Add circular arc approximator for "perfect" sliders.

osu.Game.Modes.Osu/Objects/CircularArcApproximator.cs

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+//Copyright (c) 2007-2016 ppy Pty Ltd <contact@ppy.sh>.
+//Licensed under the MIT Licence - https://raw.githubusercontent.com/ppy/osu/master/LICENCE
+
+using OpenTK;
+using osu.Framework.MathUtils;
+using System;
+using System.Collections.Generic;
+
+namespace osu.Game.Modes.Osu.Objects
+{
+    public class CircularArcApproximator
+    {
+        private Vector2 A;
+        private Vector2 B;
+        private Vector2 C;
+
+        private int amountPoints;
+
+        private const float TOLERANCE = 0.1f;
+
+        public CircularArcApproximator(Vector2 A, Vector2 B, Vector2 C)
+        {
+            this.A = A;
+            this.B = B;
+            this.C = C;
+        }
+
+        /// <summary>
+        /// Creates a piecewise-linear approximation of a circular arc curve.
+        /// </summary>
+        /// <returns>A list of vectors representing the piecewise-linear approximation.</returns>
+        public List<Vector2> CreateArc()
+        {
+            float aSq = (B - C).LengthSquared;
+            float bSq = (A - C).LengthSquared;
+            float cSq = (A - B).LengthSquared;
+
+            // If we have a degenerate triangle where a side-length is almost zero, then give up and fall
+            // back to a more numerically stable method.
+            if (Precision.AlmostEquals(aSq, 0) || Precision.AlmostEquals(bSq, 0) || Precision.AlmostEquals(cSq, 0))
+                return new List<Vector2>();
+
+            float s = aSq * (bSq + cSq - aSq);
+            float t = bSq * (aSq + cSq - bSq);
+            float u = cSq * (aSq + bSq - cSq);
+
+            float sum = s + t + u;
+
+            // If we have a degenerate triangle with an almost-zero size, then give up and fall
+            // back to a more numerically stable method.
+            if (Precision.AlmostEquals(sum, 0))
+                return new List<Vector2>();
+
+            Vector2 centre = (s * A + t * B + u * C) / sum;
+            Vector2 dA = A - centre;
+            Vector2 dC = C - centre;
+
+            float r = dA.Length;
+
+            double thetaStart = Math.Atan2(dA.Y, dA.X);
+            double thetaEnd = Math.Atan2(dC.Y, dC.X);
+
+            while (thetaEnd < thetaStart)
+                thetaEnd += 2 * Math.PI;
+
+            double dir = 1;
+            double thetaRange = thetaEnd - thetaStart;
+
+            // Decide in which direction to draw the circle, depending on which side of 
+            // AC B lies.
+            Vector2 orthoAC = C - A;
+            orthoAC = new Vector2(orthoAC.Y, -orthoAC.X);
+            if (Vector2.Dot(orthoAC, B - A) < 0)
+            {
+                dir = -dir;
+                thetaRange = 2 * Math.PI - thetaRange;
+            }
+
+            // We select the amount of points for the approximation by requiring the discrete curvature
+            // to be smaller than the provided tolerance. The exact angle required to meet the tolerance
+            // is: 2 * Math.Acos(1 - TOLERANCE / r)
+            if (2 * r <= TOLERANCE)
+                // This special case is required for extremely short sliders where the radius is smaller than
+                // the tolerance. This is a pathological rather than a realistic case.
+                amountPoints = 2;
+            else
+                amountPoints = Math.Max(2, (int)Math.Ceiling(thetaRange / (2 * Math.Acos(1 - TOLERANCE / r))));
+
+            List<Vector2> output = new List<Vector2>(amountPoints);
+
+            for (int i = 0; i < amountPoints; ++i)
+            {
+                double fract = (double)i / (amountPoints - 1);
+                double theta = thetaStart + dir * fract * thetaRange;
+                Vector2 o = new Vector2((float)Math.Cos(theta), (float)Math.Sin(theta)) * r;
+                output.Add(centre + o);
+            }
+
+            return output;
+        }
+    }
+}