Fix srtail, Sswash and sswash. Add zero.reverse-slashed.

Closes #458.

README.md

@@ -304,6 +304,7 @@ The current available styles for `design`/`upright`/`italic`/`oblique` options a
     * `v-zero-slashed`, `cv13`: Slashed Zero `0` (default).
     * `v-zero-dotted`, `cv14`: Dotted Zero `0`.
     * `v-zero-unslashed`, `cv15`: O-like `0`.
+    * `v-zero-reverse-slashed`, `cv93`: Reverse-slashed `0`.
   * Styles for `1`:
     * `v-one-nobase`, `cv50`: `1` with bottom serif (default for Sans).
     * `v-one-base`, `cv51`: `1` without bottom serif (default for Slab).

gen/build-glyphs.ptl

@@ -4,7 +4,6 @@ import '../support/spirokit' as spirokit
 import '../support/transform' as : Transform && [object [transformPoint tp] [unTransform utp] inverse]
 import '../support/anchor' as Anchor
 import '../support/monotonic-interpolate' as smoothreg
-import '../support/fairify' as fairify
 
 import [mix linreg clamp fallback TempFont includeGlyphPart compsiteMarkSet] from '../support/utils'
 import [calculateMetrics setFontMetrics MarksetDiv GenDivFrame] from '../meta/aesthetics'

glyphs/common-shapes.ptl

@@ -1,7 +1,6 @@
 $$include '../meta/macros.ptl'
 
 import '../support/transform' as : Transform && [object [transformPoint tp] [unTransform utp] inverse]
-import '../support/fairify' as fairify
 
 import [mix linreg clamp fallback] from '../support/utils'
 import [designParameters] from '../meta/aesthetics'

glyphs/letters-unified-basic.ptl

@@ -4211,13 +4211,11 @@ glyph-block LetterUnified-Basic : begin
 			define tension : 0.46 - 0.005 * strokeCoeff + slantCoeff * TANSLANT
 			return : alsothruthem {{0.4 tension} {0.6 (1 - tension)}} [widths.center stroke]
 
-		sketch # S
+		define [SStroke] : begin
-			set-width WIDTH
-			include MarkSet.capital
 			local stroke : adviceBlackness2 2.875 2 CAP
 			local ess : ESS * stroke / STROKE
 			define smooth : adviceSSmooth CAP (-1) stroke
-			include : dispiro
+			return : dispiro
 				widths.lhs stroke
 				g4   RIGHTSB (CAP - HOOK)
 				hookstart CAPO (sw -- stroke) (swItalicAdj -- STROKE)
@@ -4226,34 +4224,12 @@ glyph-block LetterUnified-Basic : begin
 				g4   RIGHTSB smooth [widths 0 stroke]
 				hookend O (sw -- stroke) (swItalicAdj -- STROKE)
 				g4   SB HOOK
-			save 'S' 'S'
-			save 'cyrDze' 0x405
 
-		sketch # s
+		define [RevSStroke] : begin
-			set-width WIDTH
-			include MarkSet.e
-			define stroke : adviceBlackness2 2.875 2 XH
-			define ess : mix stroke (ESS * stroke / STROKE) 0.5
-			define strokeCoeff : linreg 18 0 126 1 stroke
-			define smooth : adviceSSmooth XH (-1) stroke
-			include : dispiro
-				widths.lhs stroke
-				g4   (RIGHTSB + OX / 4) (XH - SHOOK)
-				hookstart XO (sw -- stroke) (swItalicAdj -- STROKE)
-				g4   (SB + OX / 8) (XH - smooth)
-				SNeck stroke (slantCoeff -- -0.01)
-				g4   (RIGHTSB - OX / 8) (smooth) [widths 0 stroke]
-				hookend O (sw -- stroke) (swItalicAdj -- STROKE)
-				g4   (SB - OX / 4) SHOOK
-			save 's' 's'
-			save 'cyrdze' 0x455
-
-		sketch # revS
-			include MarkSet.capital
 			local stroke : adviceBlackness2 2.875 2 CAP
 			local ess : ESS * stroke / STROKE
 			local smooth : adviceSSmooth CAP 0.5 stroke
-			include : dispiro
+			return : dispiro
 				widths.rhs stroke
 				g4   SB (CAP - HOOK)
 				hookstart CAPO (sw -- stroke) (swItalicAdj -- STROKE)
@@ -4263,15 +4239,27 @@ glyph-block LetterUnified-Basic : begin
 				hookend O (sw -- stroke) (swItalicAdj -- STROKE)
 				g4   RIGHTSB HOOK
 
-			save 'revS' 0x1A7
+		define [sStroke] : begin
+			define stroke : adviceBlackness2 2.875 2 XH
+			define ess : mix stroke (ESS * stroke / STROKE) 0.5
+			define strokeCoeff : linreg 18 0 126 1 stroke
+			define smooth : adviceSSmooth XH (-1) stroke
+			return : dispiro
+				widths.lhs stroke
+				g4   (RIGHTSB + OX / 4) (XH - SHOOK)
+				hookstart XO (sw -- stroke) (swItalicAdj -- STROKE)
+				g4   (SB + OX / 8) (XH - smooth)
+				SNeck stroke (slantCoeff -- -0.01)
+				g4   (RIGHTSB - OX / 8) (smooth) [widths 0 stroke]
+				hookend O (sw -- stroke) (swItalicAdj -- STROKE)
+				g4   (SB - OX / 4) SHOOK
 
-		sketch # revs
+		define [RevsStroke] : begin
-			include MarkSet.e
 			define stroke : adviceBlackness2 2.875 2 XH
 			define ess : ESS * stroke / STROKE
 			define strokeCoeff : linreg 18 0 126 1 stroke
 			define smooth : adviceSSmooth XH 1 stroke
-			include : dispiro
+			return : dispiro
 				widths.rhs stroke
 				g4   SB (XH - SHOOK)
 				hookstart XO (sw -- stroke) (swItalicAdj -- STROKE)
@@ -4281,43 +4269,73 @@ glyph-block LetterUnified-Basic : begin
 				hookend O (sw -- stroke) (swItalicAdj -- STROKE)
 				g4   RIGHTSB SHOOK
 
+		sketch # S
+			include MarkSet.capital
+			include : SStroke
+			save 'S' 'S'
+			save 'cyrDze' 0x405
+
+		sketch # s
+			include MarkSet.e
+			include : sStroke
+			save 's' 's'
+			save 'cyrdze' 0x455
+
+		sketch # revS
+			include MarkSet.capital
+			include : RevSStroke
+			save 'revS' 0x1A7
+
+		sketch # revs
+			include MarkSet.e
+			include : RevsStroke
 			save 'revs' 0x1A8
 			save 'cyrghe.italic'
 
 		sketch # srtail
-			include glyphs.rtailBR
-			local stroke : adviceBlackness2 2.875 2 XH
-			apply-transform : Upright
-			apply-transform : Translate (SB + stroke * HVCONTRAST + OXHOOK) 0
-			apply-transform : Italify
 			include MarkSet.p
-			include glyphs.s
+			local stroke : [sStroke].call currentGlyph
-			include : VBarLeft (SB + OXHOOK) 0 SHOOK
+			local start : utp currentGlyph.gizmo stroke.lhsKnots.(stroke.lhsKnots.length - 1)
+
+			include : create-glyph : glyph-construction
+				local sw : adviceBlackness2 2.875 2 XH
+				include glyphs.rtailBR
+				apply-transform : Upright
+				apply-transform : Translate (start.x + sw * HVCONTRAST) 0
+				apply-transform : Italify
+
+			include : VBarLeft start.x 0 start.y
 
 			save 'srtail' 0x282
 
 		sketch # Sswash
 			include MarkSet.if
-			include glyphs.S
+
-			local stroke : adviceBlackness2 2.875 2 CAP
+			local stroke : [SStroke].call currentGlyph
+			local start : utp currentGlyph.gizmo stroke.lhsKnots.(stroke.lhsKnots.length - 1)
+
+			local sw : adviceBlackness2 2.875 2 CAP
 			include : dispiro
 				widths.lhs [adviceBlackness 4.5]
-				g4 (SB + OXHOOK) HOOK
+				g4 start.x start.y
 				alsothru 0.15 0.6 important
-				flat (RIGHTSB - 1) DESCENDER [widths stroke 0]
+				flat (RIGHTSB - 1) DESCENDER [widths sw 0]
 				curl RIGHTSB DESCENDER
 
 			save 'Sswash' 0x2C7E
 
 		sketch # sswash
 			include MarkSet.p
-			include glyphs.s
+
-			local stroke : adviceBlackness2 2.875 2 XH
+			local stroke : [sStroke].call currentGlyph
+			local start : utp currentGlyph.gizmo stroke.lhsKnots.(stroke.lhsKnots.length - 1)
+
+			local sw : adviceBlackness2 2.875 2 XH
 			include : dispiro
 				widths.lhs [adviceBlackness 4.5]
-				g4 (SB + OXHOOK) SHOOK
+				g4 start.x start.y
-				alsothru 0.2 0.6 important
+				alsothru 0.15 0.6 important
-				flat (RIGHTSB - 1) DESCENDER [widths stroke 0]
+				flat (RIGHTSB - 1) DESCENDER [widths sw 0]
 				curl RIGHTSB DESCENDER
 			save 'sswash' 0x23F
 

glyphs/numbers.ptl

@@ -24,6 +24,14 @@ glyph-block Numbers : begin
 				flat 0 (top * (1 - 0.77)) [widths fine fine]
 				curl WIDTH (top * 0.77)
 
+	define [ZeroReverseSlashShape top] : begin
+		local fine : 0.5 * OVERLAYSTROKE
+		return : intersection
+			OShapeOutline (top + O) (0 - O) (SB - O) (RIGHTSB + O)
+			dispiro
+				flat 0 (top * 0.77) [widths fine fine]
+				curl WIDTH (top * (1 - 0.77))
+
 	define [ZeroDotShape top] : begin
 		local radius : Math.min DOTRADIUS ((RIGHTSB - SB - STROKE * 2) / 4)
 		return : Ring (top / 2 + radius) (top / 2 - radius) (MIDDLE + radius) (MIDDLE - radius)
@@ -39,6 +47,12 @@ glyph-block Numbers : begin
 		include : ZeroSlashShape CAP
 		save 'zero.lnum.slashed'
 
+	sketch # zero.lnum.reverse-slashed
+		include MarkSet.capital
+		include glyphs.'zero.lnum.unslashed'
+		include : ZeroReverseSlashShape CAP
+		save 'zero.lnum.reverse-slashed'
+
 	sketch # zero.lnum.dotted
 		include MarkSet.capital
 		include glyphs.'zero.lnum.unslashed'
@@ -56,6 +70,12 @@ glyph-block Numbers : begin
 		include : ZeroSlashShape XH
 		save 'zero.onum.slashed'
 
+	sketch # zero.onum.reverse-slashed
+		include MarkSet.e
+		include glyphs.'zero.onum.unslashed'
+		include : ZeroReverseSlashShape XH
+		save 'zero.onum.reverse-slashed'
+
 	sketch # zero.onum.dotted
 		include MarkSet.e
 		include glyphs.'zero.onum.unslashed'

snapshot/index.html

@@ -394,6 +394,11 @@
 							<span class="sample" style="font-feature-settings:'cv15'">0</span>
 							<span class="sample italic" style="font-feature-settings:'cv15'">0</span>
 				</li>
+				<li>
+					<span class="tag">cv93</span>
+							<span class="sample" style="font-feature-settings:'cv93'">0</span>
+							<span class="sample italic" style="font-feature-settings:'cv93'">0</span>
+				</li>
 			</ol></li>
 			<li><ol class="group">
 				<li>

support/curve-util.js

@@ -92,43 +92,49 @@ function removeMids(contour) {
 		contour = contour.filter(x => !x.rem);
 
 		last = contour.length - 1;
-		for (let j = 1; j < contour.length - 1; j++) {
+		let zPre, z, zPost;
-			if (!contour[j - 1].on && contour[j].on && !contour[j + 1].on) {
+		for (let j = 1; j < contour.length; j++) {
-				const mx = contour[j - 1].x + contour[j + 1].x;
+			if (j < contour.length - 1) {
-				const my = contour[j - 1].y + contour[j + 1].y;
+				zPre = contour[j - 1];
-				const dy = contour[j - 1].y - contour[j + 1].y;
+				z = contour[j];
-				if (
+				zPost = contour[j + 1];
-					Math.abs(dy) >= 1 &&
+			} else {
-					Math.abs(contour[j].x * 2 - mx) < 1 &&
+				zPre = contour[last];
-					Math.abs(contour[j].y * 2 - my) < 1
+				z = contour[0];
-				) {
+				zPost = contour[1];
-					contour[j].rem = true;
+			}
+			// if (!zPre || !zPost) continue;
+			const mx = zPre.x + zPost.x;
+			const my = zPre.y + zPost.y;
+			const dx = zPre.x - zPost.x;
+			const dy = zPre.y - zPost.y;
+
+			if (!zPre.on && z.on && !zPost.on) {
+				if (Math.abs(dy) >= 1 && Math.abs(z.x * 2 - mx) < 1 && Math.abs(z.y * 2 - my) < 1) {
+					z.rem = true;
+				}
+			} else if (!zPre.rem && zPre.on && z.on && !zPost.rem && zPost.on) {
+				if (Math.abs(dy) >= 1 && Math.abs(z.x * 2 - mx) < 1 && Math.abs(dx) < 1) {
+					z.rem = true;
+				} else if (Math.abs(dx) >= 1 && Math.abs(z.y * 2 - my) < 1 && Math.abs(dy) < 1) {
+					z.rem = true;
 				}
 			}
 		}
-		if (!contour[last].rem && !contour[last].on && contour[0].on && !contour[1].on) {
+
-			const mx = contour[last].x + contour[1].x;
-			const my = contour[last].y + contour[1].y;
-			if (Math.abs(contour[0].x * 2 - mx) < 1 && Math.abs(contour[0].y * 2 - my) < 1) {
-				contour[0].rem = true;
-			}
-		}
 		contour = contour.filter(x => !x.rem);
 		const n = contour.length;
 		if (n >= n0) break;
 	}
 	return contour;
 }
+
 function extPrior(a, b) {
 	return a.y < b.y || (a.y === b.y && ((a.on && !b.on) || (a.on === b.on && a.x < b.x)));
 }
 
 function canonicalStart(_points) {
-	const points = _points.reverse().map(z => {
+	const points = _points.reverse();
-		z.x = Math.round(z.x * 1024) / 1024;
-		z.y = Math.round(z.y * 1024) / 1024;
-		return z;
-	});
 	let jm = 0;
 	for (var j = 0; j < points.length * 2; j++) {
 		if (extPrior(points[j % points.length], points[jm])) {
@@ -138,93 +144,8 @@ function canonicalStart(_points) {
 	return points.slice(jm).concat(points.slice(0, jm));
 }
 
-function colinear(x1, y1, x2, y2, x3, y3, err) {
+function cleanupQuadContour(c) {
-	const det = x1 * (y2 - y3) + x2 * (y3 - y1) + x3 * (y1 - y2);
+	return canonicalStart(removeMids(c));
-	return det <= err && det >= -err;
-}
-
-function inspan(a, b, c) {
-	if (a > c) return inspan(c, b, a);
-	return a <= b && b <= c;
-}
-
-function handle(z1, z2, z3, z4, err) {
-	if (
-		colinear(z1.x, z1.y, z2.x, z2.y, z4.x, z4.y, err) &&
-		colinear(z1.x, z1.y, z3.x, z3.y, z4.x, z4.y, err) &&
-		inspan(z1.x, z2.x, z4.x) &&
-		inspan(z1.y, z2.y, z4.y) &&
-		inspan(z1.x, z3.x, z4.x) &&
-		inspan(z1.y, z3.y, z4.y)
-	) {
-		return [];
-	}
-
-	const curve = new typoGeom.Curve.Bez3(z1, z2, z3, z4);
-	const offPoints = typoGeom.Quadify.auto(curve, err);
-	const ans = [];
-	for (const z of offPoints) {
-		ans.push(Point.offFrom(z));
-	}
-	return ans;
-}
-
-function convertContourToTt(contour, err) {
-	if (contour.length === 0) return [];
-	if (contour.length === 1) return [contour[0]];
-	err = err || 1 / 4;
-
-	const newContour = [];
-	let z0 = contour[0];
-	newContour.push(Point.cornerFrom(z0));
-
-	for (let j = 1; j < contour.length; j++) {
-		const z = contour[j];
-		if (z.on) {
-			newContour.push(Point.cornerFrom(z));
-			z0 = z;
-		} else if (z.cubic) {
-			const z1 = z;
-			const z2 = contour[j + 1];
-			const z3 = contour[j + 2];
-			const quadZs = handle(z0, z1, z2, z3, err);
-			for (const z of quadZs) newContour.push(z);
-			newContour.push(Point.cornerFrom(z3));
-			z0 = z3;
-			j += 2;
-		} else {
-			const zc = z;
-			let zf = contour[j + 1] ? contour[j + 1] : contour[0];
-			if (!zf.on) {
-				zf = Point.cornerFromXY(mix(zc.x, zf.x, 0.5), mix(zc.y, zf.y, 0.5));
-			}
-			newContour.push(Point.offFrom(zc));
-			newContour.push(Point.cornerFrom(zf));
-			z0 = zf;
-			j++;
-		}
-	}
-	return newContour;
-}
-
-function byFirstPointCoord(a, b) {
-	if (!a.length) return -1;
-	if (!b.length) return 1;
-	let z1 = a[0];
-	let z2 = b[0];
-	return z1.y !== z2.y
-		? z1.y - z2.y
-		: z1.x !== z2.x
-		? z1.x - z2.x
-		: byFirstPointCoord(a.slice(1), b.slice(1));
-}
-function convertContourListToTt(contours, err) {
-	err = err || 1 / 4;
-	let ans = [];
-	for (let c of contours) {
-		ans.push(canonicalStart(removeMids(convertContourToTt(c, err))));
-	}
-	return ans.sort(byFirstPointCoord);
 }
 
 function convertContourToCubic(contour) {
@@ -243,34 +164,67 @@ function convertContourToCubic(contour) {
 			const z1 = z;
 			const z2 = contour[j + 1];
 			const z3 = contour[j + 2];
+
 			newContour.push(Point.cubicOffFrom(z1));
 			newContour.push(Point.cubicOffFrom(z2));
 			newContour.push(Point.cornerFrom(z3));
+
 			z0 = z3;
 			j += 2;
 		} else {
 			const zc = z;
-			let zf = contour[j + 1] ? contour[j + 1] : contour[0];
+			let zf = contour[j + 1] || contour[0];
-			if (!zf.on) {
+			const zfIsCorner = zf.on;
-				zf = Point.cornerFromXY(mix(zc.x, zf.x, 0.5), mix(zc.y, zf.y, 0.5));
+			if (!zfIsCorner) zf = Point.cornerFrom(zc).mix(0.5, zf);
-			}
 
-			const x1 = mix(z0.x, zc.x, 2 / 3);
+			newContour.push(Point.cubicOffFrom(z0).mix(2 / 3, zc));
-			const y1 = mix(z0.y, zc.y, 2 / 3);
+			newContour.push(Point.cubicOffFrom(zf).mix(2 / 3, zc));
-			const x2 = mix(zf.x, zc.x, 2 / 3);
-			const y2 = mix(zf.y, zc.y, 2 / 3);
-
-			newContour.push(Point.cubicOffFromXY(x1, y1));
-			newContour.push(Point.cubicOffFromXY(x2, y2));
 			newContour.push(Point.cornerFrom(zf));
+
 			z0 = zf;
-			j++;
+			if (zfIsCorner) j++;
 		}
 	}
 
 	return newContour;
 }
 
-exports.convertContourToTt = convertContourToTt;
+function autoCubify(arc, err) {
+	const MaxSegments = 16;
+	const Hits = 64;
+	let offPoints = [];
+	for (let nSeg = 1; nSeg <= MaxSegments; nSeg++) {
+		const perSegHits = Math.ceil(Hits / nSeg);
+		offPoints.length = 0;
+		let good = true;
+		out: for (let s = 0; s < nSeg; s++) {
+			const tBefore = s / nSeg;
+			const tAfter = (s + 1) / nSeg;
+			const z0 = Point.cornerFrom(arc.eval(tBefore));
+			const z3 = Point.cornerFrom(arc.eval(tAfter));
+			const z1 = Point.cubicOffFrom(z0).addScale(1 / (3 * nSeg), arc.derivative(tBefore));
+			const z2 = Point.cubicOffFrom(z3).addScale(-1 / (3 * nSeg), arc.derivative(tAfter));
+
+			if (s > 0) offPoints.push(z0);
+			offPoints.push(z1, z2);
+
+			const bezArc = new typoGeom.Curve.Bez3(z0, z1, z2, z3);
+
+			for (let k = 1; k < perSegHits; k++) {
+				const tk = k / perSegHits;
+				const zTest = arc.eval(mix(tBefore, tAfter, tk));
+				const zBez = bezArc.eval(tk);
+				if (Math.hypot(zTest.x - zBez.x, zTest.y - zBez.y) > err) {
+					good = false;
+					break out;
+				}
+			}
+		}
+		if (good) break;
+	}
+	return offPoints;
+}
+
+exports.cleanupQuadContour = cleanupQuadContour;
 exports.convertContourToCubic = convertContourToCubic;
-exports.convertContourListToTt = convertContourListToTt;
+exports.autoCubify = autoCubify;

support/fairify.js

@@ -1,9 +1,11 @@
 "use strict";
 
-const Transform = require("./transform.js");
+const Transform = require("./transform");
 const typoGeom = require("typo-geom");
+const Point = require("./point");
+const curveUtil = require("./curve-util");
 
-const SMALL = 1e-4;
+const SMALL = 1e-6;
 
 function solveTS(a, b, c, out, flag) {
 	const delta = b * b - 4 * a * c;
@@ -43,7 +45,7 @@ function fineAllExtrema(z1, z2, z3, z4) {
 
 	return exs.sort(ASCEND);
 }
-function mix(z1, z2, t) {
+function bez1(z1, z2, t) {
 	if (t <= 0) return z1;
 	if (t >= 1) return z2;
 	let x = (1 - t) * z1.x + t * z2.x,
@@ -74,10 +76,10 @@ function bez3(z1, z2, z3, z4, t) {
 	};
 }
 function splitBefore(z1, z2, z3, z4, t) {
-	return [z1, mix(z1, z2, t), bez2(z1, z2, z3, t), bez3(z1, z2, z3, z4, t)];
+	return [z1, bez1(z1, z2, t), bez2(z1, z2, z3, t), bez3(z1, z2, z3, z4, t)];
 }
 function splitAfter(z1, z2, z3, z4, t) {
-	return [bez3(z1, z2, z3, z4, t), bez2(z2, z3, z4, t), mix(z3, z4, t), z4];
+	return [bez3(z1, z2, z3, z4, t), bez2(z2, z3, z4, t), bez1(z3, z4, t), z4];
 }
 function splitAtExtrema(z1, z2, z3, z4, curve) {
 	const ts = fineAllExtrema(z1, z2, z3, z4);
@@ -109,41 +111,33 @@ function veryClose(z1, z2) {
 	return (z1.x - z2.x) * (z1.x - z2.x) + (z1.y - z2.y) * (z1.y - z2.y) <= SMALL;
 }
 
-function splitCurve(sourceCurve) {
+function toSpansForm(sourceCurve, fSplit) {
 	const curve = [sourceCurve[0]];
 	let last = sourceCurve[0];
 	for (let j = 1; j < sourceCurve.length; j++) {
 		if (sourceCurve[j].on) {
 			const z1 = last,
 				z4 = sourceCurve[j];
-			//	const z2 = mix(z1, z4, 1 / 3);
-			//const z3 = mix(z1, z4, 2 / 3);
 			if (!veryClose(z1, z4)) {
 				curve.push(z4);
-				//	splitAtExtrema(z1, z2, z3, z4, ANGLES, curve);
 				last = z4;
 			}
 		} else if (sourceCurve[j].cubic) {
 			const z1 = last,
 				z2 = sourceCurve[j],
-				z3 = sourceCurve[j + 1],
+				z3 = sourceCurve[(j + 1) % sourceCurve.length],
-				z4 = sourceCurve[j + 2];
+				z4 = sourceCurve[(j + 2) % sourceCurve.length];
 			if (!(veryClose(z1, z2) && veryClose(z2, z3) && veryClose(z3, z4))) {
-				splitAtExtrema(z1, z2, z3, z4, curve);
+				if (fSplit) {
+					splitAtExtrema(z1, z2, z3, z4, curve);
+				} else {
+					curve.push(z2, z3, z4);
+				}
 				last = z4;
 			}
 			j += 2;
 		} else {
-			const z1 = last,
+			throw new Error("Unreachable.");
-				zm = sourceCurve[j],
-				z4 = sourceCurve[j + 1];
-			if (!(veryClose(z1, zm) && veryClose(zm, z4))) {
-				const z2 = mix(zm, z1, 1 / 3);
-				const z3 = mix(zm, z4, 1 / 3);
-				splitAtExtrema(z1, z2, z3, z4, curve);
-				last = z4;
-			}
-			j += 1;
 		}
 	}
 	return curve;
@@ -157,35 +151,47 @@ function dot(z1, z2, z3) {
 }
 
 function markCorners(curve) {
+	for (const z of curve) z.mark = 0;
 	for (let j = 0; j < curve.length; j++) {
 		if (!curve[j].on) continue;
 		const z1 = curve[j],
 			z0 = curve[(j - 1 + curve.length) % curve.length],
 			z2 = curve[(j + 1) % curve.length];
-		if (Math.abs(cross(z1, z0, z2)) < 1e-6) {
+
+		const almostLinear = Math.abs(cross(z1, z0, z2)) < SMALL;
+		const inBetween = dot(z1, z0, z2) < 0;
+		if (z0.on && z2.on && almostLinear && inBetween) {
+			z1.mark = 0;
+		} else if (z0.on || z2.on) {
+			z1.mark = 1;
+		} else if (almostLinear && inBetween) {
 			// Z0 -- Z1 -- Z2 are linear
-			if (!z0.on && !z2.on && dot(z1, z0, z2) < 0) {
+			const angle = Math.abs(Math.atan2(z2.y - z0.y, z2.x - z0.x)) % Math.PI;
-				const angle0 = Math.atan2(z2.y - z0.y, z2.x - z0.x);
+			if (Math.abs(angle) <= SMALL || Math.abs(angle - Math.PI) <= SMALL) {
-				const angle = Math.abs(((angle0 / Math.PI) * 2) % 1);
+				z1.mark = 4;
-				if (
+			} else if (Math.abs(angle - Math.PI / 2) <= SMALL) {
-					Math.abs(Math.abs(angle0) - Math.PI / 2) <= SMALL ||
+				z1.mark = 2;
-					angle <= SMALL ||
-					angle >= 1 - SMALL
-				) {
-					z1.mark = true; // curve extremum
-				}
-			} else if (z0.on && z2.on && dot(z1, z0, z2) < 0) {
-				// Colinear on-knots
-				// Remove
-			} else {
-				z1.mark = true; // also corner
 			}
 		} else {
-			z1.mark = true; // corner
+			z1.mark = 1;
 		}
 	}
 }
 
+function canonicalStart(curve) {
+	let jm = 0,
+		rank = 0;
+	for (let j = 0; j < curve.length; j++) {
+		const zRank = (curve[j].on ? 1 : 0) + (2 * curve[j].mark || 0);
+		if (zRank > rank) {
+			jm = j;
+			rank = zRank;
+		}
+	}
+
+	return toSpansForm(curve.slice(jm).concat(curve.slice(0, jm)), false);
+}
+
 class BezierCurveCluster {
 	constructor(zs) {
 		let segments = [];
@@ -195,9 +201,7 @@ class BezierCurveCluster {
 			if (zs[j].on) {
 				const z1 = last,
 					z4 = zs[j];
-				const z2 = mix(z1, z4, 1 / 3);
+				const seg = new typoGeom.Curve.StraightSegment(z1, z4);
-				const z3 = mix(z1, z4, 2 / 3);
-				const seg = new typoGeom.Curve.Bez3(z1, z2, z3, z4);
 				segments.push(seg);
 				lengths.push(this.measureLength(seg));
 				last = z4;
@@ -212,16 +216,7 @@ class BezierCurveCluster {
 				last = z4;
 				j += 2;
 			} else {
-				const z1 = last,
+				throw new Error("Unreachable.");
-					zm = zs[j],
-					z4 = zs[j + 1];
-				const z2 = mix(zm, z1, 1 / 3);
-				const z3 = mix(zm, z4, 1 / 3);
-				const seg = new typoGeom.Curve.Bez3(z1, z2, z3, z4);
-				segments.push(seg);
-				lengths.push(this.measureLength(seg));
-				last = z4;
-				j += 1;
 			}
 		}
 
@@ -281,7 +276,7 @@ class BezierCurveCluster {
 		return det < err * err && det > -err * err;
 	}
 	isAlmostLinear(err) {
-		const N = 16;
+		const N = 64;
 		let z0 = this.eval(0);
 		let z1 = this.eval(1);
 		for (let k = 1; k < N; k++) {
@@ -292,60 +287,72 @@ class BezierCurveCluster {
 	}
 }
 
-function buildCurve(curve) {
+const QuadBuilder = {
-	let exitPoints = [];
+	corner(sink, gizmo, z) {
+		sink.push(Transform.transformPoint(gizmo, Point.cornerFrom(z)).round(1024));
+	},
+	arc(sink, gizmo, arc) {
+		if (arc.isAlmostLinear(1 / 4)) return;
+		const offPoints = typoGeom.Quadify.auto(arc, 1 / 4);
+		if (!offPoints) return;
+		for (const z of offPoints) {
+			sink.push(Transform.transformPoint(gizmo, Point.offFrom(z)).round(1024));
+		}
+	},
+	split: false,
+	canonicalStart: true,
+	duplicateStart: true
+};
+
+const SpiroBuilder = {
+	corner(sink, gizmo, z) {
+		sink.push(Transform.transformPoint(gizmo, Point.cornerFrom(z)));
+	},
+	arc(sink, gizmo, arc) {
+		if (arc.isAlmostLinear(1 / 4)) return;
+		const offPoints = curveUtil.autoCubify(arc, 1 / 4);
+		for (const z of offPoints) {
+			sink.push(Transform.transformPoint(gizmo, z));
+		}
+	},
+	split: true,
+	canonicalStart: false,
+	duplicateStart: false
+};
+
+function buildCurve(curve, gizmo, builder) {
+	let sink = [];
 	for (let j = 0; j < curve.length; j++) {
 		if (!curve[j].mark) continue;
+		builder.corner(sink, gizmo, curve[j]);
+
 		let k = j;
 		for (; k < curve.length && (k === j || !curve[k].mark); k++);
-		exitPoints.push(curve[j]);
 		const pts = curve.slice(j, k + 1);
-		let nPtsOffPoints = 0;
+		if (pts.length > 1) builder.arc(sink, gizmo, new BezierCurveCluster(pts));
-		for (const z of pts) {
-			if (!z.on) nPtsOffPoints += 1;
-		}
-		if (nPtsOffPoints > 0) {
-			const curve = new BezierCurveCluster(pts);
-			if (curve.isAlmostLinear(1)) continue;
-			const offPoints = typoGeom.Quadify.auto(curve, 1 / 4);
-			if (!offPoints) continue;
-			for (let k = 0; k < offPoints.length; k++) {
-				const z = offPoints[k];
-				if (k > 0) {
-					const z0 = offPoints[k - 1];
-					exitPoints.push({
-						x: (z.x + z0.x) / 2,
-						y: (z.y + z0.y) / 2,
-						on: true
-					});
-				}
-				exitPoints.push({
-					x: z.x,
-					y: z.y,
-					cubic: false,
-					on: false
-				});
-			}
-		}
 		j = k - 1;
 	}
-	return exitPoints;
+	return sink;
 }
 
-module.exports = function(sourceCurve, gizmo) {
+function fairifyImpl(sourceCubicContour, gizmo, builder) {
-	for (let j = 0; j < sourceCurve.length; j++) {
+	for (let j = 0; j < sourceCubicContour.length; j++) {
-		if (!isFinite(sourceCurve[j].x)) sourceCurve[j].x = 0;
+		if (!isFinite(sourceCubicContour[j].x)) sourceCubicContour[j].x = 0;
-		if (!isFinite(sourceCurve[j].y)) sourceCurve[j].y = 0;
+		if (!isFinite(sourceCubicContour[j].y)) sourceCubicContour[j].y = 0;
-		sourceCurve[j] = Transform.unTransform(gizmo, sourceCurve[j]);
+		sourceCubicContour[j] = Transform.unTransform(gizmo, sourceCubicContour[j]);
 	}
-	const curve = splitCurve(sourceCurve);
+	let splitContour = toSpansForm(sourceCubicContour, builder.split);
-	markCorners(curve);
+	markCorners(splitContour);
-	const builtCurve = buildCurve(curve);
+	if (builder.canonicalStart) {
-	const ans = [];
+		splitContour = canonicalStart(splitContour);
-	for (let j = 0; j < builtCurve.length; j++) {
+		markCorners(splitContour);
-		if (builtCurve[j] && !builtCurve[j].remove) {
-			ans.push(Transform.transformPoint(gizmo, builtCurve[j]));
-		}
 	}
-	return ans;
+	return buildCurve(splitContour, gizmo, builder);
+}
+
+exports.fairifyQuad = function(sourceCubicContour, gizmo) {
+	return fairifyImpl(sourceCubicContour, gizmo, QuadBuilder);
+};
+exports.fairifySpiro = function(sourceCubicContour, gizmo) {
+	return fairifyImpl(sourceCubicContour, gizmo, SpiroBuilder);
 };

support/point.ptl

@@ -6,6 +6,41 @@ export all : class Point
 		this.cubic = cubic || false
 		this.subdivided = subdivided || false
 
+	public [add z2] : new Point
+		this.x + z2.x
+		this.y + z2.y
+		* this.on
+		* this.cubic
+		* this.subdivided
+
+	public [addScale scale z2] : new Point
+		this.x + scale * z2.x
+		this.y + scale * z2.y
+		* this.on
+		* this.cubic
+		* this.subdivided
+
+	public [mix scale z2] : new Point
+		this.x + scale * (z2.x - this.x)
+		this.y + scale * (z2.y - this.y)
+		* this.on
+		* this.cubic
+		* this.subdivided
+
+	public [scale t] : new Point
+		t * this.x
+		t * this.y
+		* this.on
+		* this.cubic
+		* this.subdivided
+
+	public [round d] : new Point
+		[Math.round (this.x * d)] / d
+		[Math.round (this.y * d)] / d
+		* this.on
+		* this.cubic
+		* this.subdivided
+
 	static [from z on cubic subdivided] : new Point
 		* z.x
 		* z.y

support/regulate-glyph.js

@@ -4,6 +4,8 @@ const Glyph = require("./glyph");
 const autoRef = require("./autoref");
 const caryllShapeOps = require("caryll-shapeops");
 const curveUtil = require("./curve-util");
+const { fairifyQuad } = require("./fairify");
+const Transform = require("./transform");
 
 function regulateGlyph(g, skew) {
 	if (!g.contours) return;
@@ -13,31 +15,11 @@ function regulateGlyph(g, skew) {
 		const contour = g.contours[k];
 		for (let p = 0; p < contour.length; p++) {
 			contour[p].x += contour[p].y * skew;
-			if (!contour[p].on) continue;
-			contour[p].x = Math.round(contour[p].x);
-		}
-		let offJ = null,
-			mx = null;
-		for (let p = 0; p < contour.length; p++) {
-			if (!contour[p].on) continue;
-			if (offJ) {
-				const origX = contour[p].x;
-				const rx = Math.round(contour[p].x * 4) / 4;
-				const origX0 = mx;
-				const rx0 = contour[offJ - 1].x;
-				if (origX === origX0) continue;
-				for (let pOff = offJ; pOff < p; pOff++) {
-					contour[pOff].x =
-						((contour[pOff].x - origX0) / (origX - origX0)) * (rx - rx0) + rx0;
-				}
-			}
-			mx = contour[p].x;
-			contour[p].x = Math.round(contour[p].x * 4) / 4;
-			offJ = p + 1;
 		}
 	}
 
 	g.contours = simplifyContours(g.contours);
+
 	for (let k = 0; k < g.contours.length; k++) {
 		const contour = g.contours[k];
 		for (let p = 0; p < contour.length; p++) {
@@ -47,16 +29,17 @@ function regulateGlyph(g, skew) {
 }
 
 function simplifyContours(contours) {
+	const gizmo = Transform.Id();
 	const source = [];
 	for (const contour of contours) {
 		if (contour.length > 2) source.push(curveUtil.convertContourToCubic(contour));
 	}
-	const simplified = curveUtil.convertContourListToTt(
+	const simplified = caryllShapeOps.removeOverlap(source, 1, 1 << 17, true);
-		caryllShapeOps.removeOverlap(source, 1, 1 << 17, true)
+
-	);
 	const result = [];
 	for (const contour of simplified) {
-		if (contour.length > 2) result.push(contour);
+		if (contour.length <= 2) continue;
+		result.push(curveUtil.cleanupQuadContour(fairifyQuad(contour, gizmo)));
 	}
 	return result;
 }

support/spirokit.ptl

@@ -1,6 +1,6 @@
 import 'spiro' as SpiroJs
 import './spiroexpand' as SpiroExpansionContext
-import './fairify' as fairify
+import [fairifySpiro] from './fairify'
 import 'caryll-shapeops' as ShapeOps
 import './curve-util' as CurveUtil
 
@@ -190,7 +190,7 @@ export : define [SetupBuilders args] : begin
 			lhs.0.type = rhs.0.type = lhs.(lhs.length - 1).type = rhs.(rhs.length - 1).type = 'corner'
 			SpiroJs.spiroToBezierOnContext [lhs.concat : rhs.reverse] true g QUAD PRECISION
 		foreach [j : range 0 g.contours.length] : begin
-			set g.contours.(j) : fairify g.contours.(j) globalTransform
+			set g.contours.(j) : fairifySpiro g.contours.(j) globalTransform
 		set g.knots knots
 		set g.lhsKnots lhs
 		set g.rhsKnots rhs
@@ -204,7 +204,7 @@ export : define [SetupBuilders args] : begin
 		SpiroJs.spiroToBezierOnContext knots closed g QUAD PRECISION
 		foreach af [items-of lastafs] : if af : af.call g
 		foreach [j : range 0 g.contours.length] : begin
-			set g.contours.(j) : fairify g.contours.(j) (g.fairGizmo || g.gizmo)
+			set g.contours.(j) : fairifySpiro g.contours.(j) (g.fairGizmo || g.gizmo)
 		if [not dontinc] : this.include g
 		return g
 

variants.toml

@@ -357,6 +357,12 @@ sampler = '0'
 description = 'O-like `0`'
 zero = 'unslashed'
 
+[simple.v-zero-reverse-slashed]
+tag = 'cv93'
+sampler = '0'
+description = 'Reverse-slashed `0`'
+zero = 'reverse-slashed'
+
 [simple.v-one-nobase]
 tag = "cv50"
 sampler = '1'