math - Sphere center point and radius from 3 points on the surface -

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is possible find center of sphere , radius 3 points on surface ?

i'm building model segmented brain structure 3 points within structure; head, tail , middle.

thank you,

express center of sphere equidistant 3 given points , coplanar them (assuming 3 given points on great circle).

(x - xa)² + (y - ya)² + (z - za)² = r²     (x - xb)² + (y - yb)² + (z - zb)² = r²     (x - xc)² + (y - yc)² + (z - zc)² = r²      |x  y  z  1| |xa ya za 1| |xb yb zb 1| = 0 |xc yc zc 1|

subtracting first equation second , third, rid of quadratic terms.

(2x - xb - xa)(xb - xa) + (2y - yb - ya)(yb - ya) + (2z - zb - za)(zb - za) = 0 (2x - xc - xa)(xc - xa) + (2y - yc - ya)(yc - ya) + (2z - zc - za)(zc - za) = 0

now have easy linear system of 3 equations in 3 unknowns.

for conciseness can translate 3 points xa=ya=za=0, , equations simplify as

|x  y  z | |xb yb zb| = 0 |xc yc zc|  (2x - xb) xb + (2y - yb) yb + (2z - zb) zb = 0 (2x - xc) xc + (2y - yc) yc + (2z - zc) zc = 0

or

(yb zc - yc zb) x + (zb xc - zc xb) y + (xb yc - xc yb) z = 0        2 xb     x +        2 yb     y +        2 zb     z = xb² + yb² + zb²        2 xc     x +        2 yc     y +        2 zc     z = xc² + yc² + zc²

then, r² = x² + y² + z², , don't forget translate back.


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