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/
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