From 30e5d3c8ec704922bd1c1ce7c0ac7dfc29188a30 Mon Sep 17 00:00:00 2001
From: Hannah Li <hannahli2010@gmail.com>
Date: Thu, 28 Jul 2022 21:07:24 -0400
Subject: [PATCH] Implement Bezier scale and offset (#718)

* Implement scale and offset

* Add tests for offset and reduce

* Added scale to documentation page

* address comments

Co-authored-by: Hannah Li <hannahli2010@gmail.com>
Co-authored-by: Thomas Cheng <Androxium@users.noreply.github.com>
Co-authored-by: Rob Nadal <RobNadal@users.noreply.github.com>
---
 bezier-rs/docs/interactive-docs/src/App.vue   |  74 ++++++++----
 .../docs/interactive-docs/wasm/src/lib.rs     |  21 ++--
 bezier-rs/lib/src/lib.rs                      | 106 +++++++++++++++++-
 bezier-rs/lib/src/utils.rs                    |  50 +++++++++
 4 files changed, 217 insertions(+), 34 deletions(-)

diff --git a/bezier-rs/docs/interactive-docs/src/App.vue b/bezier-rs/docs/interactive-docs/src/App.vue
index d93c630c..eb6fabf7 100644
--- a/bezier-rs/docs/interactive-docs/src/App.vue
+++ b/bezier-rs/docs/interactive-docs/src/App.vue
@@ -25,7 +25,7 @@
 <script lang="ts">
 import { defineComponent, markRaw } from "vue";
 
-import { drawText, drawPoint, drawBezier, drawLine, getContextFromCanvas, drawBezierHelper, COLORS } from "@/utils/drawing";
+import { drawBezier, drawBezierHelper, drawCurve, drawLine, drawPoint, drawText, getContextFromCanvas, COLORS } from "@/utils/drawing";
 import { BezierCurveType, Point, WasmBezierInstance, WasmSubpathInstance } from "@/utils/types";
 
 import ExamplePane from "@/components/ExamplePane.vue";
@@ -194,8 +194,8 @@ export default defineComponent({
 						const normal = JSON.parse(bezier.normal(options.t));
 
 						const normalEnd = {
-							x: intersection.x - normal.x * SCALE_UNIT_VECTOR_FACTOR,
-							y: intersection.y - normal.y * SCALE_UNIT_VECTOR_FACTOR,
+							x: intersection.x + normal.x * SCALE_UNIT_VECTOR_FACTOR,
+							y: intersection.y + normal.y * SCALE_UNIT_VECTOR_FACTOR,
 						};
 
 						drawPoint(context, intersection, 3, COLORS.NON_INTERACTIVE.STROKE_1);
@@ -283,6 +283,31 @@ export default defineComponent({
 						],
 					},
 				},
+				{
+					name: "Bounding Box",
+					callback: (canvas: HTMLCanvasElement, bezier: WasmBezierInstance): void => {
+						const context = getContextFromCanvas(canvas);
+						const bboxPoints: Point[] = JSON.parse(bezier.bounding_box());
+						const minPoint = bboxPoints[0];
+						const maxPoint = bboxPoints[1];
+						drawLine(context, minPoint, { x: minPoint.x, y: maxPoint.y }, COLORS.NON_INTERACTIVE.STROKE_1);
+						drawLine(context, minPoint, { x: maxPoint.x, y: minPoint.y }, COLORS.NON_INTERACTIVE.STROKE_1);
+						drawLine(context, maxPoint, { x: minPoint.x, y: maxPoint.y }, COLORS.NON_INTERACTIVE.STROKE_1);
+						drawLine(context, maxPoint, { x: maxPoint.x, y: minPoint.y }, COLORS.NON_INTERACTIVE.STROKE_1);
+					},
+				},
+				{
+					name: "Inflections",
+					callback: (canvas: HTMLCanvasElement, bezier: WasmBezierInstance): void => {
+						const context = getContextFromCanvas(canvas);
+						const inflections: number[] = JSON.parse(bezier.inflections());
+						inflections.forEach((t) => {
+							const point = JSON.parse(bezier.evaluate(t));
+							drawPoint(context, point, 4, COLORS.NON_INTERACTIVE.STROKE_1);
+						});
+					},
+					curveDegrees: new Set([BezierCurveType.Cubic]),
+				},
 				{
 					name: "De Casteljau Points",
 					callback: (canvas: HTMLCanvasElement, bezier: WasmBezierInstance, options: Record<string, number>): void => {
@@ -357,29 +382,32 @@ export default defineComponent({
 					},
 				},
 				{
-					name: "Bounding Box",
-					callback: (canvas: HTMLCanvasElement, bezier: WasmBezierInstance): void => {
+					name: "Offset",
+					callback: (canvas: HTMLCanvasElement, bezier: WasmBezierInstance, options: Record<string, number>): void => {
 						const context = getContextFromCanvas(canvas);
-						const bboxPoints: Point[] = JSON.parse(bezier.bounding_box());
-						const minPoint = bboxPoints[0];
-						const maxPoint = bboxPoints[1];
-						drawLine(context, minPoint, { x: minPoint.x, y: maxPoint.y }, COLORS.NON_INTERACTIVE.STROKE_1);
-						drawLine(context, minPoint, { x: maxPoint.x, y: minPoint.y }, COLORS.NON_INTERACTIVE.STROKE_1);
-						drawLine(context, maxPoint, { x: minPoint.x, y: maxPoint.y }, COLORS.NON_INTERACTIVE.STROKE_1);
-						drawLine(context, maxPoint, { x: maxPoint.x, y: minPoint.y }, COLORS.NON_INTERACTIVE.STROKE_1);
-					},
-				},
-				{
-					name: "Inflections",
-					callback: (canvas: HTMLCanvasElement, bezier: WasmBezierInstance): void => {
-						const context = getContextFromCanvas(canvas);
-						const inflections: number[] = JSON.parse(bezier.inflections());
-						inflections.forEach((t) => {
-							const point = JSON.parse(bezier.evaluate(t));
-							drawPoint(context, point, 4, COLORS.NON_INTERACTIVE.STROKE_1);
+						const curves: Point[][] = JSON.parse(bezier.offset(options.distance));
+						curves.forEach((points, index) => {
+							if (points.length === 2) {
+								drawLine(context, points[0], points[1], `hsl(${40 * index}, 100%, 50%)`);
+							} else {
+								drawCurve(context, points, `hsl(${40 * index}, 100%, 50%)`);
+							}
 						});
+						drawPoint(context, curves[0][0], 4, "hsl(0, 100%, 50%)");
+						drawPoint(context, curves[curves.length - 1][curves[0].length - 1], 4, `hsl(${40 * (curves.length - 1)}, 100%, 50%)`);
+					},
+					template: markRaw(SliderExample),
+					templateOptions: {
+						sliders: [
+							{
+								variable: "distance",
+								min: -50,
+								max: 50,
+								step: 1,
+								default: 20,
+							},
+						],
 					},
-					curveDegrees: new Set([BezierCurveType.Cubic]),
 				},
 			],
 			subpathFeatures: [
diff --git a/bezier-rs/docs/interactive-docs/wasm/src/lib.rs b/bezier-rs/docs/interactive-docs/wasm/src/lib.rs
index 90f1658c..c855cb95 100644
--- a/bezier-rs/docs/interactive-docs/wasm/src/lib.rs
+++ b/bezier-rs/docs/interactive-docs/wasm/src/lib.rs
@@ -128,6 +128,16 @@ impl WasmBezier {
 		to_js_value(local_extrema)
 	}
 
+	pub fn bounding_box(&self) -> JsValue {
+		let bbox_points: [Point; 2] = self.0.bounding_box().map(|p| Point { x: p.x, y: p.y });
+		to_js_value(bbox_points)
+	}
+
+	pub fn inflections(&self) -> JsValue {
+		let inflections = self.0.inflections();
+		to_js_value(inflections)
+	}
+
 	pub fn de_casteljau_points(&self, t: f64) -> JsValue {
 		let hull = self
 			.0
@@ -152,13 +162,8 @@ impl WasmBezier {
 		to_js_value(bezier_points)
 	}
 
-	pub fn bounding_box(&self) -> JsValue {
-		let bbox_points: [Point; 2] = self.0.bounding_box().map(|p| Point { x: p.x, y: p.y });
-		to_js_value(bbox_points)
-	}
-
-	pub fn inflections(&self) -> JsValue {
-		let inflections = self.0.inflections();
-		to_js_value(inflections)
+	pub fn offset(&self, distance: f64) -> JsValue {
+		let bezier_points: Vec<Vec<Point>> = self.0.offset(distance).into_iter().map(bezier_to_points).collect();
+		to_js_value(bezier_points)
 	}
 }
diff --git a/bezier-rs/lib/src/lib.rs b/bezier-rs/lib/src/lib.rs
index 89d3760b..0b69929e 100644
--- a/bezier-rs/lib/src/lib.rs
+++ b/bezier-rs/lib/src/lib.rs
@@ -10,7 +10,7 @@ pub use subpath::*;
 use glam::{DMat2, DVec2};
 
 /// Representation of the handle point(s) in a bezier segment.
-#[derive(Copy, Clone)]
+#[derive(Copy, Clone, PartialEq)]
 enum BezierHandles {
 	Linear,
 	/// Handles for a quadratic curve.
@@ -688,6 +688,9 @@ impl Bezier {
 	/// 2. The on-curve point for `t = 0.5` must occur roughly in the center of the polygon defined by the curve's endpoint normals.
 	/// See [the offset section](https://pomax.github.io/bezierinfo/#offsetting) of Pomax's bezier curve primer for more details.
 	fn is_scalable(&self) -> bool {
+		if self.handles == BezierHandles::Linear {
+			return true;
+		}
 		// Verify all the handles are located on a single side of the curve.
 		if let BezierHandles::Cubic { handle_start, handle_end } = self.handles {
 			let angle_1 = (self.end - self.start).angle_between(handle_start - self.start);
@@ -758,6 +761,8 @@ impl Bezier {
 					if f64::abs(t1 - t2) >= step_size {
 						segment = subcurve.trim(t1, t2);
 						result.push(segment);
+					} else {
+						return;
 					}
 					t1 = t2;
 				}
@@ -832,18 +837,72 @@ impl Bezier {
 	pub fn inflections(&self) -> Vec<f64> {
 		self.unrestricted_inflections().into_iter().filter(|&t| t > 0. && t < 1.).collect::<Vec<f64>>()
 	}
+
+	/// Scale will translate a bezier curve a fixed distance away from its original position, and stretch/compress the transformed curve to match the translation ratio.
+	/// Note that not all bezier curves are possible to scale, so this function asserts that the provided curve is scalable.
+	/// A proof for why this is true can be found in the [Curve offsetting section](https://pomax.github.io/bezierinfo/#offsetting) of Pomax's bezier curve primer.
+	/// `scale` takes the parameter `distance`, which is the distance away from the curve that the new one will be scaled to. Positive values will scale the curve in the
+	/// same direction as the endpoint normals, while negative values will scale in the opposite direction.
+	fn scale(&self, distance: f64) -> Bezier {
+		assert!(self.is_scalable(), "The curve provided to scale is not scalable. Reduce the curve first.");
+
+		let normal_start = self.normal(0.);
+		let normal_end = self.normal(1.);
+
+		// If normal unit vectors are equal, then the lines are parallel
+		if normal_start.abs_diff_eq(normal_end, MAX_ABSOLUTE_DIFFERENCE) {
+			return self.translate(distance * normal_start);
+		}
+
+		// Find the intersection point of the endpoint normals
+		let intersection = utils::line_intersection(self.start, normal_start, self.end, normal_end);
+
+		let should_flip_direction = (self.start - intersection).normalize().abs_diff_eq(normal_start, MAX_ABSOLUTE_DIFFERENCE);
+		self.apply_transformation(&|point| {
+			let mut direction_unit_vector = (intersection - point).normalize();
+			if should_flip_direction {
+				direction_unit_vector *= -1.;
+			}
+			point + distance * direction_unit_vector
+		})
+	}
+
+	/// Offset will get all the reduceable subcurves, and for each subcurve, it will scale the subcurve a set distance away from the original curve.
+	/// Note that not all bezier curves are possible to offset, so this function first reduces the curve to scalable segments and then offsets those segments.
+	/// A proof for why this is true can be found in the [Curve offsetting section](https://pomax.github.io/bezierinfo/#offsetting) of Pomax's bezier curve primer.
+	/// Offset takes the following parameter:
+	/// - `distance` - The distance away from the curve that the new one will be offset to. Positive values will offset the curve in the same direction as the endpoint normals,
+	/// while negative values will offset in the opposite direction.
+	pub fn offset(&self, distance: f64) -> Vec<Bezier> {
+		let mut reduced = self.reduce(None);
+		reduced.iter_mut().for_each(|bezier| *bezier = bezier.scale(distance));
+		reduced
+	}
 }
 
 #[cfg(test)]
 mod tests {
 	use super::*;
 	use crate::consts::MAX_ABSOLUTE_DIFFERENCE;
-	use crate::utils;
 
 	use glam::DVec2;
 
+	// Compare points by allowing some maximum absolute difference to account for floating point errors
 	fn compare_points(p1: DVec2, p2: DVec2) -> bool {
-		utils::dvec2_compare(p1, p2, MAX_ABSOLUTE_DIFFERENCE).all()
+		p1.abs_diff_eq(p2, MAX_ABSOLUTE_DIFFERENCE)
+	}
+
+	// Compare vectors of points by allowing some maximum absolute difference to account for floating point errors
+	fn compare_vector_of_points(a: Vec<DVec2>, b: Vec<DVec2>) -> bool {
+		a.len() == b.len() && a.into_iter().zip(b.into_iter()).all(|(p1, p2)| p1.abs_diff_eq(p2, MAX_ABSOLUTE_DIFFERENCE))
+	}
+
+	// Compare vectors of beziers by allowing some maximum absolute difference between points to account for floating point errors
+	fn compare_vector_of_beziers(beziers: Vec<Bezier>, expected_bezier_points: Vec<Vec<DVec2>>) -> bool {
+		beziers
+			.iter()
+			.zip(expected_bezier_points.iter())
+			.all(|(&a, b)| compare_vector_of_points(a.get_points().collect::<Vec<DVec2>>(), b.to_vec()))
 	}
 
 	#[test]
@@ -947,4 +1006,45 @@ mod tests {
 		assert!(compare_points(intersections2[0], p1));
 		assert!(compare_points(intersections2[1], DVec2::new(85.84, 85.84)));
 	}
+
+	#[test]
+	fn offset() {
+		let p1 = DVec2::new(30., 50.);
+		let p2 = DVec2::new(140., 30.);
+		let p3 = DVec2::new(160., 170.);
+		let bezier1 = Bezier::from_quadratic_dvec2(p1, p2, p3);
+		let expected_bezier_points1 = vec![
+			vec![DVec2::new(31.7888, 59.8387), DVec2::new(44.5924, 57.46446), DVec2::new(56.09375, 57.5)],
+			vec![DVec2::new(56.09375, 57.5), DVec2::new(94.94197, 56.5019), DVec2::new(117.6473, 84.5936)],
+			vec![DVec2::new(117.6473, 84.5936), DVec2::new(142.3985, 113.403), DVec2::new(150.1005, 171.4142)],
+		];
+		assert!(compare_vector_of_beziers(bezier1.offset(10.), expected_bezier_points1));
+
+		let p4 = DVec2::new(32., 77.);
+		let p5 = DVec2::new(169., 25.);
+		let p6 = DVec2::new(164., 157.);
+		let bezier2 = Bezier::from_quadratic_dvec2(p4, p5, p6);
+		let expected_bezier_points2 = vec![
+			vec![DVec2::new(42.6458, 105.04758), DVec2::new(75.0218, 91.9939), DVec2::new(98.09357, 92.3043)],
+			vec![DVec2::new(98.09357, 92.3043), DVec2::new(116.5995, 88.5479), DVec2::new(123.9055, 102.0401)],
+			vec![DVec2::new(123.9055, 102.0401), DVec2::new(136.6087, 116.9522), DVec2::new(134.1761, 147.9324)],
+			vec![DVec2::new(134.1761, 147.9324), DVec2::new(134.1812, 151.7987), DVec2::new(134.0215, 155.86445)],
+		];
+		assert!(compare_vector_of_beziers(bezier2.offset(30.), expected_bezier_points2));
+	}
+
+	#[test]
+	fn reduce() {
+		let p1 = DVec2::new(0., 0.);
+		let p2 = DVec2::new(50., 50.);
+		let p3 = DVec2::new(0., 0.);
+		let bezier = Bezier::from_quadratic_dvec2(p1, p2, p3);
+
+		let expected_bezier_points = vec![
+			vec![DVec2::new(0., 0.), DVec2::new(0.5, 0.5), DVec2::new(0.989, 0.989)],
+			vec![DVec2::new(0.989, 0.989), DVec2::new(2.705, 2.705), DVec2::new(4.2975, 4.2975)],
+			vec![DVec2::new(4.2975, 4.2975), DVec2::new(5.6625, 5.6625), DVec2::new(6.9375, 6.9375)],
+		];
+		assert!(compare_vector_of_beziers(bezier.reduce(None), expected_bezier_points));
+	}
 }
diff --git a/bezier-rs/lib/src/utils.rs b/bezier-rs/lib/src/utils.rs
index 2b6bac5a..39ac14b3 100644
--- a/bezier-rs/lib/src/utils.rs
+++ b/bezier-rs/lib/src/utils.rs
@@ -146,6 +146,33 @@ pub fn solve_cubic(a: f64, b: f64, c: f64, d: f64) -> Vec<f64> {
 	}
 }
 
+/// Returns the intersection of two lines. The lines are given by a point on the line and its slope (represented by a vector).
+pub fn line_intersection(point1: DVec2, point1_slope_vector: DVec2, point2: DVec2, point2_slope_vector: DVec2) -> DVec2 {
+	assert!(point1_slope_vector.normalize() != point2_slope_vector.normalize());
+
+	// Find the intersection when the first line is vertical
+	if f64_compare(point1_slope_vector.x, 0., MAX_ABSOLUTE_DIFFERENCE) {
+		let m2 = point2_slope_vector.y / point2_slope_vector.x;
+		let b2 = point2.y - m2 * point2.x;
+		DVec2::new(point1.x, point1.x * m2 + b2)
+	}
+	// Find the intersection when the second line is vertical
+	else if f64_compare(point2_slope_vector.x, 0., MAX_ABSOLUTE_DIFFERENCE) {
+		let m1 = point1_slope_vector.y / point1_slope_vector.x;
+		let b1 = point1.y - m1 * point1.x;
+		DVec2::new(point2.x, point2.x * m1 + b1)
+	}
+	// Find the intersection where neither line is vertical
+	else {
+		let m1 = point1_slope_vector.y / point1_slope_vector.x;
+		let b1 = point1.y - m1 * point1.x;
+		let m2 = point2_slope_vector.y / point2_slope_vector.x;
+		let b2 = point2.y - m2 * point2.x;
+		let intersection_x = (b2 - b1) / (m1 - m2);
+		DVec2::new(intersection_x, intersection_x * m1 + b1)
+	}
+}
+
 /// Compare two `f64` numbers with a provided max absolute value difference.
 pub fn f64_compare(f1: f64, f2: f64, max_abs_diff: f64) -> bool {
 	(f1 - f2).abs() < max_abs_diff
@@ -215,4 +242,27 @@ mod tests {
 		let roots7 = solve_cubic(0., 0., 1., -1.);
 		assert!(roots7 == vec![1.]);
 	}
+
+	#[test]
+	fn test_find_intersection() {
+		// y = 2x + 10
+		// y = 5x + 4
+		// intersect at (2, 14)
+
+		let start1 = DVec2::new(0., 10.);
+		let end1 = DVec2::new(0., 4.);
+		let start_direction1 = DVec2::new(1., 2.);
+		let end_direction1 = DVec2::new(1., 5.);
+		assert!(line_intersection(start1, start_direction1, end1, end_direction1) == DVec2::new(2., 14.));
+
+		// y = x
+		// y = -x + 8
+		// intersect at (4, 4)
+
+		let start2 = DVec2::new(0., 0.);
+		let end2 = DVec2::new(8., 0.);
+		let start_direction2 = DVec2::new(1., 1.);
+		let end_direction2 = DVec2::new(1., -1.);
+		assert!(line_intersection(start2, start_direction2, end2, end_direction2) == DVec2::new(4., 4.));
+	}
 }