3d - Tetrahedron height vertices -

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if given irregular tetrahedron's vertices coordinates a(x1,y1,z1) b(x2,y2,z2) c(x3,y3,z3) d(x4,y4,z4) , need compute 3d coordinate h(x,y,z) of height vertex a. after many google search able find barycentric coordinate not vertex of height. please help.

three points b, c, d define plane a*x+b*y+c*z-d=0, where

a = y2 * (z3 - z4) + y3 * (z4 - z2) + y4 * (z2 - z3) b = z2 * (x3 - x4) + z3 * (x4 - x2) + z4 * (x2 - x3)  c = x2 * (y3 - y4) + x3 * (y4 - y2) + x4 * (y2 - y3)  d = x2 * (y3 * z4 - y4 * z3) + x3 * (y4 * z2 - y2 * z4) + x4 * (y2 * z3 - y3 * z2)

(derived mathworld formula (18)) normal vector plane (a,b,c).

h point - height vertex a, ah vector normal bcd plane, , h lies in plane bcd. h = (x1+k*a, y1+k*b, z1+k*c), , have substitute these coordinates in plane equation, solve k coefficient, calc h point coordinates

substitution: a*(x1+k*a) + b*(y1+k*b)+c*(z1+k*c)-d=0


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