Intersecting Circles in 3 Space
Date: 04/19/2006 at 18:23:10 From: Tilak Subject: Intersecting circles in 3 space. Hi Dr. Math, I have 2 circles in 3 space, P1 = C1 + R1cos(t)U + R1sin(t)V P2 = C2 + R2cos(s)U + R2sin(s)V where C = centre R1 and R2 = the respective radii U = a unit vector from C to the circle V = NxU N = a vector normal to the plane both circles are in Where do the circles intersect (assuming they do at 1 or 2 points)? Since both variables are "stuck" inside the cos and sin terms, I can't solve the equations even though I do have 2 equations and 2 unknowns. I'm actually trying to find the coordinates of the knee given locations of toe, ankle and hip. I just can't solve the non-linear equations. Thanks for your help.
Date: 04/20/2006 at 08:30:20 From: Doctor George Subject: Re: Intersecting circles in 3 space. Hi Tilak, Thanks for writing to Doctor Math. Here are two approaches to this problem. First Solution -------------- We need the points where P1 = P2. C1 + R1cos(t)U + R1sin(t)V = C2 + R2cos(s)U + R2sin(s)V R1cos(t)U + R1sin(t)V = C2 - C1 + R2cos(s)U + R2sin(s)V I will assume that N, and therefore V, are also unit vectors. If we take the dot product of both sides with U we get R1cos(t) = (C2 - C1).U + R2cos(s) (1) If we take the dot product of both sides with V we get R1sin(t) = (C2 - C1).V + R2sin(s) (2) If we square both sides of (1) and (2) and add them we can get t to drop out, leaving us with an equation to solve for s. With a little rearranging we could use the same technique to find an equation to solve for t, but you only need s or t to get the intersection points. Second Solution --------------- I had written this solution earlier for someone else. It uses different notation. Let C1 and C2 be the centers of the circles with radii r1 and r2, and let d be the distance between C1 and C2. Now let V1 be the unit vector from C1 to C2, and let V2 be a unit vector perpendicular to V1. Also let V3 be the vector from C1 to one of the intersection points. Finally, let A be the angle between V1 and V3. From the law of cosines we know that r2^2 = r1^2 + d^2 - 2*r1*d*cos(A) With this equation we can solve for 'A'. The intersection points will be C1 + [r1*cos(A)]*V1 + [r1*sin(A)]*V2 C1 + [r1*cos(A)]*V1 - [r1*sin(A)]*V2 I hope this helps. It looks like your actual problem might be more complex than the intersection of two circles. If you can describe it I would be glad to give it some thought. - Doctor George, The Math Forum http://mathforum.org/dr.math/
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