parabola.$
Code:
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F1
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$\text{If }\color{purple}{d\,>\,\infty}\text{, we have a }hyperbola.$
Code:

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→ → o   *    *   o
F2 * * F1
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This one has been "stretched around the Universe".
Hence, we can again see both ends.