2e:null 2f:["$","title",null,{"children":"Cordiform map projections 💛💗💖 | Fil | Observable"}] 34:T466,function stabWernerRaw(ratio, max) { /* * simplified Bonne, with phi0 = halfPi ; * https://github.com/d3/d3-geo-projection/blob/master/src/bonne.js */ function r(lambda, phi) { const rho = halfPi - phi, e = rho ? (lambda * cos(phi)) / rho : rho; return [rho * sin(e), -rho * cos(e)]; } r.invert = function(x, y) { var rho = sqrt(x ** 2 + y ** 2), phi = halfPi - rho; return [(rho / cos(phi)) * atan2(x, -y), phi]; }; /* azimuthal equidistant for the pole part */ const az = d3 .geoAzimuthalEquidistant() .rotate([0, -90]) .translate([0, 0]) .scale(1); /* mix */ function forward(lambda, phi) { if (phi < max * radians) { lambda *= ratio; return r(lambda, phi); } else { const p = az([lambda * degrees, phi * degrees]); return [p[0], -p[1]]; } } function invert(x, y) { let i = az.invert([x, -y]); if (i[1] < max) { i = r.invert(x, y); i[0] /= ratio; } else { i[0] *= radians; i[1] *= radians; } return [i[0], i[1]]; } forward.invert = invert; return forward; }35:[] 32:{"id":"e2cd77a0e9a30840","slug":"cordiform-map-projections","title":"Cordiform map projections 💛💗💖","thumbnail":"c5d45e03687a04e48b4651ba730ac41f5c098064f006acc2de56abc6f660e5d4","access_level":"public","create_time":"$D2019-02-14T14:19:13.532Z","update_time":"$D2026-05-23T20:14:52.832Z","view_time":null,"first_public_time":"$D2019-02-14T16:08:22.469Z","has_unpublished_edits":null,"fork":true,"stars":10,"starred":false,"owner":{"type":"team","id":"da2a32154638c457","login":"fil","name":"Fil","avatar_url":"https://avatars-next.observableusercontent.com/d5b1777907117a2bdf2a07eb98600d7a9beb21639fde42b9753e95c88c7731a6","tier":"starter_2024"},"version":1944,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"45a379fcfcb14253","login":"fil","name":"Fil","avatar_url":"https://avatars-next.observableusercontent.com/d5b1777907117a2bdf2a07eb98600d7a9beb21639fde42b9753e95c88c7731a6","tier":null}],"members":[{"type":"individual","id":"45a379fcfcb14253","login":"fil","name":"Fil","avatar_url":"https://avatars-next.observableusercontent.com/d5b1777907117a2bdf2a07eb98600d7a9beb21639fde42b9753e95c88c7731a6","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":4,"collection":{"id":"8d7a994f5bc56ea2","slug":"projections","title":"Projections","owner":{"id":"da2a32154638c457","login":"fil","name":"Fil"}},"type":"notebook","body":{"title":"Cordiform map projections 💛💗💖","cells":[{"id":0,"value":"md`# Cordiform map projections 💛💗💖`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":216,"value":"md`Cordiform means “in the shape of a heart”.\n\nOronce Finé made this [gracious map](#orontius) in 1534/36. The same projection was reused by John Bartholomew & sons in this [very beautiful page of their 1913 atlas](#bartholomew).\n\nThe projection was invented circa 1500 by Johann Stab (or Stabius, ~1460–1522†) from a map designed by [Albrecht Dürer](https://fr.wikipedia.org/wiki/Albrecht_D%C3%BCrer), then popularized by Johannes Werner (1468–1522†), and is now known as the Werner projection. However, as John Snyder details (p.34 of “Flattening the earth”), there are _three_ different cordiform Stab–Werner projections. The Wikipedia page about the [Werner projection](https://en.wikipedia.org/wiki/Werner_projection) is rather confusing, so let’s try to get this right.\n\nHappy Valentine's day!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1907,"value":"md`${third([-10, 0])}${third([-160, 0])}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1520,"value":"md`## The first Stab–Werner projection\n\nThe _first_ Stab–Werner projection shows just one hemisphere. Each parallel is mapped on a circle centered on the North pole, with a scale that varies with latitude in order to keep the lengths proportional. The half-equator is mapped to a half-circle.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1388,"value":"first = r =>\n map(\n d3\n .geoProjection(\n stabWernerRaw(halfPi /* map half-equator to a half-circle */, 90)\n )\n .rotate(r)\n .preclip(\n d3.geoClipPolygon({\n type: \"Polygon\",\n coordinates: [\n [[-90, -90], [-90, 90], [90 - 1e-4, 90], [90 - 1e-4, 0], [-90, -90]]\n ]\n })\n )\n .precision(0.1)\n )","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1528,"value":"md`${first([180 - 70.01, 0])} ${first([0 - 70.01, 0])}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1518,"value":"md`## The second Stab–Werner projection\n\nIn D3, the Werner projection is described as “[a limiting form of the Bonne projection, having its standard parallel at ±90°](https://bl.ocks.org/mbostock/a7ae83252305ed4d54d4).” This is in fact the _second_ Stab–Werner projection.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1535,"value":"second = r =>\n map(\n d3\n .geoProjection(stabWernerRaw(1, 90)) // identical to d3.geoBonneRaw(halfPi)\n .rotate(r)\n .precision(0.1)\n )","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":572,"value":"md`${second([-10, 0])}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1410,"value":"md`## The third Stab–Werner projection\n\nThe _third_ Stab–Werner projection expands the equator more, but that expansion has to stop at 60°N, to avoid overlaps — the polar region is then drawn with an azimuthal projection. A specific [clipPolygon](https://github.com/d3/d3-geo-polygon) is necessary to avoid an ugly line going to the pole when drawing the Sphere. This one is my favorite.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":218,"value":"third = r =>\n map(\n d3\n .geoProjection(stabWernerRaw(1 + 1 / 21, 60))\n .rotate(r)\n .preclip(\n d3.geoClipPolygon({\n type: \"Polygon\",\n coordinates: [\n [\n [0, -90],\n [-180 + 1e-4, 0],\n [-180 + 1e-4, 60],\n [-180, 60],\n [180 - 1e-4, 60],\n [180 - 1e-4, 0],\n [0, -90]\n ]\n ]\n })\n )\n .precision(0.1)\n )","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1872,"value":"md`${third([-10.01, 0])}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1878,"value":"md`Finally, here are the second and third projections, displayed with the historical [Ferro Meridian](https://en.wikipedia.org/wiki/Ferro_Meridian) so we can compare with Oronce Finé’s map below:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1876,"value":"md`${second([17 + 40 / 60, 0])} ${third([17 + 40 / 60, 0])}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1856,"value":"md`----\n\n_ define the Stab-Werner raw projection_`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1583,"value":"$34","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1866,"value":"orontius = md`

Source: https://fr.wikipedia.org/wiki/Carte_d%27Oronce_Fine

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1896,"value":"bartholomew = md`

Source: https://mdl.library.utoronto.ca/collections/scanned-maps/atlas-world ([PDF](http://maps.library.utoronto.ca/datapub/digital/G_1019_1913/G_1019_1913_large.pdf)).

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1754,"value":"md`----\n_boring zone_`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6,"value":"import {abs, epsilon, sign, degrees, radians, atan2, cos, halfPi, pi, sin, sqrt} from \"@fil/math\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1408,"value":"w = Math.max(400, width / 2 - 10)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":22,"value":"import { map } from \"@fil/base-map\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":20,"value":"d3 = require(\"d3-array\", \"d3-geo\", \"d3-geo-projection\", \"d3-fetch\", \"d3-geo-polygon\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1939,"value":"import {signature} from \"@fil/signature\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1942,"value":"signature","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W35","stdlib":"1","titling":"auto"}} 33:"$32:body"