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Topic: Intersection between two cones
Replies: 6   Last Post: Feb 7, 2013 4:36 AM

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Bill Whiten

Posts: 8
Registered: 6/18/10
Re: Intersection between two cones
Posted: Feb 6, 2013 6:21 AM
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"Doctor61" wrote in message <ket0to$214$>...
> I have two 3d circles (centre coordiantes and radii) with their normals passing through origin. If you consider the origin to be the vertex of a right cone having the circle as the base, how can I determine if there is an intersection between these cones?

if centre is x1,y1,z1 and radius r set d^2=x1^2+y1^2+z1^2
Cone is intersection of spheres a^2 (d^2+r^2)=x^2+y^2+z^2 and
a^2 r^2 = (a x1-x)^2+(a y1-y)^2+(a z1-z)^2

Eliminate a to get equation of cone:
x^2+y^2+z^2 = (r^2+d^2) (x1 x+y1 y+z1 z)^2/d^4
(check this)

Set x^2+y^2+z^2=1 and solve for points on two cones and this sphere.
E.g. Solve linear equations (1= (r^2+d^2) (x1 x+y1 y+z1 z)^2/d^4 etc)
for x and y then substitute into x^2+y^2+z^2=1 to get quadratic for z.


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