Use unit vectors instead of angles for arcs.

This makes things faster and will also simplify the implementation of round
joins.

geometry/src/basic/line_segment.rs

@@ -87,6 +87,11 @@ impl LineSegmentF32 {
         self.0[3] = y
     }
 
+    #[inline]
+    pub fn translate(&self, offset: Point2DF32) -> LineSegmentF32 {
+        LineSegmentF32(self.0 + offset.0.xyxy())
+    }
+
     #[inline]
     pub fn scale(&self, factor: f32) -> LineSegmentF32 {
         LineSegmentF32(self.0 * F32x4::splat(factor))

geometry/src/basic/transform2d.rs

@@ -15,6 +15,7 @@ use crate::basic::point::Point2DF32;
 use crate::basic::rect::RectF32;
 use crate::basic::transform3d::Transform3DF32;
 use crate::segment::Segment;
+use crate::unit_vector::UnitVector;
 use pathfinder_simd::default::F32x4;
 use std::ops::Sub;
 
@@ -37,8 +38,12 @@ impl Matrix2x2F32 {
 
     #[inline]
     pub fn from_rotation(theta: f32) -> Matrix2x2F32 {
-        let (sin_theta, cos_theta) = (theta.sin(), theta.cos());
+        Matrix2x2F32::from_rotation_vector(UnitVector::from_angle(theta))
-        Matrix2x2F32(F32x4::new(cos_theta, sin_theta, -sin_theta, cos_theta))
+    }
+
+    #[inline]
+    pub fn from_rotation_vector(vector: UnitVector) -> Matrix2x2F32 {
+        Matrix2x2F32((vector.0).0.xyyx() * F32x4::new(1.0, 1.0, -1.0, 1.0))
     }
 
     #[inline]
@@ -140,6 +145,14 @@ impl Transform2DF32 {
         }
     }
 
+    #[inline]
+    pub fn from_rotation_vector(vector: UnitVector) -> Transform2DF32 {
+        Transform2DF32 {
+            matrix: Matrix2x2F32::from_rotation_vector(vector),
+            vector: Point2DF32::default(),
+        }
+    }
+
     #[inline]
     pub fn from_translation(vector: Point2DF32) -> Transform2DF32 {
         Transform2DF32 { matrix: Matrix2x2F32::default(), vector }

geometry/src/lib.rs

@@ -27,3 +27,4 @@ pub mod stroke;
 pub mod util;
 
 mod dilation;
+mod unit_vector;

geometry/src/outline.rs

@@ -19,12 +19,11 @@ use crate::clip::{self, ContourPolygonClipper, ContourRectClipper};
 use crate::dilation::ContourDilator;
 use crate::orientation::Orientation;
 use crate::segment::{Segment, SegmentFlags, SegmentKind};
-use std::f32::consts::{FRAC_PI_2, PI};
+use crate::unit_vector::UnitVector;
+use std::f32::consts::PI;
 use std::fmt::{self, Debug, Formatter};
 use std::mem;
 
-const TWO_PI: f32 = PI * 2.0;
-
 #[derive(Clone)]
 pub struct Outline {
     pub(crate) contours: Vec<Contour>,
@@ -369,33 +368,44 @@ impl Contour {
         self.push_point(segment.baseline.to(), PointFlags::empty(), update_bounds);
     }
 
-    pub fn push_arc(&mut self,
+    pub fn push_arc(&mut self, center: Point2DF32, radius: f32, start_angle: f32, end_angle: f32) {
-                    center: Point2DF32,
+        let start = Point2DF32::new(f32::cos(start_angle), f32::sin(start_angle)).scale(radius);
-                    radius: f32,
+        let end = Point2DF32::new(f32::cos(end_angle), f32::sin(end_angle)).scale(radius);
-                    mut start_angle: f32,
+        let chord = LineSegmentF32::new(start, end).translate(center);
-                    mut end_angle: f32) {
+        let full_circle = end_angle - start_angle >= PI * 2.0;
-        start_angle %= TWO_PI;
+        self.push_arc_from_chord_full(center, chord, full_circle);
-        end_angle %= TWO_PI;
+    }
-        if end_angle <= start_angle {
+
-            end_angle += TWO_PI;
+    #[inline]
-        }
+    pub fn push_arc_from_chord(&mut self, center: Point2DF32, chord: LineSegmentF32) {
+        self.push_arc_from_chord_full(center, chord, false);
+    }
+
+    fn push_arc_from_chord_full(&mut self,
+                                center: Point2DF32,
+                                mut chord: LineSegmentF32,
+                                full_circle: bool) {
+        chord = chord.translate(-center);
+        let radius = chord.from().length();
+        chord = chord.scale(radius.recip());
+        let (mut vector, end_vector) = (UnitVector(chord.from()), UnitVector(chord.to()));
 
         let scale = Transform2DF32::from_scale(Point2DF32::splat(radius));
         let translation = Transform2DF32::from_translation(center);
 
-        let (mut angle, mut first_segment) = (start_angle, true);
+        let mut first_segment = true;
-        while angle < end_angle {
+        loop {
-            let sweep_angle = f32::min(FRAC_PI_2, end_angle - angle);
+            let mut sweep_vector = end_vector.rev_rotate_by(vector);
-            let mut segment = Segment::arc(sweep_angle);
+            let last = !(full_circle && first_segment) &&
-            let rotation = Transform2DF32::from_rotation(sweep_angle * 0.5 + angle);
+                sweep_vector.0.x() >= -EPSILON && sweep_vector.0.y() >= -EPSILON;
-            segment = segment.transform(&scale.post_mul(&rotation).post_mul(&translation));
+            if !last {
+                sweep_vector = UnitVector(Point2DF32::new(0.0, 1.0));
+            }
 
-            debug!("angle={} start_angle={} end_angle={} sweep_angle={} segment={:?}",
+            let mut segment = Segment::arc_from_cos(sweep_vector.0.x());
-                   angle,
+            let rotation =
-                   start_angle,
+                Transform2DF32::from_rotation_vector(sweep_vector.halve_angle().rotate_by(vector));
-                   end_angle,
+            segment = segment.transform(&scale.post_mul(&rotation).post_mul(&translation));
-                   sweep_angle,
-                   segment);
 
             if first_segment {
                 self.push_full_segment(&segment, true);
@@ -404,8 +414,14 @@ impl Contour {
                 self.push_segment(segment, true);
             }
 
-            angle += sweep_angle;
+            if last {
+                break;
+            }
+
+            vector = sweep_vector.rotate_by(vector);
         }
+
+        const EPSILON: f32 = 0.001;
     }
 
     #[inline]

geometry/src/segment.rs

@@ -70,14 +70,22 @@ impl Segment {
 
     /// Approximates an unit-length arc with a cubic Bézier curve.
     /// 
-    /// The maximum supported `sweep_angle` is π/2 (i.e. 90°).
+    /// The maximum supported sweep angle is π/2 (i.e. 90°).
     pub fn arc(sweep_angle: f32) -> Segment {
+        Segment::arc_from_cos(f32::cos(sweep_angle))
+    }
+
+    /// Approximates an unit-length arc with a cubic Bézier curve, given the cosine of the sweep
+    /// angle.
+    ///
+    /// The maximum supported sweep angle is π/2 (i.e. 90°).
+    pub fn arc_from_cos(cos_sweep_angle: f32) -> Segment {
         // Aleksas Riškus, "Approximation of a Cubic Bézier Curve by Circular Arcs and Vice Versa"
         // 2006.
         //
         // https://pdfs.semanticscholar.org/1639/0db1a470bd13fe428e0896671a9a5745070a.pdf
-        let phi = 0.5 * sweep_angle;
+        let term = F32x4::new(cos_sweep_angle, -cos_sweep_angle, 0.0, 0.0);
-        let p0 = Point2DF32::new(f32::cos(phi), f32::sin(phi));
+        let p0 = Point2DF32((F32x4::splat(0.5) * (F32x4::splat(1.0) + term)).sqrt());
         let p3 = p0.scale_xy(Point2DF32::new(1.0, -1.0));
         let p1 = p0 - p3.yx().scale(K);
         let p2 = p3 + p0.yx().scale(K);

geometry/src/stroke.rs

@@ -15,7 +15,6 @@ use crate::basic::point::Point2DF32;
 use crate::basic::rect::RectF32;
 use crate::outline::{Contour, Outline};
 use crate::segment::Segment;
-use std::f32::consts::FRAC_PI_2;
 use std::f32;
 use std::mem;
 
@@ -124,10 +123,9 @@ impl OutlineStrokeToFill {
                 contour.push_endpoint(p4);
             }
             LineCap::Round => {
-                // FIXME(pcwalton): Should we really be using angles here at all?
+                let offset = gradient.yx().scale_xy(Point2DF32::new(-width, width));
-                let offset = gradient.yx().scale_xy(Point2DF32::new(-width * 0.5, width * 0.5));
+                let chord = LineSegmentF32::new(p1, p1 + offset);
-                let angle = f32::atan2(gradient.y(), gradient.x());
+                contour.push_arc_from_chord(p1 + offset.scale(0.5), chord);
-                contour.push_arc(p1 + offset, width * 0.5, angle - FRAC_PI_2, angle + FRAC_PI_2);
             }
         }
     }

geometry/src/unit_vector.rs

@@ -0,0 +1,46 @@
+// pathfinder/geometry/src/unit_vector.rs
+//
+// Copyright © 2019 The Pathfinder Project Developers.
+//
+// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
+// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
+// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
+// option. This file may not be copied, modified, or distributed
+// except according to those terms.
+
+//! A utility module that allows unit vectors to be treated like angles.
+
+use crate::basic::point::Point2DF32;
+use pathfinder_simd::default::F32x4;
+
+#[derive(Clone, Copy, Debug)]
+pub struct UnitVector(pub Point2DF32);
+
+impl UnitVector {
+    #[inline]
+    pub fn from_angle(theta: f32) -> UnitVector {
+        UnitVector(Point2DF32::new(theta.cos(), theta.sin()))
+    }
+
+    /// Angle addition formula.
+    #[inline]
+    pub fn rotate_by(&self, other: UnitVector) -> UnitVector {
+        let products = (self.0).0.xyyx() * (other.0).0.xyxy();
+        UnitVector(Point2DF32::new(products[0] - products[1], products[2] + products[3]))
+    }
+
+    /// Angle subtraction formula.
+    #[inline]
+    pub fn rev_rotate_by(&self, other: UnitVector) -> UnitVector {
+        let products = (self.0).0.xyyx() * (other.0).0.xyxy();
+        UnitVector(Point2DF32::new(products[0] + products[1], products[2] - products[3]))
+    }
+
+    /// Half angle formula.
+    #[inline]
+    pub fn halve_angle(&self) -> UnitVector {
+        let x = self.0.x();
+        let term = F32x4::new(x, -x, 0.0, 0.0);
+        UnitVector(Point2DF32((F32x4::splat(0.5) * (F32x4::splat(1.0) + term)).sqrt()))
+    }
+}

simd/src/arm/mod.rs

@@ -76,6 +76,11 @@ impl F32x4 {
         unsafe { F32x4(round_v4f32(self.0)) }
     }
 
+    #[inline]
+    pub fn sqrt(self) -> F32x4 {
+        unsafe { F32x4(sqrt_v4f32(self.0)) }
+    }
+
     // Packed comparisons
 
     #[inline]
@@ -414,6 +419,8 @@ extern "C" {
     fn ceil_v4f32(a: float32x4_t) -> float32x4_t;
     #[link_name = "llvm.round.v4f32"]
     fn round_v4f32(a: float32x4_t) -> float32x4_t;
+    #[link_name = "llvm.sqrt.v4f32"]
+    fn sqrt_v4f32(a: float32x4_t) -> float32x4_t;
 
     #[link_name = "llvm.aarch64.neon.frecpe.v4f32"]
     fn vrecpe_v4f32(a: float32x4_t) -> float32x4_t;

simd/src/scalar/mod.rs

@@ -99,6 +99,16 @@ impl F32x4 {
         ])
     }
 
+    #[inline]
+    pub fn sqrt(self) -> F32x4 {
+        F32x4([
+            self[0].sqrt(),
+            self[1].sqrt(),
+            self[2].sqrt(),
+            self[3].sqrt(),
+        ])
+    }
+
     // Packed comparisons
 
     #[inline]

simd/src/x86/mod.rs

@@ -83,6 +83,11 @@ impl F32x4 {
         unsafe { F32x4(x86_64::_mm_round_ps(self.0, _MM_FROUND_TO_NEAREST_INT)) }
     }
 
+    #[inline]
+    pub fn sqrt(self) -> F32x4 {
+        unsafe { F32x4(x86_64::_mm_sqrt_ps(self.0)) }
+    }
+
     // Packed comparisons
 
     #[inline]