Add circular arc approximator for "perfect" sliders.

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

@@ -0,0 +1,102 @@
+//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;
+        }
+    }
+}

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

@@ -83,7 +83,7 @@ namespace osu.Game.Modes.Osu.Objects
 
                     s.Curve = new SliderCurve
                     {
-                        Path = points,
+                        ControlPoints = points,
                         Length = length,
                         CurveType = curveType
                     };

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

@@ -4,9 +4,8 @@
 using System.Collections.Generic;
 using OpenTK;
 using System.Linq;
-using System.Diagnostics;
 using osu.Framework.MathUtils;
-using System;
+using System.Diagnostics;
 
 namespace osu.Game.Modes.Osu.Objects
 {
@@ -14,21 +13,39 @@ namespace osu.Game.Modes.Osu.Objects
     {
         public double Length;
 
-        public List<Vector2> Path;
+        public List<Vector2> ControlPoints;
 
         public CurveTypes CurveType;
 
         private List<Vector2> calculatedPath = new List<Vector2>();
         private List<double> cumulativeLength = new List<double>();
 
-        private List<Vector2> calculateSubpath(List<Vector2> subpath)
+        private List<Vector2> calculateSubpath(List<Vector2> subControlPoints)
         {
             switch (CurveType)
             {
                 case CurveTypes.Linear:
+                    return subControlPoints;
+                case CurveTypes.PerfectCurve:
+                    // If we have a different amount than 3 control points, use bezier for perfect curves.
+                    if (ControlPoints.Count != 3)
+                        return new BezierApproximator(subControlPoints).CreateBezier();
+                    else
+                    {
+                        Debug.Assert(subControlPoints.Count == 3);
+
+                        // Here we have exactly 3 control points. Attempt to fit a circular arc.
+                        List<Vector2> subpath = new CircularArcApproximator(subControlPoints[0], subControlPoints[1], subControlPoints[2]).CreateArc();
+
+                        if (subpath.Count == 0)
+                            // For some reason a circular arc could not be fit to the 3 given points. Fall back
+                            // to a numerically stable bezier approximation.
+                            subpath = new BezierApproximator(subControlPoints).CreateBezier();
+
                         return subpath;
+                    }
                 default:
-                    return new BezierApproximator(subpath).CreateBezier();
+                    return new BezierApproximator(subControlPoints).CreateBezier();
             }
         }
 
@@ -39,21 +56,19 @@ namespace osu.Game.Modes.Osu.Objects
             // Sliders may consist of various subpaths separated by two consecutive vertices
             // with the same position. The following loop parses these subpaths and computes
             // their shape independently, consecutively appending them to calculatedPath.
-            List<Vector2> subpath = new List<Vector2>();
-            for (int i = 0; i < Path.Count; ++i)
+            List<Vector2> subControlPoints = new List<Vector2>();
+            for (int i = 0; i < ControlPoints.Count; ++i)
             {
-                subpath.Add(Path[i]);
-                if (i == Path.Count - 1 || Path[i] == Path[i + 1])
+                subControlPoints.Add(ControlPoints[i]);
+                if (i == ControlPoints.Count - 1 || ControlPoints[i] == ControlPoints[i + 1])
                 {
-                    // If we already constructed a subpath previously, then the new subpath
-                    // will have as starting position the end position of the previous subpath.
-                    // Hence we can and should remove the previous endpoint to avoid a segment
-                    // with 0 length.
-                    if (calculatedPath.Count > 0)
-                        calculatedPath.RemoveAt(calculatedPath.Count - 1);
+                    List<Vector2> subpath = calculateSubpath(subControlPoints);
+                    for (int j = 0; j < subpath.Count; ++j)
+                        // Only add those vertices that add a new segment to the path.
+                        if (calculatedPath.Count == 0 || calculatedPath.Last() != subpath[j])
+                            calculatedPath.Add(subpath[j]);
 
-                    calculatedPath.AddRange(calculateSubpath(subpath));
-                    subpath.Clear();
+                    subControlPoints.Clear();
                 }
             }
         }

osu.Game.Modes.Osu/osu.Game.Modes.Osu.csproj

@@ -42,6 +42,7 @@
   </ItemGroup>
   <ItemGroup>
     <Compile Include="Objects\BezierApproximator.cs" />
+    <Compile Include="Objects\CircularArcApproximator.cs" />
     <Compile Include="Objects\Drawables\DrawableOsuHitObject.cs" />
     <Compile Include="Objects\Drawables\Pieces\ApproachCircle.cs" />
     <Compile Include="Objects\Drawables\Pieces\CirclePiece.cs" />