Questions tagged [integral]

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The ploygon width parallel to the x axis as a function of the y ordonate?

Considering a polygon with n vertices as input. I need to calculate the integral of the form \[\int_A p(y) dA \] where $p(y)$ is a piecewise polynomial function of $y$. May be if I could find the ...
3
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1answer
89 views

How to compute the following integral over a polygon?

I am implementing an algorithm which requires me to compute the following integral, $$\int_{Poly(P_i)}||P-P_i||^2 dP$$ where $P_i=(X_i, Y_i)$ is a point in 2D and $Poly(P_i)$ is a polygon containing $...
0
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0answers
20 views

invcdf for spherical function

There are recent papers on theme of analytic solution of integrals on 2-sphere over polygonal domains. For importance sampling (sample warping) it is attractive to use the analytic solution to derive ...
3
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1answer
71 views

Adding two fogs

When calculating fog along the ray we have two main part- transmittance and scattering. What happens when we have two different fogs? With different extinction and color? Both extinction and color ...
8
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2answers
774 views

Why random monte carlo sampling instead of uniform sampling?

Why is it that it's so common to use monte carlo randomized sample locations, instead of uniform sampling? I'm assuming that taking randomized samples gives some benefit but I don't know what they ...
11
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2answers
2k views

In a physically based BRDF, what vector should be used to compute the Fresnel coefficient?

The well known Schlick approximation of the Fresnel coefficient gives the equation: $F=F_0+(1 - F_0)(1 - cos(\theta))^5$ And $cos(\theta)$ is equal to the dot product of the surface normal vector ...
4
votes
1answer
103 views

Why does the integral of NDF over a solid angle equals the area where micronormals belong to that angle?

In Understanding the Masking-Shadowing Function in Microfacet-Based BRDFs, Section 2.2, Mr Eric Heitz defines the distribution of normals as: And then, he goes on with the following assertion: I ...