2b:null 2c:["$","title",null,{"children":"How Do I Sample...? | Introduction to Computer Graphics MSState | Observable"}] 31:[] 2f:{"id":"5033dadded1358cd","slug":"how-do-i-sample","title":"How Do I Sample...?","thumbnail":"133840e068349a6cf5f26863f2190e24f8789737a432211ff744a0ff9c0229b7","access_level":"public","create_time":"$D2026-03-26T19:24:49.227Z","update_time":"$D2026-03-30T15:19:50.010Z","view_time":null,"first_public_time":"$D2026-03-30T11:52:52.366Z","fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"7c0dab8b587e2ed2","login":"cse4413msstate","name":"Introduction to Computer Graphics MSState","avatar_url":"https://avatars-next.observableusercontent.com/343b1e48cbb2480c08063f107b4631944c5bb97a5258576dd939a5be852f30bc","tier":"pro_2024"},"version":571,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"e83a97795d4ad3a2","login":"infowantstobeseen","name":"T.J. Jankun-Kelly","avatar_url":"https://avatars-next.observableusercontent.com/df9d7f817f9ad92a763b56ee2e891ca0716a6c2f6c398811547f89f0448fa181","tier":null}],"members":[{"type":"individual","id":"e83a97795d4ad3a2","login":"infowantstobeseen","name":"T.J. Jankun-Kelly","avatar_url":"https://avatars-next.observableusercontent.com/df9d7f817f9ad92a763b56ee2e891ca0716a6c2f6c398811547f89f0448fa181","tier":null,"role":"editor"},{"type":"team","id":"7c0dab8b587e2ed2","login":"cse4413msstate","name":"Introduction to Computer Graphics MSState","avatar_url":"https://avatars-next.observableusercontent.com/343b1e48cbb2480c08063f107b4631944c5bb97a5258576dd939a5be852f30bc","tier":"pro_2024","role":"viewer"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":1,"collection":{"id":"16798e6b406d73db","slug":"ray-tracing-notes","title":"Ray Tracing Notes","owner":{"id":"7c0dab8b587e2ed2","login":"cse4413msstate","name":"Introduction to Computer Graphics MSState"}},"type":"notebook","body":{"title":"How Do I Sample...?","cells":[{"id":0,"value":"# How Do I Sample...?","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":5,"value":"Herein is a list of different sampling methods for objects & distributions often found in rendering. Each provides a method to sample the object/distribution which also provides the probability associated with picking that sample. ${tex`\\xi_0, \\dots, \\xi_n`} are the uniform samples between ${tex`0\\dots 1`} needed for the sampling.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":147,"value":"

Towards the ${tex`\\mathrm{\\mathbf n}`}orth pole

The coordinate system used here is based upon the common canonical graphic system: By convention, the pole of the system ${tex`\\mathrm{\\mathbf{\\hat{z}}}`} is aligned with the surface normal ${tex`\\mathrm{\\mathbf{n}}`} with the ${tex`\\mathrm{\\mathbf{\\hat{x}}}`} and ${tex`\\mathrm{\\mathbf{\\hat{y}}}`} axes in the plane orthogonal. Spherical coordinates are used when discussing spherical systems. Because of this, given any direction ${tex`\\omega`}:\n

\n \n${tex.block`\\begin{aligned}\n \\theta &= \\omega_z\\\\\n \\varphi & = \\arctan \\frac{\\omega_y}{\\omega_x}\\\\\n \\cos \\omega &= \\mathrm{\\mathbf n} \\cdot \\omega = \\omega_z\\\\\n \\sin \\omega &= \\sqrt{1 - \\left(\\mathrm{\\mathbf n} \\cdot \\omega\\right)^2}\\\\\n \\tan \\omega &= \\frac{\\cos \\omega}{\\sin \\omega}\\\\\n\\end{aligned}`}\n\n

All vectors/directions are assumed normalized.

","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":217,"value":"toc({\n headers: \"h2,h3\",\n title: null\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":12,"value":"## Directions (${tex`\\omega`})\n\nThese describe methods for sampling directions/solid angles. Probabilities are with respect to solid angle unless otherwise mentioned.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":38,"value":"### Direction, Dirac-delta Function","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":75,"value":"There is only one direction that is valid for a Dirac-delta distribution ${tex`\\delta\\left(\\omega'-\\omega\\right)`} (in this case ${tex`\\omega'`}), thus, if sampling, we return ${tex`\\omega'`} as the sample with probability 1.\n\nNote, the PDF of sampling any direction is always 0 for the Dirac-delta function (the infinite amount of misses obliterates the chance of selecting ${tex`\\omega'`} at random).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":9,"value":"### Direction, Uniform Unit Hemisphere","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":82,"value":"To sample on a hemisphere, we first sample the appropriate azimuthal (${tex`\\theta`}) and polar angles (${tex`\\varphi`}) first:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":87,"value":"$${tex.block`\\begin{aligned}\n\\theta &= \\cos^{-1} \\xi_0\\\\\n\\varphi &= 2\\pi\\xi_1\n\\end{aligned}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":92,"value":"We are following the convention of substituting ${tex`\\xi_0`} for its difference ${tex`1-\\xi_0`} if following the common derivation.\n\nUsing the conversion from spherical to Cartesian coordinates and some trigonometry, the sample point is then","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":99,"value":"$${tex.block`\\begin{aligned}\n \\omega_x &= \\cos\\left(2\\pi\\xi_1\\right)\\sqrt{1-\\xi_0^2}\\\\\n \\omega_y &= \\sin\\left(2\\pi\\xi_1\\right)\\sqrt{1-\\xi_0^2}\\\\\n \\omega_z &= \\xi_0\\\\\n\\end{aligned}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":105,"value":"The chance of selecting this or any other direction uniformly is ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":107,"value":"$${tex.block`p\\left(\\omega\\right) = \\frac{1}{2\\pi}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":16,"value":"### Direction, Uniform Sphere","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":113,"value":"Similar to the hemisphere, but over a larger ${tex`\\theta`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":115,"value":"$${tex.block`\\begin{aligned}\n \\omega_x &= 2\\cos\\left(2\\pi\\xi_1\\right)\\sqrt{\\xi\\left(1-\\xi_0\\right)}\\\\\n \\omega_y &= 2\\sin\\left(2\\pi\\xi_1\\right)\\sqrt{\\xi\\left(1-\\xi_0\\right)}\\\\\n \\omega_z &= 1-2\\xi_0\\\\\n\\end{aligned}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":118,"value":"Similarly, the chance of sampling and direction is constant","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":120,"value":"$${tex.block`p\\left(\\omega\\right) = \\frac{1}{4\\pi}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":7,"value":"### Direction, Cosine-weighted Unit Hemisphere","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":124,"value":"With a cosine-weighted distribution, we bias directions towards the zenith of the hemisphere. Due to Lambert's law, radiance falls off proportionally to ${tex`\\cos \\theta`}, thus, importance sampling with a cosine-weighted distribution will generally have less variance than a uniform one.\n\nTo sample, we first sample a disc uniformly ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":151,"value":"$${tex.block`\\begin{aligned}\n \\xi'_i &= 2\\xi_i - 0.5\\\\\n r &= \\min \\left(|\\xi'_0|, |\\xi'_1|\\right)\\\\\n \\theta &= \\begin{cases}\n \\frac{1}{4\\pi}\\frac{\\xi'_1}{\\xi'_0} & |\\xi'_0| \\geq |\\xi'_1|\\\\\n \\frac{1}{2\\pi}-\\frac{1}{4\\pi}\\frac{\\xi'_0}{\\xi'_1} & \\text{otherwise}\\\\\n \\end{cases}\n\\end{aligned}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":129,"value":"We then project this point onto the hemisphere","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":163,"value":"tex.block`\\begin{aligned}\n \\omega_x &= r\\cos\\theta\\\\\n \\omega_y &= r\\sin\\theta\\\\\n \\omega_z &= \\sqrt{1-\\omega_x^2-\\omega_y^2}\\\\\n &= \\sqrt{1-r^2}\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":132,"value":"The probability of sampling any of the directions on the cosine-weighted hemisphere is, unsurprisingly, cosine-weighted","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":134,"value":"$${tex.block`p\\left(\\omega\\right) = \\frac{\\omega \\cdot \\hat{\\mathbf{z}}}{\\pi} = \\frac{\\cos \\theta}{\\pi}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":24,"value":"### Direction, EON Clipped Linearly Transformed Cosine\n\nThe EON rough microfacet model for diffuse materials has a diffuse lobe with two parts: a uniform hemispherical part and a clipped linearly transformed cosine lobe that captures the glancing angle properties of the distribution.\n\nTo sample this distribution, we first determine we are sampling the uniform lobe or the glancing one. The probability of sampling the uniform lobe ${tex`p_{lu}`} with roughness ${tex`r`} given outgoing direction ${tex`\\omega_o`} is","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":189,"value":"tex.block`\\begin{aligned}\n p_{lu}\\left(\\omega_o\\right) &= r^{0.1}\\left(a+\\left(b+\\left(c-d\\mu\\right)\\mu\\right)\\mu\\right)\\\\\n \\mu &= \\mathbf{n}\\cdot \\omega_o = \\omega_{o,z}\\\\\n a &= 0.162925\\\\\n b &= -0.372058\\\\\n c &= 0.538233\\\\\n d &= 0.290822\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":211,"value":"#### Sampling Uniform Lobe\n\nIf we do sample the uniform lobe (${tex`\\xi_0 \\leq p_{lu}\\left(\\omega_o\\right)`}), we sample a uniform hemisphere as above after normalizing the uniform samples to get ${tex`\\omega`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":213,"value":"tex.block`\\begin{aligned}\n \\xi_0 &\\rightarrow \\frac{\\xi_0}{p_{lu}(\\omega_o)}\\\\\n \\xi_1 &\\rightarrow \\xi_1\\\\\n \\omega_x &= \\cos\\left(2\\pi\\xi_1\\right)\\sqrt{1-\\xi_0^2}\\\\\n \\omega_y &= \\sin\\left(2\\pi\\xi_1\\right)\\sqrt{1-\\xi_0^2}\\\\\n \\omega_z &= \\xi_0\\\\\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":238,"value":"#### Sampling Glancing Lobe\nIf we sample from the glancing lobe, we first normalize the uniform samples for this decision","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":242,"value":"tex.block`\\begin{aligned}\n \\xi_0 &\\rightarrow \\frac{\\xi_0-p_{lu}(\\omega_o)}{1-p_{lu}(\\omega_o)}\\\\\n \\xi_1 &\\rightarrow \\xi_1\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":248,"value":"We then sample from the unit disc to find the planar projection of the sampled half-way vector ${tex`\\omega_h`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":251,"value":"tex.block`\\begin{aligned}\n \\omega_{h,x} &= \\sqrt{\\xi_0^2}\\cos 2\\pi\\xi_1\\\\\n \\omega_{h,y} &= \\sqrt{\\xi_0^2}\\cos 2\\pi\\xi_1\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":262,"value":"The glancing lobe's projection consists of a uniform half-disc and a scaled ellipse part with semi-major axis of 1 (to join the half-disc) and semi-minor axis based on the linear transform cosine matrix we will introduce soon. We shift the ${tex`\\mathbf{\\hat{x}}`} coordinate based on the half-way point between the semi-minor axis endpoint and the half-sphere endpoint so it is contained","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":264,"value":"tex.block`\\begin{aligned}\n s &= \\frac{1}{2}\\left(1+\\frac{1}{\\sqrt{1+m_{20}^2}}\\right)\\\\\n \\omega_{h,x} &\\rightarrow -\\operatorname{lerp}\\left(s,\\sqrt{1-\\omega_{h,y}^2}, \\omega_{h,x}\\right)\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":272,"value":"$${tex`m_{20}`} will be defined shortly.\n\nWe finish the half-way vector by projecting the planar point onto the unit hemisphere","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":274,"value":"tex.block`\\omega_{h,z} = \\sqrt{1-\\omega_{h,x}^2-\\omega_{h,y}^2}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":279,"value":"We use the linear cosine transform to generate the final orientation ${tex`\\omega`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":281,"value":"tex.block`\\begin{aligned}\n \\omega &= \\frac{\\mathbf{M}\\omega_h}{\\|\\mathbf{M}\\omega_h\\|}\\\\\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":288,"value":"where ${tex`\\mathbf{M}`} is","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":319,"value":"tex.block` \\mathbf{M} = \\begin{bmatrix}\n m_{00} & 0 & m_{02}\\\\\n 0 & m_{11} & 0 \\\\\n m_{20} & 0 & 1 \\end{bmatrix}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":317,"value":"given","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":290,"value":"tex.block`\\begin{aligned}\n m_{00} &= 1+ r(0.303392 + (-0.518982 + 0.111709\\mu)\\mu + (-0.276266 + 0.335918\\mu)r)\\\\\n m_{02} &= \\frac{-1.16407 + 1.15859\\mu + (0.150815 - 0.150105\\mu)r}{\\mu^3 - 1.43545}r\\\\\n m_{11} &= 1.0 + r(0.20013 + (-0.506373 + 0.261777\\mu)\\mu)\\\\\n m_{20} &= \\frac{0.540852 + (-1.01625 + 0.475392\\mu)\\mu}{-1.074f + (0.0725628 + \\mu)\\mu}r\\\\\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":326,"value":"#### PDF\n\nAs we have two paths to generate the orientation, we consider the probability of generating the direction under both sampling conditions. \n\nThe probability of sampling the uniform lobe is the same for both paths","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":337,"value":"tex.block`p_u\\left(\\omega\\right) = \\frac{1}{\\pi}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":347,"value":"The probability of sampling the chosen half-way vector ${tex`\\omega_h`} when sampling the glancing lobe is ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":349,"value":"tex.block`p_c\\left(\\omega_h\\right) = \\frac{\\mathbf{n}\\cdot\\omega_h}{s\\pi}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":379,"value":"If we are given ${tex`\\omega`} only (if we sampled the uniform lobe), we can use the inverse of ${tex`\\mathbf{M}`} to calculate it","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":381,"value":"tex.block`\\begin{aligned}\n \\omega_h &= \\frac{\\mathbf{M^{-1}}\\omega}{\\|\\mathbf{M^{-1}}\\omega\\|}\\\\\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":358,"value":"We then use the Jacobian ${tex`\\frac{\\partial \\omega_h}{\\partial \\omega}`} to transform from measuring the half-way vector to the final sample ${tex`\\omega`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":363,"value":"tex.block`\\begin{aligned}\n p_c\\left(\\omega\\right) &= p_c\\left(\\omega_h\\right)\\frac{\\partial \\omega_h}{\\partial \\omega}\\\\\n &= p_c\\left(\\omega_h\\right)\\frac{\\mathbf{M}^{-1}}{\\|\\mathbf{M}^{-1}\\omega\\|^3}\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":341,"value":"The overall MIS PDF combines the chances of sampling both lobes with the chances that the glancing lobe would have been sampled","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":343,"value":"tex.block`p(\\omega)=p_{lu}(\\omega_o)p_u(\\omega)+(1-p_{lu}(\\omega_o))p_c(\\omega)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":22,"value":"### Visible Microfacet Normal, Trowbridge-Reitz (GGX) Conductor\n\nUsing the Torrance-Sparrow microfacet model, we first sample the microfacet normal (the half-way vector ${tex`\\omega_h`}) and then transform it to the direction of interest ${tex`\\omega`}. The distribution we sample is the distribution of visible microfacet normals ${tex`D_{\\omega_o}(\\omega_h)`} from ${tex`\\omega_o`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":388,"value":"tex.block`\\begin{aligned}\n D_{\\omega_o}(\\omega_h) &= \\frac{G(\\omega_o)}{\\mathbf{n} \\cdot \\omega_o}D(\\omega_h)\\max(0, \\omega_o \\cdot \\omega_h)\\\\\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":394,"value":"This distribution is based upon the microfacet masking function ${tex`G(\\omega_o)`} and the normal distribution function (NDF) ${tex`D(\\omega_h)`}. If we use the Trowbridge-Reitz NDF for specular microfacets, ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":399,"value":"tex.block`\\begin{aligned}\n D\\left(\\omega_h\\right) &= \\frac{1}{\\pi\\alpha_x\\alpha_y\\cos^4\\theta_h\\left(1+\\tan^2\\theta_h\\left(\\frac{\\cos^2\\varphi_h}{\\alpha^2_x}+\\frac{\\sin^2\\varphi_h}{\\alpha^2_y}\\right)\\right)^2}\\\\\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":406,"value":"in the anisotropic case and ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":409,"value":"tex.block`\\begin{aligned}\n D\\left(\\omega_h\\right) &= \\frac{1}{\\pi\\alpha^2\\cos^4\\theta_h\\left(1+\\frac{\\tan^2\\theta_h}{\\alpha^2}\\right)^2}\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":414,"value":"in the isotropic case where ${tex`\\alpha_x, \\alpha_y, \\alpha`} are roughness values. We will focus on the isotropic case.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":397,"value":"The isotropic masking function is","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":419,"value":"tex.block`\\begin{aligned}G\\left(\\omega\\right)&=\\frac{1}{1+\\Lambda(\\omega)}\\\\\n \\Lambda(\\omega) &= \\frac{\\sqrt{1+\\alpha^2\\tan^2\\theta}-1}{2}\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":428,"value":"To sample this distribution, given an outgoing direction ${tex`\\omega_o`}, we first sample uniformly from a unit disc and deform it based upon the half-way vector's azimuth to represent how the visible normal space is compressed approaching glancing angles. We then project back onto the hemisphere.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":430,"value":"tex.block`\\begin{aligned}\n \\omega_{h,x} &= \\sqrt{\\xi_0^2}\\cos 2\\pi\\xi_1\\\\\n \\omega_{h,y} &= \\sqrt{\\xi_0^2}\\cos 2\\pi\\xi_1\\\\\n h & = \\sqrt{1-\\omega_{h,x}^2}\\\\\n \\omega_{h,y} &\\rightarrow \\operatorname{lerp}\\left(\\frac{1+\\mathbf{n}\\cdot\\omega_{o}}{2},h, \\omega_{h,y}\\right)\\\\\n \\omega_{h,z} &= \\sqrt{1-\\omega_{h,x}^2-\\omega_{h,y}^2}\\\\\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":444,"value":"We then transform ${tex`\\omega_h`} back into undeformed space via an orthnormal basis formed from ${tex`\\omega_o`}, a vector ${tex`\\mathbf{b}_1`} in the plane and ${tex`\\mathbf{b}_2`} orthogonal to both. The final outgoing direction is then found by reflection. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":447,"value":"tex.block`\\begin{aligned}\n \\mathbf{b}_1 &= \\mathbf{n} \\times \\omega_o\\\\\n \\mathbf{b}_2 &= \\omega_o \\times \\mathbf{b}_1\\\\\n \\omega_h &\\rightarrow \\begin{bmatrix}\n \\vert & \\vert & \\vert\\\\\n \\mathbf{b}_1 & \\mathbf{b}_2 & \\omega_o\\\\\n \\vert & \\vert & \\vert\\\\\n \\end{bmatrix}\\omega_h\\\\\n \\omega &= \\operatorname{reflect}(\\omega_o, \\omega_h)\\\\\n &= 2\\left(\\omega_o \\cdot \\omega_h\\right)\\omega_h - \\omega_o\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":457,"value":"The PDF for selecting ${tex`\\omega`} is based upon the PDF for sampling ${tex`\\omega_h`} and the half-vector transform from ${tex`\\omega_h`} to ${tex`\\omega`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":462,"value":"tex.block`\\begin{aligned}\n p\\left(\\omega\\right) &= p\\left(\\omega_h\\right)\\frac{\\partial \\omega_h}{\\partial \\omega}\\\\\n &= \\frac{D_{\\omega_o}(\\omega_h)}{4\\omega_o\\cdot\\omega_h}\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":502,"value":"### Visible Microfacet Normal, Trowbridge-Reitz (GGX) Dielectric\n\nFor dielectrics, we once again use the distribution of visible normals to sample a half-way vector ${tex`\\omega_h`}. We then choose which configuration to sample (reflected or transmitted) and return the appropriate values.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":511,"value":"#### Reflection\n\nThe chance to reflect is based upon the Fresnel effect that determines the reflectance","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":515,"value":"tex.block`p_r\\left(\\omega_o\\right) = F_r\\left(\\omega_o \\cdot \\omega_h, \\frac{\\eta_t}{\\eta_i}\\right)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":520,"value":"where ${tex`\\eta_i, \\eta_t`} are the incident and transmissive indices of refraction respectively. If we choose to reflect (${tex`\\xi_0 \\leq p_r\\left(\\omega_o\\right)`}), we sample as above for the conductor case and correct for the choice of reflection.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":524,"value":"tex.block`\\begin{aligned}\n p\\left(\\omega\\right) &=p_r\\left(\\omega_o\\right) p\\left(\\omega_h\\right)\\frac{\\partial \\omega_h}{\\partial \\omega}\\\\\n &= p_r\\left(\\omega_o\\right)\\frac{D_{\\omega_o}(\\omega_h)}{4\\omega_o\\cdot\\omega_h}\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":530,"value":"Note, when sampling for a dielectric, you will need three uniform samples: ${tex`\\xi_0`} for the choice of whether to reflect or transmit, and ${tex`\\xi_1,\\xi_2`} for the actual ${tex`\\omega_h`} sampling.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":528,"value":"#### Transmission\nIf we are transmitting, then we use Snell's Law to calculate ${tex`\\omega`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":533,"value":"tex.block`\\begin{aligned}\n \\omega &= \\operatorname{refract}(\\omega_o, \\omega_h, \\eta_i, \\eta_t)\\\\\n &= -\\frac{\\eta_i}{\\eta_t}\\omega_o+\\left[\\frac{\\eta_i}{\\eta_t}\\left(\\omega_o \\cdot \\omega_h\\right)-\\cos \\theta_t\\right]\\omega_h\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":540,"value":"where","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":542,"value":"tex.block`\\begin{aligned}\n \\cos\\theta_t &= \\sqrt{1-\\sin^2\\theta_t}\\\\\n \\sin^2\\theta_t &= \\frac{\\eta^2_i}{\\eta^2_t}\\left(1-\\left(\\omega_o\\cdot\\omega_h\\right)^2\\right)\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":548,"value":"The PDF for transmission takes into account the probability of transmitting with the probability of selection ${tex`\\omega`} given ${tex`\\omega_o`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":552,"value":"tex.block`\\begin{aligned}\n p\\left(\\omega\\right) &= \\left(1-p_r(\\omega_o)\\right)p_t\\left(\\omega_h\\right)\\frac{\\partial \\omega_h}{\\partial \\omega}\\\\\n &= \\left(1-p_r(\\omega_o)\\right)\\frac{D_{\\omega_o}(\\omega_h)\\left|\\omega_o\\cdot\\omega_h\\right|}{\\left(\\omega_i\\cdot\\omega_h+\\frac{\\eta_i}{\\eta_t}\\left(\\omega_o\\cdot\\omega_h\\right)\\right)^2}\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":550,"value":"All of this assumes transmission from a less to more denser medium and no internal reflection. In the former case, the indices of refraction and the incident direction/half-way angle must be reversed; it the later, no transmission occurs.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":68,"value":"### Visible Microfacet Normal, Blinn-Phong Specular \n\nThe Torrance-Sparrow model for microfacets is what is used for the Trowbridge-Reitz normal distribution function above. The same sampling process could be used for the specular part of the Blinn-Phong model using the correct NDF and geometric terms.\n\nThe NDF for Blinn-Phong is","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":174,"value":"tex.block`D\\left(\\omega\\right) = \\frac{s+2}{2\\pi}\\left(\\mathbf{n}\\cdot\\omega\\right)^s`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":178,"value":"where ${tex`s`} is the Blinn-Phong glossiness term. We can assume the implicit masking function","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":467,"value":"tex.block`G(\\omega) = \\mathbf{\\omega}\\cdot\\omega_h`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":470,"value":"We sample this distribution the same as above, and the PDF is the same.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":32,"value":"## Surface Points (${tex`\\mathrm{p}`})\n\nThese describe methods for sampling points on a surface. Probabilities are with respect to object's surface area unless otherwise mentioned.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":41,"value":"### Unit Hemisphere\n\nSame as the Direction sampling above (the point ${tex`\\mathrm{p}`} is at the end of the offset ${tex`\\omega`}). You can make this into a hemi-ellipsoid of any size by scaling appropriately.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":45,"value":"### Unit Sphere\n\nSame as the Direction sampling above (the point ${tex`\\mathrm{p}`} is at the end of the offset ${tex`\\omega`}). You can make this into an ellipsoid of any size by scaling appropriately.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":53,"value":"### Unit Disc\n\nWe take care to use an area-balanced distribution after that in PBR. You can make this into an ellipse of any size by scaling appropriately.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":485,"value":"tex.block`\\begin{aligned}\n \\xi'_i &= 2\\xi_i - 0.5\\\\\n r &= \\min \\left(|\\xi'_0|, |\\xi'_1|\\right)\\\\\n \\theta &= \\begin{cases}\n \\frac{1}{4\\pi}\\frac{\\xi'_1}{\\xi'_0} & |\\xi'_0| \\geq |\\xi'_1|\\\\\n \\frac{1}{2\\pi}-\\frac{1}{4\\pi}\\frac{\\xi'_0}{\\xi'_1} & \\text{otherwise}\\\\\n \\end{cases}\\\\\n \\mathrm{p}_x &= r\\cos\\theta\\\\\n \\mathrm{p}_y &= r\\sin\\theta\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":495,"value":"The PDF is","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":497,"value":"tex.block`\\begin{aligned}\n p(\\mathrm{p}) &= \\frac{1}{\\pi}\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":47,"value":"### Unit Square\n\nYou can make this into a square or rectangle of any size by scaling appropriately. To sample","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":476,"value":"tex.block`\\begin{aligned}\n \\mathrm{p}_{x} &= \\xi_0\\\\\n \\mathrm{p}_{y} &=\\xi_1\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":480,"value":"The PDF is","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":482,"value":"tex.block`\\begin{aligned}\n p(\\mathrm{p}) &= 1\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":55,"value":"## Light Points ${tex`\\mathrm{p}`} \n\nThese describe methods for sampling points on a surface; most are based upon a sampling method above. Probabilities are with respect to object's surface area unless otherwise mentioned.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":58,"value":"### Point Light","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":60,"value":"### Directional Light","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":62,"value":"### Area Light","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":64,"value":"### Environmental Light","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":222,"value":"## TOC Generation \n\nGenerates a table of contents of easy navigation.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":219,"value":"import {toc} from \"@nebrius/indented-toc\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W31","stdlib":"1","titling":"auto"}} 30:"$2f:body"