Point on an Ellipse

Date: 05/16/97 at 23:44:16
From: Rich Kadel
Subject: Calculate point on an ellipse given angle

None of the physics, geometry, or calculus books I have give me this 
formula, but it seems as if it should be simple.

I have an ellipse defined by major and minor axis, 2a, and 2b, 
respectively. Assume 'a' is along the x-axis and 'b' is along the 
y-axis. Given an arbitrary angle theta from either axis (you pick), 
I need either the point on the ellipse that a vector with that angle 
intersects, or the distance from the origin to the edge of the ellipse 
along the vector.

Thanks

Date: 05/17/97 at 11:19:13
From: Doctor Jerry
Subject: Re: Calculate point on an ellipse given angle

Hi Rich,

Here's a method that works in the first quadrant and can be adapted 
for other quadrants. I'm assuming that, given an angle theta (I'll 
use t), you want to locate the point common to the ellipse and line 
with equation y = tan(t)*x. You could just substitute from this 
equation into the equation x^2/a^2 + y^2/b^2 = 1 and solve for the 
remaining variable.  

It may be slightly easier to use the parametric equations of the 
ellipse. Suppose the ellipse is described by the parametric equations
x = a*cos(q) and y = b*sin(q), where 0 <= q < 2*pi.

We want to calculate q so that y = tan(t)*x, that is,
b*sin(q) = tan(t)*a*cos(q).

So, tan(q) = a*tan(t)/b.

In the first quadrant, q = arctan(a*tan(t)/b).

This value can be substituted into the parametric equations for the 
ellipse, to evaluate x and y.

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