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Re: Cirlce in 3D?

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Using John Browne's Grassmann algebra application and with the three points
at Cartesian coordinates {ax, ay, az}, {bx, by, bz}, and {cx, cy, cz} I
calculated the center of the circle at:

{x -> -(4 (az (by - cy) + bz cy - by cz + 
         ay (-bz + 
            cz)) (-(ax^2 + ay^2 + az^2 - bx^2 - by^2 - bz^2) (-by cx +
             ay (-bx + cx) + ax (by - cy) + bx cy) - 
         2 (az - bz) (az by cx - ay bz cx - az bx cy + ax bz cy + 
            ay bx cz - ax by cz)) - (2 (ax^2 + ay^2 + az^2 - bx^2 - 
            by^2 - bz^2) (bz - cz) - 
         2 (az - bz) (bx^2 + by^2 + bz^2 - cx^2 - cy^2 - 
            cz^2)) (-2 (ay - by) (ay (bx - cx) + by cx - bx cy + 
            ax (-by + cy)) - 
         2 (az - bz) (az (bx - cx) + bz cx - bx cz + 
            ax (-bz + cz))))/(-4 (az (bx - cx) + bz cx - bx cz + 
         ax (-bz + cz)) (-2 (ay - by) (ay (bx - cx) + by cx - bx cy + 
            ax (-by + cy)) - 
         2 (az - bz) (az (bx - cx) + bz cx - bx cz + 
            ax (-bz + cz))) + 
      4 (az (by - cy) + bz cy - by cz + 
         ay (-bz + cz)) (2 (ax - bx) (-by cx + ay (-bx + cx) + 
            ax (by - cy) + bx cy) + 
         2 (az - bz) (az (by - cy) + bz cy - by cz + ay (-bz + cz)))),
  y -> (-2 (ax^2 + ay^2 + az^2 - bx^2 - by^2 - bz^2) (bz - cz) + 
     2 (az - bz) (bx^2 + by^2 + bz^2 - cx^2 - cy^2 - 
        cz^2) + (4 (az (bx - cx) + bz cx - bx cz + 
          ax (-bz + cz)) (4 (az (by - cy) + bz cy - by cz + 
             ay (-bz + 
                cz)) (-(ax^2 + ay^2 + az^2 - bx^2 - by^2 - 
                 bz^2) (-by cx + ay (-bx + cx) + ax (by - cy) + 
                bx cy) - 
             2 (az - bz) (az by cx - ay bz cx - az bx cy + ax bz cy + 
                ay bx cz - ax by cz)) - (2 (ax^2 + ay^2 + az^2 - 
                bx^2 - by^2 - bz^2) (bz - cz) - 
             2 (az - bz) (bx^2 + by^2 + bz^2 - cx^2 - cy^2 - 
                cz^2)) (-2 (ay - by) (ay (bx - cx) + by cx - bx cy + 
                ax (-by + cy)) - 
             2 (az - bz) (az (bx - cx) + bz cx - bx cz + 
                ax (-bz + cz)))))/(-4 (az (bx - cx) + bz cx - bx cz + 
           ax (-bz + cz)) (-2 (ay - by) (ay (bx - cx) + by cx - 
              bx cy + ax (-by + cy)) - 
           2 (az - bz) (az (bx - cx) + bz cx - bx cz + 
              ax (-bz + cz))) + 
        4 (az (by - cy) + bz cy - by cz + 
           ay (-bz + cz)) (2 (ax - bx) (-by cx + ay (-bx + cx) + 
              ax (by - cy) + bx cy) + 
           2 (az - bz) (az (by - cy) + bz cy - by cz + 
              ay (-bz + cz)))))/(4 (az (by - cy) + bz cy - by cz + 
       ay (-bz + cz))), 
 z -> (-az^2 by + by cx^2 + az^2 cy - bx^2 cy - by^2 cy - bz^2 cy + 
      by cy^2 + ax^2 (-by + cy) + ay^2 (-by + cy) + by cz^2 + 
      ay (bx^2 + by^2 + bz^2 - cx^2 - cy^2 - cz^2))/(2 (-bz cy + 
        az (-by + cy) + ay (bz - cz) + by cz)) + ((-by cx + 
        ay (-bx + cx) + ax (by - cy) + 
        bx cy) (4 (az (by - cy) + bz cy - by cz + 
           ay (-bz + 
              cz)) (-(ax^2 + ay^2 + az^2 - bx^2 - by^2 - 
               bz^2) (-by cx + ay (-bx + cx) + ax (by - cy) + 
              bx cy) - 
           2 (az - bz) (az by cx - ay bz cx - az bx cy + ax bz cy + 
              ay bx cz - ax by cz)) - (2 (ax^2 + ay^2 + az^2 - bx^2 - 
              by^2 - bz^2) (bz - cz) - 
           2 (az - bz) (bx^2 + by^2 + bz^2 - cx^2 - cy^2 - 
              cz^2)) (-2 (ay - by) (ay (bx - cx) + by cx - bx cy + 
              ax (-by + cy)) - 
           2 (az - bz) (az (bx - cx) + bz cx - bx cz + 
              ax (-bz + cz)))))/((az (by - cy) + bz cy - by cz + 
        ay (-bz + cz)) (-4 (az (bx - cx) + bz cx - bx cz + 
           ax (-bz + cz)) (-2 (ay - by) (ay (bx - cx) + by cx - 
              bx cy + ax (-by + cy)) - 
           
           2 (az - bz) (az (bx - cx) + bz cx - bx cz + 
              ax (-bz + cz))) + 
        4 (az (by - cy) + bz cy - by cz + 
           ay (-bz + cz)) (2 (ax - bx) (-by cx + ay (-bx + cx) + 
              ax (by - cy) + bx cy) + 
           2 (az - bz) (az (by - cy) + bz cy - by cz + 
              ay (-bz + cz)))))}

I have a notebook on it but it uses the GrassmannCalculus and Presentations
Applications. Presentations has Circle3D and Disk3D primitives. The notebook
also calculates and displays random cases.


David Park
djmpark@comcast.net 
http://home.comcast.net/~djmpark/index.html 



From: Ste[hen Gray [mailto:stevebg@roadrunner.com] 

I'm looking for a neat formula to find the center of a circle in 3D through
3 points. I also need a good way to display it, preferably thickened so I
can show several and see whether they are linked, etc. To my surprise I did
not find anything on the Wolfram sites about these problems. (I have
Mathematica 7, if that matters.)

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Re: Cirlce in 3D? "djmpark" <djmpark@comcast.net> - 2014-05-13 02:26 +0000

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