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Recently I read NVidia's Mipmapping_Normal_Maps which says we can used the un-renormalized averaged normal to compute the standard deviation of the angle between averaged normal and sample normals. By the first step, it assumes a Gaussian distribution of the angular deviation and give a figure (sorry but I cannot post an image as a new user, please refer to Figure_2 in that paper).

Then my question is, how the length of averaged normal is represented by a function of Standard Deviation of the angle(original function of Gaussian distribution, red curve in the figure)?

sHeldON
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I believe the answer to your question is equation (1) in the paper. It shows how the averaged normal is equal to the reciprocal of 1 + sigma^2. Sigma is the standard deviation. Sometimes sigma^2 is called the variance.

At any rate, if you know the standard deviation, that's your value for sigma in the equations. Square it to get the variance, sigma^2.

OrangeWombat
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  • Thanks for your reply, but equation(1) in the paper is the approximation of Gaussian distribution, which is green curve in Figure(2), while the original Gaussian distribution is red curve. Sorry my question is not clearly, but what i want to know is the function to the original Gaussian distribution(red curve). – sHeldON Oct 18 '12 at 03:06
  • Ah, I understand. A Gaussian is typically of the form e^(-x^2). In this case the variable is the deviation, sigma. The first two terms of a Taylor expansion would give 1 + sigma^2, in reciprocal of which is the expression given in the paper for averaged normal. Now, there are going to be particulars (e.g., why does this Gaussian asymptotically approach 0.2, rather than 0, as you'd expect) that only the paper's authors can answer - you may want to contact them. See [link](http://mathworld.wolfram.com/NormalDistribution.html) – OrangeWombat Oct 22 '12 at 13:35