Fix broken geometry of tailed i/l under heavy oblique quasi-proportional (#2274)

* Fix broken geometry of tailed `i`/`l` under heavy oblique quasi-proportional.

* Fix remaining broken geometries

changes/29.0.6.md

@@ -4,3 +4,4 @@
 * Make presence of descender serif automatic for LATIN CAPTITAL LETTER BETA (`U+A7B4`).
 * Remove tailless variants for TURNED GREEK SMALL LETTER IOTA (`U+2129`).
 * Make presence of top-right serif automatic for CYRILLIC SMALL LIGATURE EN GHE (`U+04A5`) under `cyrl/en`=`tailed-top-left-serifed`.
+* Fix broken geometry of tailed `i`/`l` under heavy oblique quasi-proportional.

packages/font-glyphs/src/common/shapes.ptl

@@ -430,12 +430,12 @@ glyph-block CommonShapes : begin
 
 	define nHookSegments 12
 	define [HookShape toStraight toFinish isStart args] : begin
-		local [object y tight sw swItalicAdj noAdjTerminalY turnSlope] args
+		local [object y tight sw swItalicAdj noAdjTerminalY turnSlope isTail] args
 
 		local atBottom : toStraight.y > y
 		local ltr : if isStart (toFinish.x < toStraight.x) (toFinish.x > toStraight.x)
 		local dtu : if isStart (y > toFinish.y) (y < toFinish.y)
-		local doSwash : !tight && !isStart && atBottom && para.isItalic && [if (para.slopeAngle >= 0) ltr [not ltr]]
+		local doSwash : !tight && !isStart && atBottom && (para.isItalic || isTail) && [if (para.slopeAngle >= 0) ltr [not ltr]]
 		local superness : if tight DesignParameters.tightHookSuperness DesignParameters.superness
 
 		# Adjust terminal's position if necessary
@@ -527,7 +527,9 @@ glyph-block CommonShapes : begin
 		local-parameter : swItalicAdj    -- sw
 		local-parameter : noAdjTerminalY -- false
 		local-parameter : turnSlope      -- nothing
-		return : Interpolator hookStartBlender : object y tight sw swItalicAdj noAdjTerminalY turnSlope
+		local-parameter : isTail         -- false
+		return : Interpolator hookStartBlender
+			object y tight sw swItalicAdj noAdjTerminalY turnSlope isTail
 
 	glyph-block-export hookend
 	define flex-params [hookend] : begin
@@ -537,7 +539,9 @@ glyph-block CommonShapes : begin
 		local-parameter : swItalicAdj    -- sw
 		local-parameter : noAdjTerminalY -- false
 		local-parameter : turnSlope      -- nothing
-		return : Interpolator hookEndBlender : object y tight sw swItalicAdj noAdjTerminalY turnSlope
+		local-parameter : isTail         -- false
+		return : Interpolator hookEndBlender
+			object y tight sw swItalicAdj noAdjTerminalY turnSlope isTail
 
 	glyph-block-export Ungizmo
 	define [Ungizmo] : glyph-proc

packages/font-glyphs/src/letter/latin/lower-a.ptl

@@ -65,7 +65,7 @@ glyph-block Letter-Latin-Lower-A : begin
 				[if isMask corner flat] (df.rightSB + O) bartop [heading Leftward]
 				curl barSmooth bartop
 				archv
-				g4 (df.leftSB + OX) (bowlArcY1 - [HSwToV Stroke] * leftSlopeS) [heading {.x HVContrast .y leftSlope}]
+				g4 (df.leftSB + OX) (bowlArcY1 - [HSwToV Stroke] * leftSlope) [heading {.x HVContrast .y leftSlope}]
 				arcvh
 				match kind
 					0 : list

packages/font-glyphs/src/letter/latin/lower-il.ptl

@@ -66,14 +66,14 @@ glyph-block Letter-Latin-Lower-I : begin
 			local shift : TailedDotlessIShift df
 			local left : xMiddle - [HSwToV : 0.5 * df.mvs]
 			local right : mix df.leftSB df.rightSB (1.1 - shift)
-			local rightTerm : Math.max right (left + HookX + df.mvs)
+			local rightTerm : Math.max right (left + HVContrast * (df.mvs + 1.1 * fine)) (left + HVContrast * df.mvs + HookX)
 			local middle : mix left right (0.55 * df.div)
 			local hookDepth : Math.max (df.mvs * 0.9) (Hook * [StrokeWidthBlend 0.85 1] * df.div)
 			include : dispiro
 				widths.lhs df.mvs
 				flat left top [heading Downward]
 				curl left (SmallArchDepthB * 0.8)
-				hookend O (turnSlope -- (0.25 * (Stroke - fine) / Stroke))
+				hookend O (turnSlope -- (0.25 * (Stroke - fine) / Stroke)) (isTail -- true)
 				g4   rightTerm hookDepth [widths.lhs fine]
 
 			local xDot : xMiddle + [StrokeWidthBlend 0.25 0] * TanSlope * df.width

packages/geometry/src/curve-util.mjs

@@ -187,14 +187,27 @@ export class RoundCapCurve {
 		const r = mix(this.r0, this.r1, t);
 		const theta = mix(this.theta0, this.theta1, t);
 
-		return {
+		return new Vec2(
-			x: centerX + r * Math.cos(theta) * this.contrast,
+			centerX + r * Math.cos(theta) * this.contrast,
-			y: centerY + r * Math.sin(theta),
+			centerY + r * Math.sin(theta),
-		};
+		);
 	}
 
 	derivative(t) {
-		// TODO: calculate an exact form instead of using finite difference
+		const theta = mix(this.theta0, this.theta1, t);
-		return derivativeFromFiniteDifference(this, t);
+		const r = mix(this.r0, this.r1, t);
+		const dx =
+			this.center1.x -
+			this.center0.x +
+			this.contrast *
+				((this.r1 - this.r0) * Math.cos(theta) -
+					(this.theta1 - this.theta0) * r * Math.sin(theta));
+		const dy =
+			this.center1.y -
+			this.center0.y +
+			((this.r1 - this.r0) * Math.sin(theta) +
+				(this.theta1 - this.theta0) * r * Math.cos(theta));
+
+		return new Vec2(dx, dy);
 	}
 }