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<title>Castle Game Engine: CastleSphereSampling</title>
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<h2>Castle Game Engine</h2><p><a href="introduction.html" class="navigation">Introduction</a></p><p><a href="AllUnits.html" class="navigation">Units</a></p><p><a href="ClassHierarchy.html" class="navigation">Class Hierarchy</a></p><p><a href="AllClasses.html" class="navigation">Classes, Interfaces, Objects and Records</a></p><p><a href="AllTypes.html" class="navigation">Types</a></p><p><a href="AllVariables.html" class="navigation">Variables</a></p><p><a href="AllConstants.html" class="navigation">Constants</a></p><p><a href="AllFunctions.html" class="navigation">Functions and Procedures</a></p><p><a href="AllIdentifiers.html" class="navigation">Identifiers</a></p></td><td class="content">
<h1 class="unit">Unit CastleSphereSampling</h1>
<table class="sections wide_list">
<tr>
<td><a class="section" href="#PasDoc-Description">Description</a></td><td><a class="section" href="#PasDoc-Uses">Uses</a></td><td>Classes, Interfaces, Objects and Records</td><td><a class="section" href="#PasDoc-FuncsProcs">Functions and Procedures</a></td><td>Types</td><td>Constants</td><td>Variables</td></tr></table>
<a name="PasDoc-Description"></a><h2 class="description">Description</h2>
<p>
Random sampling of points (directions) on a sphere and hemisphere. Useful e.g. for ray-tracers.
<p>Most of the implementation based on "Global Illumination Compendium" IV.B [<a href="http://www.cs.kuleuven.ac.be/~phil/GI/">http://www.cs.kuleuven.ac.be/˜phil/GI/</a>]. See also images there, they help to illustrate the meaning of some functions here.
<p>We use two ways to represent points on hemisphere:
<p></p>
<ol class="paragraph_spacing">
<li value="1"><p>Functions <i>without "XYZ" suffix</i> return vector of 2 floats = two angles, called phi and theta (in this order). Phi (in [0, 2*Pi]) is an angle from some chosen meridian. Theta (in [0, Pi/2] for hemisphere and [0, Pi] for sphere) is an angle from chosen vector pointing outward from the (hemi)sphere. See images in "Global Illumination Compendium", they are probably much easier to understand than this definition.</p></li>
<li value="2"><p>Functions <i>with "XYZ" suffix</i> return 3D point x, y, z.
<p></p>
<ul class="paragraph_spacing">
<li><p>(0, 0, 0) is the center of (hemi)sphere,</p></li>
<li><p>(0, 0, 1) is the chosen outward vector (i.e. Theta = 0 there),</p></li>
<li><p>(1, 0, 0) is the direction where Phi = 0 and Theta = Pi/2,</p></li>
<li><p>(0, 1, 0) is the direction where Phi = Pi/2 and Theta = Pi/2.</p></li>
</ul>
<p>
<p>This is matching conventions in "Global Illumination Compendium", see there (point 21).</p></li>
</ol>
<p>
<p>Functions with Density <> Const return PdfValue for returned point, i.e. for density p(Theta) it's PfdValue = p(Result[1]). These functions try to calculate PdfValue smartly (often calculating PfdValue and calculating Result uses the same intermediate calculation, so we can save some computation). PdfValue is needed for importance sampling.</p>
<a name="PasDoc-Uses"></a><h2 class="uses">Uses</h2>
<ul class="useslist"><li><a href="CastleVectors.html">CastleVectors</a></li><li><a href="CastleUtils.html">CastleUtils</a></li></ul><h2 class="overview">Overview</h2>
<a name="PasDoc-FuncsProcs"></a><h3 class="summary">Functions and Procedures</h3>
<table class="summary wide_list">
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<td class="itemcode"><code>function <b><a href="CastleSphereSampling.html#PhiThetaToXYZ">PhiThetaToXYZ</a></b>(const PhiTheta: <a href="CastleVectors.html#TVector2Single">TVector2Single</a>; const SphereRadius: Single) :<a href="CastleVectors.html#TVector3Single">TVector3Single</a>; overload;</code></td>
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<td class="itemcode"><code>function <b><a href="CastleSphereSampling.html#PhiThetaToXYZ">PhiThetaToXYZ</a></b>(const PhiTheta: <a href="CastleVectors.html#TVector2Single">TVector2Single</a>; const SphereTheta0: <a href="CastleVectors.html#TVector3Single">TVector3Single</a>): <a href="CastleVectors.html#TVector3Single">TVector3Single</a>; overload;</code></td>
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<td class="itemcode"><code>function <b><a href="CastleSphereSampling.html#XYZToPhiTheta">XYZToPhiTheta</a></b>(const XYZ: <a href="CastleVectors.html#TVector3Single">TVector3Single</a>): <a href="CastleVectors.html#TVector2Single">TVector2Single</a>;</code></td>
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<tr class="list2">
<td class="itemcode"><code>function <b><a href="CastleSphereSampling.html#RandomHemispherePointConst">RandomHemispherePointConst</a></b>: <a href="CastleVectors.html#TVector2Single">TVector2Single</a>;</code></td>
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<td class="itemcode"><code>function <b><a href="CastleSphereSampling.html#RandomHemispherePointConstXYZ">RandomHemispherePointConstXYZ</a></b>: <a href="CastleVectors.html#TVector3Single">TVector3Single</a>;</code></td>
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<tr class="list2">
<td class="itemcode"><code>function <b><a href="CastleSphereSampling.html#RandomHemispherePointCosTheta">RandomHemispherePointCosTheta</a></b>( out PdfValue: Single): <a href="CastleVectors.html#TVector2Single">TVector2Single</a>;</code></td>
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<tr class="list">
<td class="itemcode"><code>function <b><a href="CastleSphereSampling.html#RandomHemispherePointCosThetaXYZ">RandomHemispherePointCosThetaXYZ</a></b>( out PdfValue: Single): <a href="CastleVectors.html#TVector3Single">TVector3Single</a>;</code></td>
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<td class="itemcode"><code>function <b><a href="CastleSphereSampling.html#RandomHemispherePointCosThetaExp">RandomHemispherePointCosThetaExp</a></b>(const n: Single; out PdfValue: Single): <a href="CastleVectors.html#TVector2Single">TVector2Single</a>;</code></td>
</tr>
<tr class="list">
<td class="itemcode"><code>function <b><a href="CastleSphereSampling.html#RandomHemispherePointCosThetaExpXYZ">RandomHemispherePointCosThetaExpXYZ</a></b>(const n: Single; out PdfValue: Single): <a href="CastleVectors.html#TVector3Single">TVector3Single</a>;</code></td>
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</table>
<h2 class="description">Description</h2>
<h3 class="detail">Functions and Procedures</h3>
<table class="detail wide_list">
<tr class="list">
<td class="itemcode"><a name="PhiThetaToXYZ"></a><code>function <b>PhiThetaToXYZ</b>(const PhiTheta: <a href="CastleVectors.html#TVector2Single">TVector2Single</a>; const SphereRadius: Single) :<a href="CastleVectors.html#TVector3Single">TVector3Single</a>; overload;</code></td>
</tr>
<tr><td colspan="1">
<p>
Convert from PhiTheta representation of (hemi)sphere direction to XYZ representation.
<p>See the beginning of this unit's documentation, <a class="normal" href="CastleSphereSampling.html">CastleSphereSampling</a>, for more precise description of XYZ representation.</p>
</td></tr>
</table>
<table class="detail wide_list">
<tr class="list">
<td class="itemcode"><a name="PhiThetaToXYZ"></a><code>function <b>PhiThetaToXYZ</b>(const PhiTheta: <a href="CastleVectors.html#TVector2Single">TVector2Single</a>; const SphereTheta0: <a href="CastleVectors.html#TVector3Single">TVector3Single</a>): <a href="CastleVectors.html#TVector3Single">TVector3Single</a>; overload;</code></td>
</tr>
<tr><td colspan="1">
<p>
Convert from PhiTheta representation of (hemi)sphere direction to XYZ representation.
<p>This is the more advanced version where you can freely specify which vector is the "main outside (hemi)sphere vector", SphereTheta0. Points with Theta = 0 are exactly on SphereTheta0. It is <i>undefined</i> where point like (Phi = 0, Theta = Pi/2) (or any other point with Theta <> 0) will be placed, i.e. it's not defined where's the "chosen meridian" for Phi = 0. However <i>it's defined that this meridian will be determined only by SphereTheta0</i>, and this is usually sufficient (since this makes sure that sampling and then converting to XYZ multiple points with the same SphereTheta0 will preserve sampled density).
<p>Note that the length of SphereTheta0 determines also the sphere radius.</p>
</td></tr>
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<table class="detail wide_list">
<tr class="list">
<td class="itemcode"><a name="XYZToPhiTheta"></a><code>function <b>XYZToPhiTheta</b>(const XYZ: <a href="CastleVectors.html#TVector3Single">TVector3Single</a>): <a href="CastleVectors.html#TVector2Single">TVector2Single</a>;</code></td>
</tr>
<tr><td colspan="1">
<p>
Convert from XYZ representation of (hemi)sphere direction to PhiTheta.</p>
</td></tr>
</table>
<table class="detail wide_list">
<tr class="list">
<td class="itemcode"><a name="RandomHemispherePointConst"></a><code>function <b>RandomHemispherePointConst</b>: <a href="CastleVectors.html#TVector2Single">TVector2Single</a>;</code></td>
</tr>
<tr><td colspan="1">
<p>
Random point (direction) on unit hemisphere, sampled with constant density (p(Theta) = 1/2*Pi). </p>
</td></tr>
</table>
<table class="detail wide_list">
<tr class="list">
<td class="itemcode"><a name="RandomHemispherePointConstXYZ"></a><code>function <b>RandomHemispherePointConstXYZ</b>: <a href="CastleVectors.html#TVector3Single">TVector3Single</a>;</code></td>
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<tr><td colspan="1">
</td></tr>
</table>
<table class="detail wide_list">
<tr class="list">
<td class="itemcode"><a name="RandomHemispherePointCosTheta"></a><code>function <b>RandomHemispherePointCosTheta</b>( out PdfValue: Single): <a href="CastleVectors.html#TVector2Single">TVector2Single</a>;</code></td>
</tr>
<tr><td colspan="1">
<p>
Random point (direction) on unit hemisphere, sampled with density p(Theta) = cos(Theta)/Pi. </p>
</td></tr>
</table>
<table class="detail wide_list">
<tr class="list">
<td class="itemcode"><a name="RandomHemispherePointCosThetaXYZ"></a><code>function <b>RandomHemispherePointCosThetaXYZ</b>( out PdfValue: Single): <a href="CastleVectors.html#TVector3Single">TVector3Single</a>;</code></td>
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<tr><td colspan="1">
</td></tr>
</table>
<table class="detail wide_list">
<tr class="list">
<td class="itemcode"><a name="RandomHemispherePointCosThetaExp"></a><code>function <b>RandomHemispherePointCosThetaExp</b>(const n: Single; out PdfValue: Single): <a href="CastleVectors.html#TVector2Single">TVector2Single</a>;</code></td>
</tr>
<tr><td colspan="1">
<p>
Random point (direction) on unit hemisphere, sampled with density p(Theta) = (n+1) * (cos(Theta))ˆn / 2*Pi. </p>
</td></tr>
</table>
<table class="detail wide_list">
<tr class="list">
<td class="itemcode"><a name="RandomHemispherePointCosThetaExpXYZ"></a><code>function <b>RandomHemispherePointCosThetaExpXYZ</b>(const n: Single; out PdfValue: Single): <a href="CastleVectors.html#TVector3Single">TVector3Single</a>;</code></td>
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