2d:null 2e:["$","title",null,{"children":"Lighting & Shading III: Light Models | Introduction to Computer Graphics MSState | Observable"}] 33:Tc1f,light_point_demo = { const start = Math.round(0.75 * Math.round(width / 12)); const end = Math.round(1.0 * Math.round(width / 12)); const angles = [0, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330]; const radius = Math.round(0.065 * Math.round(width / 12)); const lerp = x => start * (1 - (x % 10) / 10) + end * (x % 10) / 10; const fig_svg = svg``; const fig = html`

${fig_svg}

The incoming flux density (irradiance ${tex`E`} or energy flux per unit area) decreases as we move away from the point light.

`; return fig; }34:T83f,light_direct_demo = { const start = Math.round(0.75 * Math.round(width / 6)); const end = Math.round(1.0 * Math.round(width / 6)); const levels = Array.from({length: 13}, (x, i) => i * width / 3 / 12); const radius = Math.round(0.065 * Math.round(width / 12)); const lerp = x => start * (1 - (x % 10) / 10) + end * (x % 10) / 10; const fig_svg = svg``; const fig = html`

${fig_svg}

The incoming flux density (the irradiance ${tex`E`}) does not change away from a directional light.

`; return fig; }35:Te77,light_spot_demo = { const start = Math.round(0.875 * Math.round(width / 6)); const end = Math.round(1.125 * Math.round(width / 6)); const light_r = 0.125 * Math.round(width / 6); const levels = Array.from({length: 13}, (x, i) => i * width / 3 / 12); const angles = Array.from({length: 13}, (x, i) => (6 - i) * Math.PI / 12); const radius = Math.round(0.065 * Math.round(width / 12)); const lerp = x => start * (1 - (x % 10) / 10) + end * (x % 10) / 10; const fig_svg = svg``; const fig = html`

${fig_svg}

The incoming flux density (the irradiance ${tex`E`}) decreases as the spot light increases; it is also attenuated by distance.

`; return fig; }36:T6f2,Since lights provide the only source of energy in the scene, setting their intensity is important. There are two approaches: - *Physically Realizable*: There are several methods here, but they tend to be variants of spectral rendering. A set of power for different wavelengths for the light are provided, and propagated into the scene (or, alternatively, sampled during tracing). RGB inputs can be given, but these are then converted based upon assumptions about the base scene illumination. These lights will base their radiance on realizable ${tex`\Phi`} and ${tex`E`} values (or as their photometric values for visible light). - *Winging It*: Light values are set in RGB based upon the scene designers intent. Pure physical realizability is not the goal, thought the images should not violate too many energy rules (unless that is the intent). These lights will used individual set ${tex`k`} values for each light. In both cases, we have to distinguish between the "colors" of the materials (the BRDFs) and those of the lights (the radiant power): - BRDFs represent the ability of the surface to scatter light as a ratio of the incoming irradiance and the radiant exitance over the surface. As such, the BRDFs should generally be in the range of ${tex`[0,1]`} so as not to emit more light than provided. - The power of the light cannot be negative, but otherwise is unbounded (in the range ${tex`[0,\infty)`}). Light values are generally greater than one (sometimes much) as they will be attenuated by each bounce in the scene. Alternatively, the light's energy may be specified by a attenuation (between 0 and 1 for each RGB or other component) and then an color intensity (in radiometric, photometric, or arbitrary units the "winging it" approach) can be given.37:[] 31:{"id":"6a3a35cb203e4ea1","slug":"lighting-shading-iii-light-models","title":"Lighting & Shading III: Light Models","thumbnail":"c4f1aeb9b29a655c9bd8c035abd8d45d20ff7d1e8e6edfcd3a3270acb0b51403","access_level":"public","create_time":"$D2024-11-08T16:32:04.471Z","update_time":"$D2025-08-14T16:16:02.992Z","view_time":null,"first_public_time":"$D2024-11-10T05:01:52.429Z","has_unpublished_edits":null,"fork":false,"stars":1,"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":345,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"e83a97795d4ad3a2","login":"infowantstobeseen","name":"T.J. 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We will introduce lights from the least to most complexity.\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":27,"value":"toc({\n headers: \"h2,h3,h4\",\n title: null\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"## Point Lights","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":33,"value":"A point light is pretty simple: At an infinitely small point in space, light just appears and emanates in all direction. The light has a given light flux ${tex`\\Phi`} which is \"constant\" over its infinitesimal surface area, so its irradiance ${tex`E=\\Phi`} is effectively its flux. Looking at any particular direction gives us a constant emitted radiance; due to integration factors, ${tex`L_e=\\frac{\\Phi}{4\\pi}`}.\n\nMost of our objects are not located where the light is. Thus, the projected irradiance decreases—the light disperses more as we get further from the point source. The *total* power is still the same, but the amount over a small area (and latter solid angle) decreases as ${tex`\\frac{1}{r^2}`} where ${tex`r`} is the distance from the light. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":30,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":52,"value":"Inputs.text({label: html`Irradiance ${tex`E`} (${tex`\\mathrm{\\frac{W}{m^2}}`})`, disabled: true, value: 50 / 4 / Math.PI / (shell*shell)})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":76,"value":"Returning to our materials, the exitance radiance from a surface is based upon the fractional part of the energy density projected onto the surface. For a point light, the incident radiance before projection is","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":87,"value":"tex.block`L_i\\left(\\mathrm{p}, \\omega_i\\right) = \\frac{\\Phi}{4\\pi r^2}=\\frac{1}{r^2}k`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":94,"value":"where ${tex`k`} can be used to set the color directly if the power ${tex`\\Phi`} spectrum is unknown or unimportant (many renderers just set ${tex`k`}, usually called the light intensity/color, directly without concern for the radiometry).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":9,"value":"## Directional Lights","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":98,"value":"Directional lights are assumed to be at a sufficient enough distance—and are sufficiently powerful—that their radiant exitance is constant along the light beam; light sources like the Sun or Moon are good examples. Directional lights are defined by the *light direction* ${tex`\\omega_l`}; as is our convention, this direction points *towards* the light and *away* from the surface.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":106,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":111,"value":"Inputs.text({label: html`Irradiance ${tex`E`} (${tex`\\mathrm{\\frac{W}{m^2}}`})`, disabled: true, value: `${50}`})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":117,"value":"Since the energy flux is constant, the incoming radiance (before projection) is also constant:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":119,"value":"tex.block`L_i\\left(\\mathrm{p}, \\omega_i\\right) = \\Phi=k`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":122,"value":"Again, we can use a physically motivated power or a empirically-based light intensity.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":14,"value":"## Spotlights","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":124,"value":"Spotlights combine aspects of point and directional lights: They emit from a specific location, but only in a small area around the point. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":126,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":162,"value":"viewof spot_angle = Inputs.range([15, 180], {label: html`Spot Angle (${tex`\\degree`})`, step: 15, value: 45})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":167,"value":"Inputs.text({label: html`Irradiance ${tex`E`} (${tex`\\mathrm{\\frac{W}{m^2}}`})`, disabled: true, value: `${50 / 2 / Math.PI / shell / shell / (1 - Math.cos(spot_angle / 360 * Math.PI))}`})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":256,"value":"Note how the irradiance *increases* as the spotlight angle *decreases*: because we are holding the total power fixed, we are focusing more power in a smaller surface area, thus increasing the radiant flux, irradiance, and radiance. That surface area is the spherical sector defined by our spot light angle ${tex`\\alpha`} (the angle between the two edges of the light cone):","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":264,"value":"tex.block`E = \\frac{\\Phi}{2\\pi r^2(1 - \\cos \\frac{\\alpha}{2})}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":273,"value":"Note that this irradiance is attenuated by distance as well (since the spherical sector size increases with the distance). To find the incoming radiance ${tex`L_i`}, we divide by the solid angle the spot light subtends (before projection):","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":280,"value":"tex.block`L_i\\left(\\mathrm{p}, \\omega_i\\right)=\\begin{cases}\n\\frac{\\Phi}{r^2},& -\\omega_s \\cdot \\omega_i \\geq \\cos \\frac{\\alpha}{2}\\\\\n0,& \\text{otherwise}\n\\end{cases}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":291,"value":"where ${tex`\\omega_s`} is the direction the spotlight faces (pointing away from the spotlight, and thus in visible locations, pointing towards our surface; this is the reverse of our convention).\n\nNote, this formulation has a sharp fall-off of the light; even after the projection, the intensity goes quickly from positive to zero. An adjustment based upon interpolating from light-cone edge to the middle is a common compromise.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":298,"value":"tex.block`L_i\\left(\\mathrm{p}, \\omega_i\\right)=\\begin{cases}\n\\frac{\\Phi}{r^2}a\\left(-\\omega_s \\cdot \\omega_i\\right)=\\frac{k}{r^2}a\\left(-\\omega_s \\cdot \\omega_i\\right),& -\\omega_s \\cdot \\omega_i \\geq \\cos \\frac{\\alpha}{2}\\\\\n0,& \\text{otherwise}\n\\end{cases}\\\\\na\\left(t\\right) = \\frac{t - \\cos\\frac{\\alpha}{2}}{1 - \\cos{\\frac{\\alpha}{2}}}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":16,"value":"## Area Lights","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":304,"value":"Area lights are more realistic than point lights and straightforward to implement in a ray tracer. Given a surface, if that surface is an area light, it has a non-zero emissive component. Assuming the radiant flux is uniform over the surface and is not angular dependent, then the emitted radiance in the hemisphere above the object is also constant:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":306,"value":"tex.block`L_e\\left(\\mathrm{p}, \\omega_i\\right) = \\frac{\\Phi}{2\\pi A}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":314,"value":"where ${tex`A`} is the surface area of the emitting object. This light is then projected on a distant object and disperses based upon that distance; if directly above the object:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":317,"value":"tex.block`L_i\\left(\\mathrm{p}, \\omega_i\\right) = \\frac{\\Phi}{2\\pi A r^2}= \\frac{k}{A r^2}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"## Environmental Lights (Environment Mapping)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":320,"value":"Even in ray tracing, rays are not generally traced to infinite distance. Thus, the light \"outside\" the scene needs some representation. If one assumes no objects outside the region of interest (the view volume), we can just use (one or more) directional light(s) that we test against of the ray leaves the view volume.\n\nHowever, it is often the case that while we render carefully inside of the volume, there do exists objects outside (perhaps at quite a distance) that should contribute light to the scene. A general solution is to represent the \"outside\" as a series of images. For example, one might gather a set of six photographs about a scene of interest, and use those as the \"environment\": if a ray exits in the direction of one of the images, find the corresponding color from the image and use that as the lighting. This approach is called environment mapping.\n\nWe will differ further discussion on environment mapping until we get to textures.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":21,"value":"## Using Lights","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":330,"value":"$36","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":23,"value":"import {toc} from \"@nebrius/indented-toc\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":44,"value":"tick = {\n //return 1;\n let i = 0;\n while (true) {\n mutable old_tick = i;\n yield Promises.tick(100, ++i);\n }\n} ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":57,"value":"shell = Math.floor((tick % 60) / 10)+1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":47,"value":"mutable old_tick = 0","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W37","stdlib":"1","titling":"auto"}} 32:"$31:body"