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Hop
Posts:
4
Registered:
6/10/10


A nice family of conics
Posted:
Jun 10, 2010 7:46 AM


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



