31:T509,# Cylindrical Equal-Area Projections The [cylindrical equal-area projection](https://en.wikipedia.org/wiki/Cylindrical_equal-area_projection) is not a single map projection, but a family of projections that share a formulation: ${tex.block`\begin{aligned} x &= (\lambda - \lambda_0)\cos\varphi_0 \\ y &= \frac{\sin \varphi}{\cos\varphi_0} \end{aligned}`} Here ${tex`\lambda`} is the longitude, ${tex`\lambda_0`} is the central meridian, ${tex`\varphi`} is the latitude, and ${tex`\varphi_0`} is the standard parallel. (The central meridian ${tex`\lambda_0`} is most often 0°, centering the map on [Greenwich, England](https://en.wikipedia.org/wiki/Greenwich).) The cylindical equal-area projection projection was first invented by [Johann Heinrich Lambert](https://en.wikipedia.org/wiki/Johann_Heinrich_Lambert) in 1772. Lambert invented many widely-used projections, including the transverse Mercator. Various cartographers have since named cylindrical equal-area projections with specific standard parallels; the [Gall–Peters projection](https://en.wikipedia.org/wiki/Gall–Peters_projection) is the cylindrical equal-area projection with standard parallels 45°N and 45°S, while the [Berhmann projection](https://en.wikipedia.org/wiki/Behrmann_projection) uses 30°N and 30°S.32:[] 2f:{"title":"Cylindrical Equal-Area Projections","cells":[{"id":1,"value":"$31","pinned":false,"hidden":false,"mode":"md"},{"id":118,"value":"viewof parallel = html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"map = {\n const context = DOM.context2d(width, height);\n const path = d3.geoPath(projection, context);\n context.beginPath(), path(graticule), context.strokeStyle = \"#ccc\", context.stroke();\n context.beginPath(), path(land), context.fill();\n context.beginPath(), path({type: \"Sphere\"}), context.strokeStyle = \"#000\", context.stroke();\n return context.canvas;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":248,"value":"The *cylindrical* property means that the meridians (lines of constant longitude) and parallels (lines of constant latitude) are straight horizontal and vertical lines, respectively. In addition, the meridians are equally-spaced. The *equal-area* property means that the relative areas of shapes are preserved; this is a necessary property for [choropleth maps](https://en.wikipedia.org/wiki/Choropleth_map), though there are many other equal-area projections to consider, notably the [Mollweide projection](https://en.wikipedia.org/wiki/Mollweide_projection) for a global map.\n\nThe family of cylindrical equal-area projections is implemented in [d3-geo-projection](https://github.com/d3/d3-geo-projection) as [d3.geoCylindricalEqualArea](https://github.com/d3/d3-geo-projection/blob/master/README.md#geoCylindricalEqualArea); the standard parallels are set using [*projection*.parallel](https://github.com/d3/d3-geo-projection/blob/master/README.md#cylindricalEqualArea_parallel).","pinned":false,"hidden":false,"mode":"md"},{"id":16,"value":"projection = d3.geoCylindricalEqualArea()\n .parallel(parallel)\n .translate([width / 2, height / 2])\n .fitExtent([[0.5, 0.5], [width - 0.5, height - 0.5]], {type: \"Sphere\"})\n .precision(0.1)","pinned":true,"hidden":false,"mode":"ojs"},{"id":172,"value":"The aspect ratio ${tex`R = \\frac{w}{h}`} of the cylindrical equal-area projection can be derived from the chosen ${tex`\\varphi_0`}. Here ${tex`w = |x|`} means the range of possible values of ${tex`x`}, *i.e.*, the width; similarly ${tex`|\\lambda| = 2\\pi`} and ${tex`|\\varphi| = \\pi`}.\n\n${tex.block`\\begin{aligned}\n|x| &= |\\lambda|\\cos\\varphi_0 = 2\\pi\\cos\\varphi_0 \\\\\n|y| &= \\textstyle \\frac{|\\sin\\varphi|}{\\cos\\varphi_0} = \\frac{2}{\\cos\\varphi_0} \\\\\nR &= \\textstyle \\frac{|x|}{|y|} = \\pi(\\cos\\varphi_0)^2\n\\end{aligned}`}","pinned":false,"hidden":false,"mode":"md"},{"id":89,"value":"height = Math.round(width / (Math.PI * Math.cos(parallel * Math.PI / 180) ** 2))","pinned":true,"hidden":false,"mode":"ojs"},{"id":304,"value":"---\n\n## Appendix","pinned":false,"hidden":false,"mode":"md"},{"id":36,"value":"graticule = d3.geoGraticule10()","pinned":true,"hidden":false,"mode":"ojs"},{"id":38,"value":"land = topojson.feature(world, world.objects.land)","pinned":true,"hidden":false,"mode":"ojs"},{"id":7,"value":"world = fetch(\"https://cdn.jsdelivr.net/npm/world-atlas@2/land-50m.json\").then(response => response.json())","pinned":true,"hidden":false,"mode":"ojs"},{"id":13,"value":"topojson = require(\"topojson-client@3\")","pinned":true,"hidden":false,"mode":"ojs"},{"id":3,"value":"d3 = require(\"d3-geo@3\", \"d3-geo-projection@4\")","pinned":true,"hidden":false,"mode":"ojs"}],"files":"$W32","stdlib":"1","titling":"auto"}