2e:null 2f:["$","title",null,{"children":"Cylindrical equal-area projections | Joshua Horowitz | Observable"}] 34:Tb92,md` ${ showSecretControls ? md` Current aspect ratio: ${diagram.getAttributeByLabel("aspect ratio").value().toFixed(2)} * ${html` setAspectRatio(3.1415)}>Lambert ratio 3.14, standard @ equator`} * ${html` setAspectRatio(2.356)}>Behrmann ratio 2.36, standard @ 30°N/S`} * ${html` setAspectRatio(2.000)}>Smyth ratio 2.00, standard @ 37.1°N/S`} * ${html` setAspectRatio(1.571)}>Gall ratio 1.57, standard @ 45°N/S`} * ${html` setAspectRatio(1.000)}>Tobler "world in a square" ratio 1.00, standard @ 55.5°N/S`}

` : md`` } If you want a projection where parallels are horizontal lines, meridians are vertical lines, and area is represented faithfully, then what you're looking for is a **cylindrical equal-area projection**. If you check [Wikipedia](https://en.wikipedia.org/wiki/Cylindrical_equal-area_projection), you'll see a whole list of these. There's the "Lambert" projection, the "Behrmann" projection, the "Gall" projection, etc. etc. etc. But I'm here to tell you that there's really only one cylindrical equal-area projection. All of these supposedly-distinct projections are just the same projection stretched into different-size rectangles. In other words: they're the same projection with different aspect ratios. This seems inappropriate. Should people really get to name aspect ratios after themselves? Though, to be fair, choice of aspect ratio has one particularly interesting consequence... As the ellipses ("[Tissot's indicatrices](https://en.wikipedia.org/wiki/Tissot%27s_indicatrix)") above show, only narrow slices of the map will ever be free from squash-and-stretch distortion. For instance, with the aspect ratio shown above (2.36, the "Behrmann" projection), land near the 30°N and 30°S parallels are represented accurately, but land around the the equator is stretched vertically and land near the poles is stretched horizontally. (Though, to be clear, these stretches are compensated for with squashes in the perpendicular direction, so there is no net distortion of *area*.) Cartographers say that the Behrmann projection has "standard parallels" at 30°N/S. By choosing different aspect ratios, you change where these standard parallels are! **Drag the map left and right to stretch it.** Watch how the standard parallels move, as different indicatrices come in and out of the distortion-free zone. (Or **try dragging the parallels themselves**, to see how the map scales accordingly.) Equal-area projections are certainly an improvement over the Mercator projection for most world-viewing purposes. But it's good to remember that they, too, make unavoidable decisions about what will be seen most clearly and what will be left distorted. ${html``} `35:[{"id":"741501e6b3ec2c08b90dd1f599e4935ea3a9e25dad16002d317ca67cc42dbda92c85599f8a4ff60da433160a0c7dfa99347e6a2910131ab421d6ffd13d0eb69e","href":"https://static.observableusercontent.com/files/741501e6b3ec2c08b90dd1f599e4935ea3a9e25dad16002d317ca67cc42dbda92c85599f8a4ff60da433160a0c7dfa99347e6a2910131ab421d6ffd13d0eb69e","name":"cylindrical equal-area projections (2).json","type":"application/json","lastModified":1591915516985.08,"size":1113103}] 32:{"id":"66aededdaaeb4c49","slug":"cylindrical-equal-area-projections","title":"Cylindrical equal-area projections","thumbnail":"071ea6317bee74d97161b92f3cb7caf3a373582f1d7ecaa0c4090021d8afe48a","access_level":"public","create_time":"$D2020-06-11T22:31:42.245Z","update_time":"$D2020-06-11T23:46:20.502Z","view_time":null,"first_public_time":"$D2020-06-11T23:47:11.887Z","has_unpublished_edits":null,"fork":false,"stars":5,"starred":false,"owner":{"type":"team","id":"4a29764d3c5ac978","login":"joshuahhh","name":"Joshua Horowitz","avatar_url":"https://avatars-next.observableusercontent.com/c2ab36a2b36058e9d5ad6cab5c830958647aa34d40616ae28603a9a9e5d0064a","tier":"starter_2024"},"version":149,"publish_version":149,"role":null,"authored":false,"authors":[{"type":"individual","id":"ddc378ed3cec53e0","login":"joshuahhh","name":"Joshua Horowitz","avatar_url":"https://avatars-next.observableusercontent.com/c2ab36a2b36058e9d5ad6cab5c830958647aa34d40616ae28603a9a9e5d0064a","tier":null}],"members":[{"type":"individual","id":"ddc378ed3cec53e0","login":"joshuahhh","name":"Joshua Horowitz","avatar_url":"https://avatars-next.observableusercontent.com/c2ab36a2b36058e9d5ad6cab5c830958647aa34d40616ae28603a9a9e5d0064a","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Cylindrical equal-area projections","cells":[{"id":0,"value":"md`# Cylindrical equal-area projections`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":9,"value":"viewof diagram = apparatus({\n width: \"100%\",\n height: 300,\n projectData: await FileAttachment(\"cylindrical equal-area projections (2).json\").json(),\n regionOfInterest: {\n x: [-10, 32],\n y: [-1, 11]\n },\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":28,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":137,"value":"md`

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":38,"value":"md`\n## Appendix\nThis interactive diagram was made with [Apparatus](http://aprt.us/). See [this notebook](https://observablehq.com/@joshuahhh/apparatus) to learn how to embed Apparatus diagrams in your own Observable notebooks.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6,"value":"import {apparatus} from '@joshuahhh/apparatus'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":73,"value":"setAspectRatio = (aspectRatio) => {\n diagram.getAttributeByLabel(\"aspect ratio\").setExpression(`${aspectRatio.toFixed(5)}{0.1-3.141592}`);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":124,"value":"mutable showSecretControls = false","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":83,"value":"htl = require(\"htl@0.2\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":85,"value":"html = htl.html","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W35","stdlib":"1","titling":"auto"}} 33:"$32:body"