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/