General Equation for Intersections of Line and Ellipse
Date: 03/13/2007 at 06:04:49 From: Baris Subject: Intersection of line and ellipse I am trying to find the intersection points (if there are any) of a line and ellipse. The line is defined by its 2 points, say (x0,y0) and (x1,y1), and the ellipse is defined as x^2/a^2 + y^2/b^2 = r^2 (center is at the origin). Is there a simple way to find any intersections? First I wrote the line's equation: y-y0 = ((y1-y0)/(x1-x0))(x-x0) y = ((y1-y0)/(x1-x0))(x-x0) + y0 then I tried to put y into the ellipse equation so I can have a second degree equation with x as the only variable. But the equation becomes too complicated.
Date: 03/13/2007 at 06:35:10 From: Doctor George Subject: Re: Intersection of line and ellipse Hi Baris, Thanks for writing to Doctor Math. It looks like you were doing the right things. See if this approach makes the algebra a little simpler. Use your two points to construct the parametric equations for the line as follows. x = x0 + at y = y0 + bt Now substitute these expressions for x and y into the ellipse equation and solve for t. Does that make sense? Write again if you need more help. - Doctor George, The Math Forum http://mathforum.org/dr.math/
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