Projective Geometry

Date: 01/13/97 at 21:47:16
From: TONY & ANDREW
Subject: Finding the equation of an ellipse with a trapezoid

Hi,

For my assignment in fractals class I've generated a fractal terrain. 
It lies on a 2D plane. I'm adding perspective and movement to it. When 
looking at it from a semi-bird's eye view, the plane looks like a 
regular trapezoid. When I rotate it right or left, the four corners 
seem to move along an ellipse.  How do you find the equation with the 
center of the ellipse equal to the center of the trapezoid?  

I've tried picking two points and setting them equal to each other: 

	b^2*x^2 + a^2*y^2 = b^2*xx^2 + a^2*yy^2

but it ends up that you have to divide by zero because the y of both 
sides is the same when squared. Also the center is not (0,0), but 
(0.5,0.5) and the range is 0 to 1 for both x and y, if that changes 
anything. 

I'd really appreciate your help. I've worked day and night on this, 
consulting everywhere, and I can't still get it to work!

Thanks!

Tony Kuo

Date: 01/14/97 at 13:30:22
From: Doctor Rob
Subject: Re: Finding the equation of an ellipse with a trapezoid

This is a very interesting question in projective geometry.

When you refer to a plane, I am assuming you mean a square region in 
an infinite plane. When it rotates about its center, its corners move 
on a circle whose center is the center of the square and whose 
diameter is the length of the diagonal of the square. 

You are looking at this circle obliquely. The lines which light 
traverses from the points on this circle to your eye form a cone whose 
base is that circle, but the line from the vertex (your eye) to the 
center of the circle is not perpendicular to the plane.  Call this 
line the "center line".  

You wish to know the equation of the intersection of the cone with a 
plane perpendicular to the center line.  This will, indeed, be an 
ellipse.  You will need to know the x-, y-, and z-coordinates of your 
eye, the x- and y-coordinates of the center of the square (we assume 
the z-coordinate is zero), and the length of the side of the square 
(from which you compute the radius of the circle by dividing by 
Sqrt[2]).  Then you can write down the equation of the circle, and the 
equation of the center line, and the equation of the cone.  Then you 
can write down the equation of a plane perpendicular to the center 
line and passing through any convenient point.  Next you rotate your 
coordinates so that the plane you constructed is the xy-plane with 
equation z = 0.  Plug that into the rotated equation of the cone, and 
you will have the equation of the ellipse you seek.

I hope you understand what I have written above.  If not, or if you
need more assistance, feel free to write back again.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/