The need to sqrt the U(0, 1) term makes sense. If we multiply a circle's radius by x, we increase its area by x^2, so if we just sample radius uniformly, we will put in the same number of samples in the much smaller circle as the one that has area that is x^2 times larger.
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If we wanted to extend this sampling to 3D could we sample inside a cube which would give us an equal distribution on a sphere?
The need to sqrt the U(0, 1) term makes sense. If we multiply a circle's radius by x, we increase its area by x^2, so if we just sample radius uniformly, we will put in the same number of samples in the much smaller circle as the one that has area that is x^2 times larger.
If we wanted to extend this sampling to 3D could we sample inside a cube which would give us an equal distribution on a sphere?