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relation between differential solid angle and surface area

Posted: Sat Feb 27, 2016 8:43 pm
by shocker_0x15

We can change the measure of PDF to another one by using Jacobian determinant.
For example, the PDF of IBL explicit sampling with respect to differential solid angle subtended to the origin is:
p(u, v) / (2 * pi^2 * sin(theta))
This is derived by dividing the original PDF with respect to texture space by the Jacobian determinant for changing variables in that space to spherial coordinate.
(By the way, I want MathJax support to be back.)

I want to know how to derive the relation between differential solid angle and differential surface area.
That is well known as:
dw = cos(theta) / ||x - x'||^2 * dA
It is intuitively easy by looking the figure depicting the relation between them, but I want to know whether we can derive this using Jacobian determinant.