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Topic: A nice family of conics
Replies: 5   Last Post: Jun 16, 2010 8:49 PM

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Hop

Posts: 4
Registered: 6/10/10
A nice family of conics
Posted: Jun 10, 2010 7:46 AM
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I have been wanting to do a perspective drawing of ripples on a lake.

To help me, I imagine a pin hole camera projecting an image of the
ripples on a film plane.

The pin hole is the origin: (0, 0, 0)
Camera face plane is y = 0
Film plane is z = -1
The lake surface lies in the plane z = -1
The ripples are concentric circles lieing in the lake plane, each
circle having the center (0, 1, -1)

I believe these things to be true:

The image of the circles' center is (0, -1, 1).

The image of each circle is a conic section. Each conic section has
(0, -1, 1) as a focus. And the horizon line is the directrix.

A circle having radius r maps into a conic section of eccentricity r.
For example, a circle of radius .5 maps into an ellipse with
eccentricity .5. The circle of radius 1 maps into a parabola.

Here is a pic:
http://www.clowder.net/hop/TMI/Ripples.jpg
The circle of radius 1 and its image, the parabola, are darkened.

I've convinced myself the above is true for circles with radius < 1.
The parabola and hyperbolas seem to fit but I haven't thought of a
demonstration.

Can anyone come up with a proof?

Regards,

Hop
http://clowder.net/hop/index.html



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