Conic Section of an Elliptical Cone
Date: 08/01/2007 at 22:25:41 From: Tom Subject: conic section of an elliptical cone If a flashlight with an elliptical beam is shining on the wall at a slant, is the spot elliptical and, if so, how can the parameters be computed? This problem involves a lot of parameters. There is the semi-major and semi-minor axes of the ellipse, the orientation of the semi-major axis, and the angle of incidence. I solved this numerically by curve-fitting the boundary of the spot to an ellipse. I used a hundred points. It worked OK, but I'm not too happy about it. There ought to be a formula for it.
Date: 08/02/2007 at 06:18:47 From: Doctor George Subject: Re: conic section of an elliptical cone Hi Tom, Thanks for writing to Doctor Math. The intersection is only an ellipse if the center of the beam intersects the wall. Otherwise you get a different conic section. I think we need to model the light beam as a cone. I'll assume that the cone can be elliptical for more generality. Start by putting a coordinate system at the origin of the cone so that the cone is in standard position. Now write the equation of the plane in that coordinate system as follows. a1 x + b1 y + c1 z + d = 0 We can also write it like this. _ _ | x | [a1 b1 c1] | y | + d = 0 (1) | z | - - Now let _ _ _ _ | x | | u | | y | = Q | v | (2) | z | | w | - - - - where Q is an orthogonal matrix such that [a1 b1 c1] Q = [0 0 m] (3) where m is the length of [a1 b1 c1]. The strategy is to simplify the problem by rotating the cone and the plane so that the plane is horizontal in the coordinate system. Substituting (2) into (1) and utilizing (3) gives us w = -d / m Now if the cone is x^2/(a2)^2 + y^2/(b2)^2 = z^2/(c2)^2 we can write it like this. _ _ _ _ | 1/(a2)^2 0 0 | | x | [x y z] | 0 1/(b2)^2 0 | | y | = 0 (4) | 0 0 -1/(c2^)2 | | z | - - - - Substituting (2) into (4) and utilizing (3) gives us _ _ _ _ | 1/(a2)^2 0 0 | | u | [u v -d/m] Q^T | 0 1/(b2)^2 0 | Q | v | = 0 | 0 0 -1/(c2)^2 | | -d/m | - - - - Carrying out the multiplication gives us a general quadratic in u and v that is the equation of the intersection. If the axis of the cone intersects the plane at a steep enough angle the result will be an ellipse. An shallower angle will lead to other conic types. [The breakdown of the cases is more complex that what I indicated in a previous message. The complement of the half angle of the cone appears to be the critical angle value.] A rotation and translation in the (u,v) plane will lead to the standard form of the intersection in (u',v') coordinates. Once you find all of the points and vectors of interest in (u',v') coordinates and then (u,v,w) coordinates, you can multiply by Q to find them in (x,y,z) coordinates. All that remains is to find Q. There are a number of ways to find such a matrix. That can be left for a separate discussion if need be. Does that make sense? Write again if you need more help. - Doctor George, The Math Forum http://mathforum.org/dr.math/
Date: 08/03/2007 at 07:28:24 From: Tom Subject: Thank you (conic section of an elliptical cone) That is a nice, simple solution. Thank you very much.
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