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POFast uniformly distributed random points on the surface of a unit hemisphere
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copied!<p>I am trying to generate uniform random points on the surface of a unit sphere for a Monte Carlo ray tracing program. When I say uniform I mean the points are uniformly distributed with respect to surface area. My current methodology is to calculate uniform random points on a hemisphere pointing in the positive z axis and base in the x-y plane.</p> <p>The random point on the hemisphere represents the direction of emission of thermal radiation for a diffuse grey emitter.</p> <p><strong>I achieve the correct result when I use the following calculation :</strong></p> <p><strong>Note : dsfmt* is will return a random number between 0 and 1.</strong></p> <pre><code>azimuthal = 2*PI*dsfmt_genrand_close_open(&dsfmtt); zenith = asin(sqrt(dsfmt_genrand_close_open(&dsfmtt))); // Calculate the cartesian point osRay.c._x = sin(zenith)*cos(azimuthal); osRay.c._y = sin(zenith)*sin(azimuthal); osRay.c._z = cos(zenith); </code></pre> <p>However this is quite slow and profiling suggests that it takes up a large proportion of run time. Therefore I sought out some alternative methods:</p> <p><strong>The Marsaglia 1972 rejection method</strong></p> <pre><code>do { x1 = 2.0*dsfmt_genrand_open_open(&dsfmtt)-1.0; x2 = 2.0*dsfmt_genrand_open_open(&dsfmtt)-1.0; S = x1*x1 + x2*x2; } while(S > 1.0f); osRay.c._x = 2.0*x1*sqrt(1.0-S); osRay.c._y = 2.0*x2*sqrt(1.0-S); osRay.c._z = abs(1.0-2.0*S); </code></pre> <p><strong>Analytical cartesian coordinate calculation</strong></p> <pre><code>azimuthal = 2*PI*dsfmt_genrand_close_open(&dsfmtt); u = 2*dsfmt_genrand_close_open(&dsfmtt) -1; w = sqrt(1-u*u); osRay.c._x = w*cos(azimuthal); osRay.c._y = w*sin(azimuthal); osRay.c._z = abs(u); </code></pre> <p>While these last two methods run serval times faster than the first, when I use them I get results which indicate that they are not generating uniform random points on the surface of a sphere but rather are giving a distribution which favours the equator.</p> <p>Additionally the last two methods give identical final results however I am certain that they are incorrect as I am comparing against an analytical solution.</p> <p>Every reference I have found indicates that these methods do produce uniform distributions however I do not achieve the correct result.</p> <p>Is there an error in my implementation or have I missed a fundamental idea in the second and third methods?</p>

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