# Solving systems of equations

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I was starting to build a small application that could build a polygon sphere, using a tetrahedron as a base. The idea is to subdivide the triangles into sets of 4 new. When subdivision is done every point should be moved to the surface of the sphere. My problem is this. I need 4 vertices to start out with, and I would like to place each vertice on the unit sphere. I choose to place the first point at {0,1,0}, and the second on in the XY plane where X > 0. I guess I would find the solution by solving the following set of equations. But how do I do that? Placement of the first point X_a = 0 Y_a = 1 Z_a = 0 //Second point on the XY plane where X_b > 0 Z_b = 0 //Every point shuld be placed on the unit sphere eq1: X_a^2+Y_a^2+Z_a^2-1=0 eq2: X_b^2+Y_b^2+Z_b^2-1=0 eq3: X_c^2+Y_c^2+Z_c^2-1=0 eq4: X_d^2+Y_d^2+Z_d^2-1=0 //The distance "d" between every point should be equal. eq5: (X_a-X_b)^2+(Y_a-Y_b)^2+(Z_a-Z_b)^2-d=0 eq6: (X_a-X_c)^2+(Y_a-Y_c)^2+(Z_a-Z_c)^2-d=0 eq7: (X_a-X_d)^2+(Y_a-Y_d)^2+(Z_a-Z_d)^2-d=0 eq8: (X_b-X_c)^2+(Y_b-Y_c)^2+(Z_b-Z_c)^2-d=0 eq9: (X_b-X_d)^2+(Y_b-Y_d)^2+(Z_b-Z_d)^2-d=0 eq10: (X_c-X_d)^2+(Y_c-Y_d)^2+(Z_c-Z_d)^2-d=0 I keep loosing my track. How can I solve the equations?

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One easy way to generate a tetrahedron is place the vertices at non-adjacent corners of a cube, e.g. at

(1, 1, 1)
(1,-1, -1)
(-1, 1, -1)
(-1, -1, 1)

The tetrahedron edges then are diagonals of the cube faces, one edge per face.

This is a cube with side 2, diameter sqrt(6) and so radius sqrt(3), so to put them on a unit sphere divide by sqrt(3).

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