Let a sphere with a unit radius lie on the x-y plane, centered (touching) the point zero. Let a single point be somewhere on the surface of the sphere, but we can only see its projected point on the x-y plane. Given the values x(t) (its distance on the x axis), y(t) (its distance on the y axis), rho(t) (its radius on the x-y plane) and t (the angle of the point from the x axis), what is z(t) (the distance from the x-y plane to the point on the sphere)?
There should be two solutions, one for when a point is on the top half of the sphere and one when its on the bottom half, as both will project to the same place on the x-y plane.
Unknowns: r(t), the actual point P and z(t).
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Since the point is on a unit sphere, its distance from the center will be $1$. And since the equation of the surface of the sphere is $1=x^2+y^2+(z-1)^2,$ the correct solution for $z$ will be $$z=1\pm\sqrt{1-x^2-y^2}.$$