points in space

[Peter J. Floriani, Ph.D.]

c07craig at sfsu.edu wrote:

>Suppose we know vector PQ which is in a plane in space.  
>We also know angle theta, which is the angle between PQ and PR.  
>PR is also in the same plane as PQ, and the unit length 
>of PQ is the same as PR.  So, given all this,
>how do we find vector PR?

>In other words, how do we solve for point R in three dimensional space?

>Another way to put it is how can one use a polar coordinate 
>system that is in an arbitrary plane in space?


This is rather straightforward with vector algebra, 
and is supplied below.

If you need further help, check out a vector algebra text.

Peter J. Floriani, Ph.D.
floriani at epix.net

"I have often thanked God for the telephone." G. K. Chesterton, 1910

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Given the following:
   N is the vector normal to the given plane
   Q is the vector in the plane along the axis of the polar system
   P (a point in the plane) is the pole

Let [S] be the matrix which rotates <0,0,1> to N.
Let [Z] be the matrix which rotates N to <0,0,1>.
Let [T] be the matrix which rotates <1,0,0> to Q*[Z].

Thus, [T]*[S] rotates the xy plane to the NQ system.
So, given a polar coordinate (r,theta),
to find it in the new system, compute
   <r*cos(theta),r*sin(theta),0> * ([T]*[S])

This is the vector R desired.

Or, to get the point in 3-space, merely offset this by P.

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To rotate v to w, compute a 3x3 matrix in the following way:

A. Rotate v to <d,0,0> where d=|v| by
     1. rotate about x axis so v' lies in the xy plane.
     2. rotate about z axis so v'' lies in the x axis.
     and record the operation in r.

Details:
If v is <anything,0,0> then r is the identity matrix.
Otherwise, let v=<x,y,z> and compute:

      d=sqrt(x*x+y*y+z*z)
      q=sqrt(y*y+z*z)
      r[1,1]=x/d
      r[1,2]=-q/d
      r[1,3]=0
      r[2,1]=y/d
      r[2,2]=x*y/(d*q)
      r[2,3]=-z/q
      r[3,1]=z/d
      r[3,2]=x*z/(d*q)
      r[3,3]=y/q

 B. Rotate <d,0,0> to w
     3. rotate about z axis
     4. rotate about x axis
     and record the operation in s.

Details:
If w is <anything, 0,0> then s is the identity matrix.
Otherwise, let w=<x,y,z> and compute:


      d=sqrt(x*x+y*y+z*z)
      q=sqrt(y*y+z*z)
      s[1,1]=x/d
      s[1,2]=y/d
      s[1,3]=z/d
      s[2,1]=-q/d
      s[2,2]=x*y/(d*q)
      s[2,3]=x*z/(d*q)
      s[3,1]=0
      s[3,2]=-z/q
      s[3,3]=y/q

C. Finally, the desired matrix is r*s.