From 512347ab9ce363e306d941e4e1f6cce06da272d1 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Philippe=20Rivi=C3=A8re?= Date: Sun, 24 Feb 2019 21:06:41 +0100 Subject: [PATCH 1/4] =?UTF-8?q?Imago=20projection,=20by=20Justin=20Kunimun?= =?UTF-8?q?e=20Inspired=20by=20Hajime=20Narukawa=E2=80=99s=20AuthaGraph?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Closes https://github.com/d3/d3-geo-projection/issues/73 --- README.md | 16 ++ img/imago.png | Bin 0 -> 101918 bytes src/imago.js | 348 ++++++++++++++++++++++++++++++++++++++++++++ src/index.js | 1 + src/newton.js | 16 ++ test/compare-images | 1 + test/invert-test.js | 13 ++ 7 files changed, 395 insertions(+) create mode 100644 img/imago.png create mode 100644 src/imago.js create mode 100644 src/newton.js diff --git a/README.md b/README.md index c3e6d62..a710d0b 100644 --- a/README.md +++ b/README.md @@ -111,6 +111,22 @@ Buckminster Fuller’s Airocean projection (also known as “Dymaxion”), based The Cahill-Keyes projection, designed by Gene Keyes (1975), is built on Bernard J. S. Cahill’s 1909 octant design. Implementation by Mary Jo Graça (2011), ported to D3 by Enrico Spinielli (2013). +# d3.geoImago() [<>](https://github.com/d3/d3-geo-polygon/blob/master/src/imago.js "Source") + +[](https://kunimune.home.blog/2017/11/23/the-secrets-of-the-authagraph-revealed/) + +The Imago projection, engineered by Justin Kunimune (2017), is inspired by Hajime Narukawa’s AuthaGraph design (1999). + +# imago.k([k]) + +Exponent. Defaults to 0.68. + +# imago.shift([shift]) + +Horizontal shift. Defaults to 1.16. + + + # d3.geoTetrahedralLee() [<>](https://github.com/d3/d3-geo-polygon/blob/master/src/tetrahedralLee.js "Source")
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pi, + pow, + sign, + sin, + sqrt, + tan +} from "./math"; +import { geoProjectionMutator as projectionMutator } from "d3-geo"; +import { default as clipPolygon } from "./clip/polygon"; +import { solve } from "./newton.js"; + +var hypot = Math.hypot; + +var ASIN_ONE_THD = asin(1 / 3), + centrums = [ + [halfPi, 0, 0, -halfPi, 0, sqrt(3)], + [-ASIN_ONE_THD, 0, pi, halfPi, 0, -sqrt(3)], + [-ASIN_ONE_THD, (2 * pi) / 3, pi, (5 * pi) / 6, 3, 0], + [-ASIN_ONE_THD, (-2 * pi) / 3, pi, pi / 6, -3, 0] + ], + TETRAHEDRON_WIDE_VERTEX = { + sphereSym: 3, + planarSym: 6, + width: 6, + height: 2 * sqrt(3), + centrums, + rotateOOB: function(x, y, xCen, yCen) { + yCen * 0; + if (abs(x) > this.width / 2) return [2 * xCen - x, -y]; + else return [-x, this.height * sign(y) - y]; + }, + inBounds: () => true + }, + configuration = TETRAHEDRON_WIDE_VERTEX; + +export function imagoRaw(k) { + function faceProject(lon, lat) { + const tht = atan(((lon - asin(sin(lon) / sqrt(3))) / pi) * sqrt(12)), + p = (halfPi - lat) / atan(sqrt(2) / cos(lon)); + + return [(pow(p, k) * sqrt(3)) / cos(tht), tht]; + } + + function faceInverse(r, th) { + const l = solve( + l => atan(((l - asin(sin(l) / sqrt(3))) / pi) * sqrt(12)), + th, + th / 2 + ), + R = r / (sqrt(3) / cos(th)); + return [halfPi - pow(R, 1 / k) * atan(sqrt(2) / cos(l)), l]; + } + + function obliquifySphc(latF, lonF, pole) { + if (pole == null) + // null pole indicates that this procedure should be bypassed + return [latF, lonF]; + + const lat0 = pole[0], + lon0 = pole[1], + tht0 = pole[2]; + + let lat1, lon1; + if (lat0 == halfPi) lat1 = latF; + else + lat1 = asin( + sin(lat0) * sin(latF) + cos(lat0) * cos(latF) * cos(lon0 - lonF) + ); // relative latitude + + if (lat0 == halfPi) + // accounts for all the 0/0 errors at the poles + lon1 = lonF - lon0; + else if (lat0 == -halfPi) lon1 = lon0 - lonF - pi; + else { + lon1 = + acos( + (cos(lat0) * sin(latF) - sin(lat0) * cos(latF) * cos(lon0 - lonF)) / + cos(lat1) + ) - pi; // relative longitude + if (isNaN(lon1)) { + if ( + (cos(lon0 - lonF) >= 0 && latF < lat0) || + (cos(lon0 - lonF) < 0 && latF < -lat0) + ) + lon1 = 0; + else lon1 = -pi; + } else if (sin(lonF - lon0) > 0) + // it's a plus-or-minus arccos. + lon1 = -lon1; + } + lon1 = lon1 - tht0; + + return [lat1, lon1]; + } + + function obliquifyPlnr(coords, pole) { + if (pole == null) + //this indicates that you just shouldn't do this calculation + return coords; + + let lat1 = coords[0], + lon1 = coords[1]; + const lat0 = pole[0], + lon0 = pole[1], + tht0 = pole[2]; + + lon1 += tht0; + let latf = asin(sin(lat0) * sin(lat1) - cos(lat0) * cos(lon1) * cos(lat1)), + lonf, + innerFunc = sin(lat1) / cos(lat0) / cos(latf) - tan(lat0) * tan(latf); + if (lat0 == halfPi) + // accounts for special case when lat0 = pi/2 + lonf = lon1 + lon0; + else if (lat0 == -halfPi) + // accounts for special case when lat0 = -pi/2 + lonf = -lon1 + lon0 + pi; + else if (abs(innerFunc) > 1) { + // accounts for special case when cos(lat1) -> 0 + if ((lon1 == 0 && lat1 < -lat0) || (lon1 != 0 && lat1 < lat0)) + lonf = lon0 + pi; + else lonf = lon0; + } else if (sin(lon1) > 0) lonf = lon0 + acos(innerFunc); + else lonf = lon0 - acos(innerFunc); + + let thtf = pole[2]; + + return [latf, lonf, thtf]; + } + + function forward(lon, lat) { + const width = configuration.width, + height = configuration.height; + const numSym = configuration.sphereSym; //we're about to be using this variable a lot + let latR = -Infinity; + let lonR = -Infinity; + let centrum = null; + for (const testCentrum of centrums) { + //iterate through the centrums to see which goes here + const relCoords = obliquifySphc(lat, lon, testCentrum); + if (relCoords[0] > latR) { + latR = relCoords[0]; + lonR = relCoords[1]; + centrum = testCentrum; + } + } + + const lonR0 = + floor((lonR + pi / numSym) / ((2 * pi) / numSym)) * ((2 * pi) / numSym); + + const rth = faceProject(lonR - lonR0, latR); + const r = rth[0]; + const th = rth[1] + centrum[3] + (lonR0 * numSym) / configuration.planarSym; + const x0 = centrum[4]; + const y0 = centrum[5]; + + let output = [r * cos(th) + x0, r * sin(th) + y0]; + if (abs(output[0]) > width / 2 || abs(output[1]) > height / 2) { + output = configuration.rotateOOB(output[0], output[1], x0, y0); + } + return output; + } + + function invert(x, y) { + if (isNaN(x) || isNaN(y)) return null; + + if (!configuration.inBounds(x, y)) return null; + + const numSym = configuration.planarSym; + + let rM = +Infinity; + let centrum = null; //iterate to see which centrum we get + for (const testCentrum of centrums) { + const rR = hypot(x - testCentrum[4], y - testCentrum[5]); + if (rR < rM) { + //pick the centrum that minimises r + rM = rR; + centrum = testCentrum; + } + } + const th0 = centrum[3], + x0 = centrum[4], + y0 = centrum[5], + r = hypot(x - x0, y - y0), + th = atan2(y - y0, x - x0) - th0, + thBase = + floor((th + pi / numSym) / ((2 * pi) / numSym)) * ((2 * pi) / numSym); + + let relCoords = faceInverse(r, th - thBase); + + if (relCoords == null) return null; + + relCoords[1] = (thBase * numSym) / configuration.sphereSym + relCoords[1]; + let absCoords = obliquifyPlnr(relCoords, centrum); + return [absCoords[1], absCoords[0]]; + } + + forward.invert = invert; + + return forward; +} + +export function imagoBlock() { + var k = 0.68, + m = projectionMutator(imagoRaw), + p = m(k); + + p.k = function(_) { + return arguments.length ? m((k = +_)) : k; + }; + + var a = -atan(1 / sqrt(2)) * degrees, + poly = { + type: "Polygon", + coordinates: [ + [ + [-180 + epsilon, a + epsilon], + [0, 90], + [180 - epsilon, a + epsilon], + [180 - epsilon, a - epsilon], + [-180 + epsilon, a - epsilon], + [-180 + epsilon, a + epsilon] + ] + ] + }; + + return p + .preclip(clipPolygon(poly)) + .scale(144.04) + .rotate([18, -12.5, 3.5]) + .center([0, 35.2644]); +} + +function imagoWideRaw(k, shift) { + var imago = imagoRaw(k); + const height = configuration.height; + + function forward(lon, lat) { + const p = imago(lon, lat), + q = [p[1], -p[0]]; + + if (q[1] > 0) { + q[0] = height - q[0]; + q[1] *= -1; + } + + q[0] += shift; + if (q[0] < 0) q[0] += height * 2; + + return q; + } + + function invert(x, y) { + x = (x - shift) / height; + + if (x > 1.5) { + x -= 2; + } + + if (x > 0.5) { + x = 1 - x; + y *= -1; + } + + return imago.invert(-y, x * height); + } + + forward.invert = invert; + return forward; +} + +export default function() { + var k = 0.68, + shift = 1.16, + m = projectionMutator(imagoWideRaw), + p = m(k, shift); + + p.shift = function(_) { + return arguments.length ? clipped(m(k, (shift = +_))) : shift; + }; + p.k = function(_) { + return arguments.length ? clipped(m((k = +_), shift)) : k; + }; + + function clipped(p) { + const N = 50 + 2 * epsilon, + border = [], + e = 4e-3; + + const scale = p.scale(), + center = p.center(), + translate = p.translate(), + rotate = p.rotate(); + p.scale(1) + .center([0, 90]) + .rotate([0, 0]) + .translate([shift, 0]); + for (let i = N - epsilon; i > 0; i--) { + border.unshift( + p.invert([ + 1.5 * configuration.height - e, + ((configuration.width / 2) * i) / N + ]) + ); + border.push( + p.invert([ + -0.5 * configuration.height + e, + ((configuration.width / 2) * i) / N + ]) + ); + } + border.push(border[0]); + + return p + .scale(scale) + .center(center) + .translate(translate) + .rotate(rotate) + .preclip( + clipPolygon({ + type: "Polygon", + coordinates: [border] + }) + ); + } + + return clipped(p) + .rotate([18, -12.5, 3.5]) + .scale(138.42) + .translate([480, 250]) + .center([-139.405, 40.5844]); +} diff --git a/src/index.js b/src/index.js index 5ef6aa6..b0833df 100644 --- a/src/index.js +++ b/src/index.js @@ -10,5 +10,6 @@ export {default as geoTetrahedralLee, leeRaw as geoLeeRaw} from "./tetrahedralLe export {default as geoGrayFullerRaw} from "./grayfuller"; export {default as geoAirocean} from "./airocean"; export {default as geoIcosahedral} from "./icosahedral"; +export {default as geoImago, imagoBlock as geoImagoBlock, imagoRaw as geoImagoRaw} from "./imago"; export {default as geoCubic} from "./cubic"; export {default as geoCahillKeyes, cahillKeyesRaw as geoCahillKeyesRaw} from "./cahillKeyes"; diff --git a/src/newton.js b/src/newton.js new file mode 100644 index 0000000..8d53099 --- /dev/null +++ b/src/newton.js @@ -0,0 +1,16 @@ +import {abs, epsilon} from "./math"; + +// Newton-Raphson +// Solve f(x) = y, start from x +export function solve(f, y, x) { + var steps = 100, delta, f0, f1; + x = x === undefined ? 0 : +x; + y = +y; + do { + f0 = f(x); + f1 = f(x + epsilon); + if (f0 === f1) f1 = f0 + epsilon; + x -= delta = (-1 * epsilon * (f0 - y)) / (f0 - f1); + } while (steps-- > 0 && abs(delta) > epsilon); + return steps < 0 ? NaN : x; +} diff --git a/test/compare-images b/test/compare-images index bf0ff0b..67d64bf 100755 --- a/test/compare-images +++ b/test/compare-images @@ -7,6 +7,7 @@ for i in \ cubic \ dodecahedral \ icosahedral \ + imago \ polyhedralButterfly \ polyhedralCollignon \ polyhedralWaterman \ diff --git a/test/invert-test.js b/test/invert-test.js index c6b4617..c07d7d9 100644 --- a/test/invert-test.js +++ b/test/invert-test.js @@ -17,3 +17,16 @@ tape("inverse polyhedrals", function(test) { test.end(); }); +tape("inverse Imago", function(test) { + + [ d3.geoImago(), d3.geoImagoBlock() ] + .forEach(function(projection) { + [ [-23, 12], [10,10], [100,-45] ] + .forEach(function(location) { + projection.angle(Math.random()*360); + test.projectionEqual(projection, location, projection(location), 1e-5); + }); + }); + test.end(); +}); + From 76020922276ea5c1dfd162ca76a1bd6f939202f9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Philippe=20Rivi=C3=A8re?= Date: Thu, 28 Feb 2019 11:01:23 +0100 Subject: [PATCH 2/4] default parameter k=0.59 gives a distinctive "almost conformal" taste to the Imago projection. MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit ImagoBlock keeps k=0.68 (which resembles the AuthaGraph original design more). Make sure the clipping works for 0.5 ≤ k ≤ 1 See discussion at https://github.com/d3/d3-geo-projection/issues/73 --- img/imago.png | Bin 101918 -> 100098 bytes src/imago.js | 32 +++++++++++++++----------------- 2 files changed, 15 insertions(+), 17 deletions(-) diff --git a/img/imago.png b/img/imago.png index bc26b49d37ddb07f402b490ee5198bdeb5d7f9b8..55affbc845ae58d9fafc7d47ef4346ec890b3e68 100644 GIT binary patch literal 100098 zcmagFV{k4^5GMM@ww;{VwoYu@wr%^wc5-6dwoYs(Z*1H9)xG;;_ttLRU(?k!&vegJ zO?6Mt(@~1@5(u!kumAu6K}u3o82|w52LM1Gq5o3=RwwY5{~JJ!Wh6uaKN0a$eE^W~ zE-6tVRnLt}KL}4_SS~cO-VeDOeyVdu~s;AEyHM;^^pQ}OKLj#puv5*`g*vL z+yzXBo{m+lwR7g9!mcesV;=uaEi zt!$8mXE;=U-X2cg9!ULvvV|w$^8X{_5CVMu)BnHI{#W^bLjDi%{|Whjmiph||98m$ zYW;tM{2wg--<|ybv3M5%(;wjBr28Z>ma6gP5KCKo4?jDsJjNd2;qZC&vvPIgy0wBu z>^RY!(0;NQlHg>2|md)e-P?YLX&o;Q3DU zalPB)^CA+@_lEDh4F>fhIC!@R7QfZ!bCV2}-}mzt!VnMUgV@QP=Y0yS!12j<{-FHv z9A0)1cwl0SfNGELyPZ1txDGjf6B^@diW~bL@C%0G3V2=;NJH}A)qg-6N4pG2dg#6O zI6BMQ{&9F2?dn4MCc>TNkK9?V(Fo~pVyLtp;X5djz&gucQ&<})Lh0Xt+sL4_ToC%P z!awW?_0PE&Z%p~n*rq@Isgp%=!*y*$Q)8#!CUxPsbi9K8n6)*jv3Al!x>rRo=d{H< zD$W*{Z>hFQ>(N`m>n;QO}oH|{8*Pd$a1dl?)*iGUVgIkJAltLwm1477BE@S zIOn0bTDwT8ZWr*_hI~2pR^+!rUX5E*c$dL1*BOaF_s;b*G5f1ueU>8>`V6mT{NHxj ztcOwUR8DSVQso3%ulveEK*b2+b+dsyEMpP1ch=Cu&WCiMBCmFQ?`-g1rR@?G#08j z8dMrNvOi;TmAKu9-a}~Y{5kw_0IbG!A{c^tlm7y=lOhwx_AlCZTW#{Z8sI1$&aV22 z3Qj75&62-ow>xCeozk<4n47eKvZQ5`kj~I(?RPnJr92AL4^68rjDl<+sdt)DQt0OM zebwW!+SNIEqQPCUOy5eW4c(8Z^?1u;*b+G?QE87#zHn7&V_(?Us;jClw>-<$%exRw z5GsWYFm4-&x88R+6lXq&)1~_U5=(nsD_Zo2mQd{LWn_U(gU$6 znJl^~mW|(~slL6%+G=>xX4%jn2;+4kz>gnlOj)c2y?s5{PK4!^LSkypB(9-f$Y{X<-G-zlm;y-83;qmf>fD|92Fio$0 z`NIov=L)|DU^~e!xgZ#f{$iv~CFsASo|^rHkz+rE|O1&SQqg^@Y~yjUX|+X#*LU5@LVH64h3)_fH=Z~ z0)OXg!SBt&8&x2L=;^VuHCwl6mjs5tik7?%-yMu8L>7>Q2$ToMmG9N{kJ|P6t61!gFWvuO z2U0fMgRSX=#`x0YI+n`%i2 z-vzvF8*IhRamsAlHQla{R(d&XX|LDTEj7nB&DAwcC%rpOGXS7FPlNog%1cyf7?O`|@ge!chBk+#EG zUk+Mp390T6zrHG}&=&Pgs@*E)$2A zFu=Q$)_dm$D+<5``3)2jxCqhT)1bC}I%k_1xqG|zZ<&`bO46pVPwSCJml5`_NN0Be zATjD0-?rH`a=~t60+muI+`dCDzP z?>QdcEwecpTD2#;uVgYMiU_d|wl$~PsZ=LLzW3t@Dx^a50JA23H&nBS+kfafs>VBH zLK%FMFna|WgCKPpySy6ByJFVP`o|kD;w9BfYlXGmsiBd6Ugphxv$pmTFuM(&I)-besXD8c&ONbRnt$(Nnd%G0W!*N(WhzU#0jbJk)Gl9~zurILg{ zwDCj{sYf?2b`uj5<5XYkOB?naFON?M)SbwhGmLcFbj@&294^41*v#NcffWC)RdL-Joy!`Zs>lU#iL4CJsxM0d+jPs@@<>p!Bp86w`0Oa=MUO*PJ`nXI3sIvkfi_e$e<_w;ax(6>@FKa=YKpv$4M<)}y@ zsDckPm&e&i`r{Xo{q*jZM+S210hXw)x5pAKsxp}-X9q4|GVL1>s8;I;zo^Z55~Ff= zyCzfQt#wlo%#;;=f*F`T>=gKrm@wsc$sq~FC@r;huc;stTPUow2pU1NbYk#ur|~X7 zl6$cV5?KBx*IfjQ*wetC8+EMH=rnG&55FFz`BvAVN7V@XueD=U=c$eb z9f{2%ygq3834OGcK*n95H}Q+dx{7*xs7ffF@#*RGap6itYbf@HPU-wd<>o?p6s7Uq zztxoz#eU8Z^jFvH(ol0Fz#6b+;)vF-{?>|(1-MjU0CMB{{1vKA_Iuh(M>tO8Morjp zB6BFyP6fD|hZO`j>+!5F5U1ina*@3>g0C1( znuk?Jw6cw+EVFLo0Gjh6FyU(&CsJ{HXh&-@1nXw3Q+V0p#U}T7wNh)l*Nmc~ba33- z9hQS=_2%$3`@LXJXozOoF|)oI@kxA1E^Ofc{pkUq=5t5%z8{k(-fRSHsvjg|bvWeL zIU4TZe2M;9PW)Eo3(oiGYWw<@hEI+W zhl)h>1cxpyhx}qiOcu>Ja^`Cd>a$upjSaEp25$+2sr0!gE2G4rxG&0jCHMFWH+8%d zyVohWEnzd&D||gnn^v_}$Xeh`P$sMUH;W@KTSr(qHX@BoBqGw&i!qn9m?L372e3P~ z&ORJbpcU7rwJG;u4u-IqEBa(jVNOt~-NVTp(P=q`q@5U#pBy^e&Qb$}k6Ec{5f7Ti z38FvSxFX;qV#HwzBMqyVTs*9wq*Rh_dFVzQ6aox!mYWaiBQvo0um83eZz$~9kwCoZ zU<5J+{X2##;fs*D%VMDYc$An9$vo2B$a|8|hDddal2!VF9yKnp*L=wt;A5|dRYG&c zkTo8LN27>rXtVI}^_qBATDMcDeio8r>E62|h0XzMG*W*hN(T{_0SqX2KM6-GFEz

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- [-180 + epsilon, a - epsilon], - [-180 + epsilon, a + epsilon] - ] - ] - }; + border = [ + [-180 + epsilon, a + epsilon], + [0, 90], + [180 - epsilon, a + epsilon], + [180 - epsilon, a - epsilon], + [-180 + epsilon, a - epsilon], + [-180 + epsilon, a + epsilon] + ]; return p - .preclip(clipPolygon(poly)) + .preclip(clipPolygon({ + type: "Polygon", + coordinates: [border] + })) .scale(144.04) .rotate([18, -12.5, 3.5]) .center([0, 35.2644]); @@ -286,7 +284,7 @@ function imagoWideRaw(k, shift) { } export default function() { - var k = 0.68, + var k = 0.59, shift = 1.16, m = projectionMutator(imagoWideRaw), p = m(k, shift); @@ -299,9 +297,9 @@ export default function() { }; function clipped(p) { - const N = 50 + 2 * epsilon, + const N = 100 + 2 * epsilon, border = [], - e = 4e-3; + e = 3e-3; const scale = p.scale(), center = p.center(), From f26ca6dde457bdba82d18c594671a58a526df071 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Philippe=20Rivi=C3=A8re?= Date: Thu, 28 Feb 2019 11:10:30 +0100 Subject: [PATCH 3/4] Imago defaults to k=0.59 --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index a710d0b..bbf4e0a 100644 --- a/README.md +++ b/README.md @@ -119,7 +119,7 @@ The Imago projection, engineered by Justin Kunimune (2017), is inspired by Hajim # imago.k([k]) -Exponent. Defaults to 0.68. +Exponent. Defaults to 0.59. # imago.shift([shift]) From 2415671af899cabbcf9011ff03a83457e617a834 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Philippe=20Rivi=C3=A8re?= Date: Mon, 11 Mar 2019 19:07:10 +0100 Subject: [PATCH 4/4] imago readme with useful k values (see https://github.com/d3/d3-geo-polygon/pull/24) --- README.md | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/README.md b/README.md index bbf4e0a..dfe91f7 100644 --- a/README.md +++ b/README.md @@ -119,14 +119,12 @@ The Imago projection, engineered by Justin Kunimune (2017), is inspired by Hajim # imago.k([k]) -Exponent. Defaults to 0.59. +Exponent. Useful values include 0.59 (default, minimizes angular distortion of the continents), 0.68 (gives the best approximation of the Authagraph) and 0.72 (prevents kinks in the graticule). # imago.shift([shift]) Horizontal shift. Defaults to 1.16. - - # d3.geoTetrahedralLee() [<>](https://github.com/d3/d3-geo-polygon/blob/master/src/tetrahedralLee.js "Source")
# d3.geoLeeRaw