MonoGame.Samples/FarseerPhysicsEngine/Collision/Collision.cs

/*
* Farseer Physics Engine based on Box2D.XNA port:
* Copyright (c) 2010 Ian Qvist
* 
* Box2D.XNA port of Box2D:
* Copyright (c) 2009 Brandon Furtwangler, Nathan Furtwangler
*
* Original source Box2D:
* Copyright (c) 2006-2009 Erin Catto http://www.gphysics.com 
* 
* This software is provided 'as-is', without any express or implied 
* warranty.  In no event will the authors be held liable for any damages 
* arising from the use of this software. 
* Permission is granted to anyone to use this software for any purpose, 
* including commercial applications, and to alter it and redistribute it 
* freely, subject to the following restrictions: 
* 1. The origin of this software must not be misrepresented; you must not 
* claim that you wrote the original software. If you use this software 
* in a product, an acknowledgment in the product documentation would be 
* appreciated but is not required. 
* 2. Altered source versions must be plainly marked as such, and must not be 
* misrepresented as being the original software. 
* 3. This notice may not be removed or altered from any source distribution. 
*/

using System;
using System.Diagnostics;
using System.Runtime.InteropServices;
using FarseerPhysics.Collision.Shapes;
using FarseerPhysics.Common;
using Microsoft.Xna.Framework;

namespace FarseerPhysics.Collision
{
    internal enum ContactFeatureType : byte
    {
        Vertex = 0,
        Face = 1,
    }

    /// <summary>
    /// The features that intersect to form the contact point
    /// This must be 4 bytes or less.
    /// </summary>
    public struct ContactFeature
    {
        /// <summary>
        /// Feature index on ShapeA
        /// </summary>
        public byte IndexA;

        /// <summary>
        /// Feature index on ShapeB
        /// </summary>
        public byte IndexB;

        /// <summary>
        /// The feature type on ShapeA
        /// </summary>
        public byte TypeA;

        /// <summary>
        /// The feature type on ShapeB
        /// </summary>
        public byte TypeB;
    }

    /// <summary>
    /// Contact ids to facilitate warm starting.
    /// </summary>
    [StructLayout(LayoutKind.Explicit)]
    public struct ContactID
    {
        /// <summary>
        /// The features that intersect to form the contact point
        /// </summary>
        [FieldOffset(0)]
        public ContactFeature Features;

        /// <summary>
        /// Used to quickly compare contact ids.
        /// </summary>
        [FieldOffset(0)]
        public uint Key;
    }

    /// <summary>
    /// A manifold point is a contact point belonging to a contact
    /// manifold. It holds details related to the geometry and dynamics
    /// of the contact points.
    /// The local point usage depends on the manifold type:
    /// -ShapeType.Circles: the local center of circleB
    /// -SeparationFunction.FaceA: the local center of cirlceB or the clip point of polygonB
    /// -SeparationFunction.FaceB: the clip point of polygonA
    /// This structure is stored across time steps, so we keep it small.
    /// Note: the impulses are used for internal caching and may not
    /// provide reliable contact forces, especially for high speed collisions.
    /// </summary>
    public struct ManifoldPoint
    {
        /// <summary>
        /// Uniquely identifies a contact point between two Shapes
        /// </summary>
        public ContactID Id;

        public Vector2 LocalPoint;

        public float NormalImpulse;

        public float TangentImpulse;
    }

    public enum ManifoldType
    {
        Circles,
        FaceA,
        FaceB
    }

    /// <summary>
    /// A manifold for two touching convex Shapes.
    /// Box2D supports multiple types of contact:
    /// - clip point versus plane with radius
    /// - point versus point with radius (circles)
    /// The local point usage depends on the manifold type:
    /// -ShapeType.Circles: the local center of circleA
    /// -SeparationFunction.FaceA: the center of faceA
    /// -SeparationFunction.FaceB: the center of faceB
    /// Similarly the local normal usage:
    /// -ShapeType.Circles: not used
    /// -SeparationFunction.FaceA: the normal on polygonA
    /// -SeparationFunction.FaceB: the normal on polygonB
    /// We store contacts in this way so that position correction can
    /// account for movement, which is critical for continuous physics.
    /// All contact scenarios must be expressed in one of these types.
    /// This structure is stored across time steps, so we keep it small.
    /// </summary>
    public struct Manifold
    {
        /// <summary>
        /// Not use for Type.SeparationFunction.Points
        /// </summary>
        public Vector2 LocalNormal;

        /// <summary>
        /// Usage depends on manifold type
        /// </summary>
        public Vector2 LocalPoint;

        /// <summary>
        /// The number of manifold points
        /// </summary>
        public int PointCount;

        /// <summary>
        /// The points of contact
        /// </summary>
        public FixedArray2<ManifoldPoint> Points;

        public ManifoldType Type;
    }

    /// <summary>
    /// This is used for determining the state of contact points.
    /// </summary>
    public enum PointState
    {
        /// <summary>
        /// Point does not exist
        /// </summary>
        Null,

        /// <summary>
        /// Point was added in the update
        /// </summary>
        Add,

        /// <summary>
        /// Point persisted across the update
        /// </summary>
        Persist,

        /// <summary>
        /// Point was removed in the update
        /// </summary>
        Remove,
    }

    /// <summary>
    /// Used for computing contact manifolds.
    /// </summary>
    public struct ClipVertex
    {
        public ContactID ID;
        public Vector2 V;
    }

    /// <summary>
    /// Ray-cast input data. The ray extends from p1 to p1 + maxFraction * (p2 - p1).
    /// </summary>
    public struct RayCastInput
    {
        public float MaxFraction;
        public Vector2 Point1, Point2;
    }

    /// <summary>
    /// Ray-cast output data.  The ray hits at p1 + fraction * (p2 - p1), where p1 and p2
    /// come from RayCastInput. 
    /// </summary>
    public struct RayCastOutput
    {
        public float Fraction;
        public Vector2 Normal;
    }

    /// <summary>
    /// An axis aligned bounding box.
    /// </summary>
    public struct AABB
    {
        private static DistanceInput _input = new DistanceInput();

        /// <summary>
        /// The lower vertex
        /// </summary>
        public Vector2 LowerBound;

        /// <summary>
        /// The upper vertex
        /// </summary>
        public Vector2 UpperBound;

        public AABB(Vector2 min, Vector2 max)
            : this(ref min, ref max)
        {
        }

        public AABB(ref Vector2 min, ref Vector2 max)
        {
            LowerBound = min;
            UpperBound = max;
        }

        public AABB(Vector2 center, float width, float height)
        {
            LowerBound = center - new Vector2(width / 2, height / 2);
            UpperBound = center + new Vector2(width / 2, height / 2);
        }

        /// <summary>
        /// Get the center of the AABB.
        /// </summary>
        /// <value></value>
        public Vector2 Center
        {
            get { return 0.5f * (LowerBound + UpperBound); }
        }

        /// <summary>
        /// Get the extents of the AABB (half-widths).
        /// </summary>
        /// <value></value>
        public Vector2 Extents
        {
            get { return 0.5f * (UpperBound - LowerBound); }
        }

        /// <summary>
        /// Get the perimeter length
        /// </summary>
        /// <value></value>
        public float Perimeter
        {
            get
            {
                float wx = UpperBound.X - LowerBound.X;
                float wy = UpperBound.Y - LowerBound.Y;
                return 2.0f * (wx + wy);
            }
        }

        /// <summary>
        /// Gets the vertices of the AABB.
        /// </summary>
        /// <value>The corners of the AABB</value>
        public Vertices Vertices
        {
            get
            {
                Vertices vertices = new Vertices();
                vertices.Add(LowerBound);
                vertices.Add(new Vector2(LowerBound.X, UpperBound.Y));
                vertices.Add(UpperBound);
                vertices.Add(new Vector2(UpperBound.X, LowerBound.Y));
                return vertices;
            }
        }

        /// <summary>
        /// first quadrant
        /// </summary>
        public AABB Q1
        {
            get { return new AABB(Center, UpperBound); }
        }

        public AABB Q2
        {
            get
            {
                return new AABB(new Vector2(LowerBound.X, Center.Y), new Vector2(Center.X, UpperBound.Y));
                ;
            }
        }

        public AABB Q3
        {
            get { return new AABB(LowerBound, Center); }
        }

        public AABB Q4
        {
            get { return new AABB(new Vector2(Center.X, LowerBound.Y), new Vector2(UpperBound.X, Center.Y)); }
        }

        public Vector2[] GetVertices()
        {
            Vector2 p1 = UpperBound;
            Vector2 p2 = new Vector2(UpperBound.X, LowerBound.Y);
            Vector2 p3 = LowerBound;
            Vector2 p4 = new Vector2(LowerBound.X, UpperBound.Y);
            return new[] { p1, p2, p3, p4 };
        }

        /// <summary>
        /// Verify that the bounds are sorted.
        /// </summary>
        /// <returns>
        /// 	<c>true</c> if this instance is valid; otherwise, <c>false</c>.
        /// </returns>
        public bool IsValid()
        {
            Vector2 d = UpperBound - LowerBound;
            bool valid = d.X >= 0.0f && d.Y >= 0.0f;
            valid = valid && LowerBound.IsValid() && UpperBound.IsValid();
            return valid;
        }

        /// <summary>
        /// Combine an AABB into this one.
        /// </summary>
        /// <param name="aabb">The aabb.</param>
        public void Combine(ref AABB aabb)
        {
            LowerBound = Vector2.Min(LowerBound, aabb.LowerBound);
            UpperBound = Vector2.Max(UpperBound, aabb.UpperBound);
        }

        /// <summary>
        /// Combine two AABBs into this one.
        /// </summary>
        /// <param name="aabb1">The aabb1.</param>
        /// <param name="aabb2">The aabb2.</param>
        public void Combine(ref AABB aabb1, ref AABB aabb2)
        {
            LowerBound = Vector2.Min(aabb1.LowerBound, aabb2.LowerBound);
            UpperBound = Vector2.Max(aabb1.UpperBound, aabb2.UpperBound);
        }

        /// <summary>
        /// Does this aabb contain the provided AABB.
        /// </summary>
        /// <param name="aabb">The aabb.</param>
        /// <returns>
        /// 	<c>true</c> if it contains the specified aabb; otherwise, <c>false</c>.
        /// </returns>
        public bool Contains(ref AABB aabb)
        {
            bool result = true;
            result = result && LowerBound.X <= aabb.LowerBound.X;
            result = result && LowerBound.Y <= aabb.LowerBound.Y;
            result = result && aabb.UpperBound.X <= UpperBound.X;
            result = result && aabb.UpperBound.Y <= UpperBound.Y;
            return result;
        }

        /// <summary>
        /// Determines whether the AAABB contains the specified point.
        /// </summary>
        /// <param name="point">The point.</param>
        /// <returns>
        /// 	<c>true</c> if it contains the specified point; otherwise, <c>false</c>.
        /// </returns>
        public bool Contains(ref Vector2 point)
        {
            //using epsilon to try and gaurd against float rounding errors.
            if ((point.X > (LowerBound.X + Settings.Epsilon) && point.X < (UpperBound.X - Settings.Epsilon) &&
                 (point.Y > (LowerBound.Y + Settings.Epsilon) && point.Y < (UpperBound.Y - Settings.Epsilon))))
            {
                return true;
            }
            return false;
        }

        public static bool TestOverlap(AABB a, AABB b)
        {
            return TestOverlap(ref a, ref b);
        }

        public static bool TestOverlap(ref AABB a, ref AABB b)
        {
            Vector2 d1 = b.LowerBound - a.UpperBound;
            Vector2 d2 = a.LowerBound - b.UpperBound;

            if (d1.X > 0.0f || d1.Y > 0.0f)
                return false;

            if (d2.X > 0.0f || d2.Y > 0.0f)
                return false;

            return true;
        }

        public static bool TestOverlap(Shape shapeA, int indexA,
                                       Shape shapeB, int indexB,
                                       ref Transform xfA, ref Transform xfB)
        {
            _input.ProxyA.Set(shapeA, indexA);
            _input.ProxyB.Set(shapeB, indexB);
            _input.TransformA = xfA;
            _input.TransformB = xfB;
            _input.UseRadii = true;

            SimplexCache cache;
            DistanceOutput output;
            Distance.ComputeDistance(out output, out cache, _input);

            return output.Distance < 10.0f * Settings.Epsilon;
        }


        // From Real-time Collision Detection, p179.
        public bool RayCast(out RayCastOutput output, ref RayCastInput input)
        {
            output = new RayCastOutput();

            float tmin = -Settings.MaxFloat;
            float tmax = Settings.MaxFloat;

            Vector2 p = input.Point1;
            Vector2 d = input.Point2 - input.Point1;
            Vector2 absD = MathUtils.Abs(d);

            Vector2 normal = Vector2.Zero;

            for (int i = 0; i < 2; ++i)
            {
                float absD_i = i == 0 ? absD.X : absD.Y;
                float lowerBound_i = i == 0 ? LowerBound.X : LowerBound.Y;
                float upperBound_i = i == 0 ? UpperBound.X : UpperBound.Y;
                float p_i = i == 0 ? p.X : p.Y;

                if (absD_i < Settings.Epsilon)
                {
                    // Parallel.
                    if (p_i < lowerBound_i || upperBound_i < p_i)
                    {
                        return false;
                    }
                }
                else
                {
                    float d_i = i == 0 ? d.X : d.Y;

                    float inv_d = 1.0f / d_i;
                    float t1 = (lowerBound_i - p_i) * inv_d;
                    float t2 = (upperBound_i - p_i) * inv_d;

                    // Sign of the normal vector.
                    float s = -1.0f;

                    if (t1 > t2)
                    {
                        MathUtils.Swap(ref t1, ref t2);
                        s = 1.0f;
                    }

                    // Push the min up
                    if (t1 > tmin)
                    {
                        if (i == 0)
                        {
                            normal.X = s;
                        }
                        else
                        {
                            normal.Y = s;
                        }

                        tmin = t1;
                    }

                    // Pull the max down
                    tmax = Math.Min(tmax, t2);

                    if (tmin > tmax)
                    {
                        return false;
                    }
                }
            }

            // Does the ray start inside the box?
            // Does the ray intersect beyond the max fraction?
            if (tmin < 0.0f || input.MaxFraction < tmin)
            {
                return false;
            }

            // Intersection.
            output.Fraction = tmin;
            output.Normal = normal;
            return true;
        }
    }

    /// <summary>
    /// Edge shape plus more stuff.
    /// </summary>
    public struct FatEdge
    {
        public bool HasVertex0, HasVertex3;
        public Vector2 Normal;
        public Vector2 V0, V1, V2, V3;
    }

    /// <summary>
    /// This lets us treate and edge shape and a polygon in the same
    /// way in the SAT collider.
    /// </summary>
    public class EPProxy
    {
        public Vector2 Centroid;
        public int Count;
        public Vector2[] Normals = new Vector2[Settings.MaxPolygonVertices];
        public Vector2[] Vertices = new Vector2[Settings.MaxPolygonVertices];
    }

    public struct EPAxis
    {
        public int Index;
        public float Separation;
        public EPAxisType Type;
    }

    public enum EPAxisType
    {
        Unknown,
        EdgeA,
        EdgeB,
    }

    public static class Collision
    {
        private static FatEdge _edgeA;

        private static EPProxy _proxyA = new EPProxy();
        private static EPProxy _proxyB = new EPProxy();

        private static Transform _xf;
        private static Vector2 _limit11, _limit12;
        private static Vector2 _limit21, _limit22;
        private static float _radius;
        private static Vector2[] _tmpNormals = new Vector2[2];

        /// <summary>
        /// Evaluate the manifold with supplied transforms. This assumes
        /// modest motion from the original state. This does not change the
        /// point count, impulses, etc. The radii must come from the Shapes
        /// that generated the manifold.
        /// </summary>
        /// <param name="manifold">The manifold.</param>
        /// <param name="transformA">The transform for A.</param>
        /// <param name="radiusA">The radius for A.</param>
        /// <param name="transformB">The transform for B.</param>
        /// <param name="radiusB">The radius for B.</param>
        /// <param name="normal">World vector pointing from A to B</param>
        /// <param name="points">Torld contact point (point of intersection).</param>
        public static void GetWorldManifold(ref Manifold manifold,
                                            ref Transform transformA, float radiusA,
                                            ref Transform transformB, float radiusB, out Vector2 normal,
                                            out FixedArray2<Vector2> points)
        {
            points = new FixedArray2<Vector2>();
            normal = Vector2.Zero;

            if (manifold.PointCount == 0)
            {
                normal = Vector2.UnitY;
                return;
            }

            switch (manifold.Type)
            {
                case ManifoldType.Circles:
                    {
                        Vector2 tmp = manifold.Points[0].LocalPoint;
                        float pointAx = transformA.Position.X + transformA.R.Col1.X * manifold.LocalPoint.X +
                                        transformA.R.Col2.X * manifold.LocalPoint.Y;

                        float pointAy = transformA.Position.Y + transformA.R.Col1.Y * manifold.LocalPoint.X +
                                        transformA.R.Col2.Y * manifold.LocalPoint.Y;

                        float pointBx = transformB.Position.X + transformB.R.Col1.X * tmp.X +
                                        transformB.R.Col2.X * tmp.Y;

                        float pointBy = transformB.Position.Y + transformB.R.Col1.Y * tmp.X +
                                        transformB.R.Col2.Y * tmp.Y;

                        normal.X = 1;
                        normal.Y = 0;

                        float result = (pointAx - pointBx) * (pointAx - pointBx) +
                                       (pointAy - pointBy) * (pointAy - pointBy);
                        if (result > Settings.Epsilon * Settings.Epsilon)
                        {
                            float tmpNormalx = pointBx - pointAx;
                            float tmpNormaly = pointBy - pointAy;
                            float factor = 1f / (float)Math.Sqrt(tmpNormalx * tmpNormalx + tmpNormaly * tmpNormaly);
                            normal.X = tmpNormalx * factor;
                            normal.Y = tmpNormaly * factor;
                        }

                        Vector2 c = Vector2.Zero;
                        c.X = (pointAx + radiusA * normal.X) + (pointBx - radiusB * normal.X);
                        c.Y = (pointAy + radiusA * normal.Y) + (pointBy - radiusB * normal.Y);

                        points[0] = 0.5f * c;
                    }
                    break;

                case ManifoldType.FaceA:
                    {
                        normal.X = transformA.R.Col1.X * manifold.LocalNormal.X +
                                   transformA.R.Col2.X * manifold.LocalNormal.Y;
                        normal.Y = transformA.R.Col1.Y * manifold.LocalNormal.X +
                                   transformA.R.Col2.Y * manifold.LocalNormal.Y;

                        float planePointx = transformA.Position.X + transformA.R.Col1.X * manifold.LocalPoint.X +
                                            transformA.R.Col2.X * manifold.LocalPoint.Y;

                        float planePointy = transformA.Position.Y + transformA.R.Col1.Y * manifold.LocalPoint.X +
                                            transformA.R.Col2.Y * manifold.LocalPoint.Y;

                        for (int i = 0; i < manifold.PointCount; ++i)
                        {
                            Vector2 tmp = manifold.Points[i].LocalPoint;

                            float clipPointx = transformB.Position.X + transformB.R.Col1.X * tmp.X +
                                               transformB.R.Col2.X * tmp.Y;

                            float clipPointy = transformB.Position.Y + transformB.R.Col1.Y * tmp.X +
                                               transformB.R.Col2.Y * tmp.Y;

                            float value = (clipPointx - planePointx) * normal.X + (clipPointy - planePointy) * normal.Y;

                            Vector2 c = Vector2.Zero;
                            c.X = (clipPointx + (radiusA - value) * normal.X) + (clipPointx - radiusB * normal.X);
                            c.Y = (clipPointy + (radiusA - value) * normal.Y) + (clipPointy - radiusB * normal.Y);

                            points[i] = 0.5f * c;
                        }
                    }
                    break;

                case ManifoldType.FaceB:
                    {
                        normal.X = transformB.R.Col1.X * manifold.LocalNormal.X +
                                   transformB.R.Col2.X * manifold.LocalNormal.Y;
                        normal.Y = transformB.R.Col1.Y * manifold.LocalNormal.X +
                                   transformB.R.Col2.Y * manifold.LocalNormal.Y;

                        float planePointx = transformB.Position.X + transformB.R.Col1.X * manifold.LocalPoint.X +
                                            transformB.R.Col2.X * manifold.LocalPoint.Y;

                        float planePointy = transformB.Position.Y + transformB.R.Col1.Y * manifold.LocalPoint.X +
                                            transformB.R.Col2.Y * manifold.LocalPoint.Y;

                        for (int i = 0; i < manifold.PointCount; ++i)
                        {
                            Vector2 tmp = manifold.Points[i].LocalPoint;

                            float clipPointx = transformA.Position.X + transformA.R.Col1.X * tmp.X +
                                               transformA.R.Col2.X * tmp.Y;

                            float clipPointy = transformA.Position.Y + transformA.R.Col1.Y * tmp.X +
                                               transformA.R.Col2.Y * tmp.Y;

                            float value = (clipPointx - planePointx) * normal.X + (clipPointy - planePointy) * normal.Y;

                            Vector2 c = Vector2.Zero;
                            c.X = (clipPointx - radiusA * normal.X) + (clipPointx + (radiusB - value) * normal.X);
                            c.Y = (clipPointy - radiusA * normal.Y) + (clipPointy + (radiusB - value) * normal.Y);

                            points[i] = 0.5f * c;
                        }
                        // Ensure normal points from A to B.
                        normal *= -1;
                    }
                    break;
                default:
                    normal = Vector2.UnitY;
                    break;
            }
        }

        public static void GetPointStates(out FixedArray2<PointState> state1, out FixedArray2<PointState> state2,
                                          ref Manifold manifold1, ref Manifold manifold2)
        {
            state1 = new FixedArray2<PointState>();
            state2 = new FixedArray2<PointState>();

            // Detect persists and removes.
            for (int i = 0; i < manifold1.PointCount; ++i)
            {
                ContactID id = manifold1.Points[i].Id;

                state1[i] = PointState.Remove;

                for (int j = 0; j < manifold2.PointCount; ++j)
                {
                    if (manifold2.Points[j].Id.Key == id.Key)
                    {
                        state1[i] = PointState.Persist;
                        break;
                    }
                }
            }

            // Detect persists and adds.
            for (int i = 0; i < manifold2.PointCount; ++i)
            {
                ContactID id = manifold2.Points[i].Id;

                state2[i] = PointState.Add;

                for (int j = 0; j < manifold1.PointCount; ++j)
                {
                    if (manifold1.Points[j].Id.Key == id.Key)
                    {
                        state2[i] = PointState.Persist;
                        break;
                    }
                }
            }
        }


        /// Compute the collision manifold between two circles.
        public static void CollideCircles(ref Manifold manifold,
                                          CircleShape circleA, ref Transform xfA,
                                          CircleShape circleB, ref Transform xfB)
        {
            manifold.PointCount = 0;

            float pAx = xfA.Position.X + xfA.R.Col1.X * circleA.Position.X + xfA.R.Col2.X * circleA.Position.Y;
            float pAy = xfA.Position.Y + xfA.R.Col1.Y * circleA.Position.X + xfA.R.Col2.Y * circleA.Position.Y;
            float pBx = xfB.Position.X + xfB.R.Col1.X * circleB.Position.X + xfB.R.Col2.X * circleB.Position.Y;
            float pBy = xfB.Position.Y + xfB.R.Col1.Y * circleB.Position.X + xfB.R.Col2.Y * circleB.Position.Y;

            float distSqr = (pBx - pAx) * (pBx - pAx) + (pBy - pAy) * (pBy - pAy);
            float radius = circleA.Radius + circleB.Radius;
            if (distSqr > radius * radius)
            {
                return;
            }

            manifold.Type = ManifoldType.Circles;
            manifold.LocalPoint = circleA.Position;
            manifold.LocalNormal = Vector2.Zero;
            manifold.PointCount = 1;

            ManifoldPoint p0 = manifold.Points[0];

            p0.LocalPoint = circleB.Position;
            p0.Id.Key = 0;

            manifold.Points[0] = p0;
        }

        /// <summary>
        /// Compute the collision manifold between a polygon and a circle.
        /// </summary>
        /// <param name="manifold">The manifold.</param>
        /// <param name="polygonA">The polygon A.</param>
        /// <param name="transformA">The transform of A.</param>
        /// <param name="circleB">The circle B.</param>
        /// <param name="transformB">The transform of B.</param>
        public static void CollidePolygonAndCircle(ref Manifold manifold,
                                                   PolygonShape polygonA, ref Transform transformA,
                                                   CircleShape circleB, ref Transform transformB)
        {
            manifold.PointCount = 0;

            // Compute circle position in the frame of the polygon.
            Vector2 c =
                new Vector2(
                    transformB.Position.X + transformB.R.Col1.X * circleB.Position.X +
                    transformB.R.Col2.X * circleB.Position.Y,
                    transformB.Position.Y + transformB.R.Col1.Y * circleB.Position.X +
                    transformB.R.Col2.Y * circleB.Position.Y);
            Vector2 cLocal =
                new Vector2(
                    (c.X - transformA.Position.X) * transformA.R.Col1.X +
                    (c.Y - transformA.Position.Y) * transformA.R.Col1.Y,
                    (c.X - transformA.Position.X) * transformA.R.Col2.X +
                    (c.Y - transformA.Position.Y) * transformA.R.Col2.Y);

            // Find the min separating edge.
            int normalIndex = 0;
            float separation = -Settings.MaxFloat;
            float radius = polygonA.Radius + circleB.Radius;
            int vertexCount = polygonA.Vertices.Count;

            for (int i = 0; i < vertexCount; ++i)
            {
                Vector2 value1 = polygonA.Normals[i];
                Vector2 value2 = cLocal - polygonA.Vertices[i];
                float s = value1.X * value2.X + value1.Y * value2.Y;

                if (s > radius)
                {
                    // Early out.
                    return;
                }

                if (s > separation)
                {
                    separation = s;
                    normalIndex = i;
                }
            }

            // Vertices that subtend the incident face.
            int vertIndex1 = normalIndex;
            int vertIndex2 = vertIndex1 + 1 < vertexCount ? vertIndex1 + 1 : 0;
            Vector2 v1 = polygonA.Vertices[vertIndex1];
            Vector2 v2 = polygonA.Vertices[vertIndex2];

            // If the center is inside the polygon ...
            if (separation < Settings.Epsilon)
            {
                manifold.PointCount = 1;
                manifold.Type = ManifoldType.FaceA;
                manifold.LocalNormal = polygonA.Normals[normalIndex];
                manifold.LocalPoint = 0.5f * (v1 + v2);

                ManifoldPoint p0 = manifold.Points[0];

                p0.LocalPoint = circleB.Position;
                p0.Id.Key = 0;

                manifold.Points[0] = p0;

                return;
            }

            // Compute barycentric coordinates
            float u1 = (cLocal.X - v1.X) * (v2.X - v1.X) + (cLocal.Y - v1.Y) * (v2.Y - v1.Y);
            float u2 = (cLocal.X - v2.X) * (v1.X - v2.X) + (cLocal.Y - v2.Y) * (v1.Y - v2.Y);

            if (u1 <= 0.0f)
            {
                float r = (cLocal.X - v1.X) * (cLocal.X - v1.X) + (cLocal.Y - v1.Y) * (cLocal.Y - v1.Y);
                if (r > radius * radius)
                {
                    return;
                }

                manifold.PointCount = 1;
                manifold.Type = ManifoldType.FaceA;
                manifold.LocalNormal = cLocal - v1;
                float factor = 1f /
                               (float)
                               Math.Sqrt(manifold.LocalNormal.X * manifold.LocalNormal.X +
                                         manifold.LocalNormal.Y * manifold.LocalNormal.Y);
                manifold.LocalNormal.X = manifold.LocalNormal.X * factor;
                manifold.LocalNormal.Y = manifold.LocalNormal.Y * factor;
                manifold.LocalPoint = v1;

                ManifoldPoint p0b = manifold.Points[0];

                p0b.LocalPoint = circleB.Position;
                p0b.Id.Key = 0;

                manifold.Points[0] = p0b;
            }
            else if (u2 <= 0.0f)
            {
                float r = (cLocal.X - v2.X) * (cLocal.X - v2.X) + (cLocal.Y - v2.Y) * (cLocal.Y - v2.Y);
                if (r > radius * radius)
                {
                    return;
                }

                manifold.PointCount = 1;
                manifold.Type = ManifoldType.FaceA;
                manifold.LocalNormal = cLocal - v2;
                float factor = 1f /
                               (float)
                               Math.Sqrt(manifold.LocalNormal.X * manifold.LocalNormal.X +
                                         manifold.LocalNormal.Y * manifold.LocalNormal.Y);
                manifold.LocalNormal.X = manifold.LocalNormal.X * factor;
                manifold.LocalNormal.Y = manifold.LocalNormal.Y * factor;
                manifold.LocalPoint = v2;

                ManifoldPoint p0c = manifold.Points[0];

                p0c.LocalPoint = circleB.Position;
                p0c.Id.Key = 0;

                manifold.Points[0] = p0c;
            }
            else
            {
                Vector2 faceCenter = 0.5f * (v1 + v2);
                Vector2 value1 = cLocal - faceCenter;
                Vector2 value2 = polygonA.Normals[vertIndex1];
                float separation2 = value1.X * value2.X + value1.Y * value2.Y;
                if (separation2 > radius)
                {
                    return;
                }

                manifold.PointCount = 1;
                manifold.Type = ManifoldType.FaceA;
                manifold.LocalNormal = polygonA.Normals[vertIndex1];
                manifold.LocalPoint = faceCenter;

                ManifoldPoint p0d = manifold.Points[0];

                p0d.LocalPoint = circleB.Position;
                p0d.Id.Key = 0;

                manifold.Points[0] = p0d;
            }
        }

        /// <summary>
        /// Compute the collision manifold between two polygons.
        /// </summary>
        /// <param name="manifold">The manifold.</param>
        /// <param name="polyA">The poly A.</param>
        /// <param name="transformA">The transform A.</param>
        /// <param name="polyB">The poly B.</param>
        /// <param name="transformB">The transform B.</param>
        public static void CollidePolygons(ref Manifold manifold,
                                           PolygonShape polyA, ref Transform transformA,
                                           PolygonShape polyB, ref Transform transformB)
        {
            manifold.PointCount = 0;
            float totalRadius = polyA.Radius + polyB.Radius;

            int edgeA = 0;
            float separationA = FindMaxSeparation(out edgeA, polyA, ref transformA, polyB, ref transformB);
            if (separationA > totalRadius)
                return;

            int edgeB = 0;
            float separationB = FindMaxSeparation(out edgeB, polyB, ref transformB, polyA, ref transformA);
            if (separationB > totalRadius)
                return;

            PolygonShape poly1; // reference polygon
            PolygonShape poly2; // incident polygon
            Transform xf1, xf2;
            int edge1; // reference edge
            bool flip;
            const float k_relativeTol = 0.98f;
            const float k_absoluteTol = 0.001f;

            if (separationB > k_relativeTol * separationA + k_absoluteTol)
            {
                poly1 = polyB;
                poly2 = polyA;
                xf1 = transformB;
                xf2 = transformA;
                edge1 = edgeB;
                manifold.Type = ManifoldType.FaceB;
                flip = true;
            }
            else
            {
                poly1 = polyA;
                poly2 = polyB;
                xf1 = transformA;
                xf2 = transformB;
                edge1 = edgeA;
                manifold.Type = ManifoldType.FaceA;
                flip = false;
            }

            FixedArray2<ClipVertex> incidentEdge;
            FindIncidentEdge(out incidentEdge, poly1, ref xf1, edge1, poly2, ref xf2);

            int count1 = poly1.Vertices.Count;

            int iv1 = edge1;
            int iv2 = edge1 + 1 < count1 ? edge1 + 1 : 0;

            Vector2 v11 = poly1.Vertices[iv1];
            Vector2 v12 = poly1.Vertices[iv2];

            float localTangentX = v12.X - v11.X;
            float localTangentY = v12.Y - v11.Y;

            float factor = 1f / (float)Math.Sqrt(localTangentX * localTangentX + localTangentY * localTangentY);
            localTangentX = localTangentX * factor;
            localTangentY = localTangentY * factor;

            Vector2 localNormal = new Vector2(localTangentY, -localTangentX);
            Vector2 planePoint = 0.5f * (v11 + v12);

            Vector2 tangent = new Vector2(xf1.R.Col1.X * localTangentX + xf1.R.Col2.X * localTangentY,
                                          xf1.R.Col1.Y * localTangentX + xf1.R.Col2.Y * localTangentY);
            float normalx = tangent.Y;
            float normaly = -tangent.X;

            v11 = new Vector2(xf1.Position.X + xf1.R.Col1.X * v11.X + xf1.R.Col2.X * v11.Y,
                              xf1.Position.Y + xf1.R.Col1.Y * v11.X + xf1.R.Col2.Y * v11.Y);
            v12 = new Vector2(xf1.Position.X + xf1.R.Col1.X * v12.X + xf1.R.Col2.X * v12.Y,
                              xf1.Position.Y + xf1.R.Col1.Y * v12.X + xf1.R.Col2.Y * v12.Y);

            // Face offset.
            float frontOffset = normalx * v11.X + normaly * v11.Y;

            // Side offsets, extended by polytope skin thickness.
            float sideOffset1 = -(tangent.X * v11.X + tangent.Y * v11.Y) + totalRadius;
            float sideOffset2 = tangent.X * v12.X + tangent.Y * v12.Y + totalRadius;

            // Clip incident edge against extruded edge1 side edges.
            FixedArray2<ClipVertex> clipPoints1;
            FixedArray2<ClipVertex> clipPoints2;

            // Clip to box side 1
            int np = ClipSegmentToLine(out clipPoints1, ref incidentEdge, -tangent, sideOffset1, iv1);

            if (np < 2)
                return;

            // Clip to negative box side 1
            np = ClipSegmentToLine(out clipPoints2, ref clipPoints1, tangent, sideOffset2, iv2);

            if (np < 2)
            {
                return;
            }

            // Now clipPoints2 contains the clipped points.
            manifold.LocalNormal = localNormal;
            manifold.LocalPoint = planePoint;

            int pointCount = 0;
            for (int i = 0; i < Settings.MaxManifoldPoints; ++i)
            {
                Vector2 value = clipPoints2[i].V;
                float separation = normalx * value.X + normaly * value.Y - frontOffset;

                if (separation <= totalRadius)
                {
                    ManifoldPoint cp = manifold.Points[pointCount];
                    Vector2 tmp = clipPoints2[i].V;
                    float tmp1X = tmp.X - xf2.Position.X;
                    float tmp1Y = tmp.Y - xf2.Position.Y;
                    cp.LocalPoint.X = tmp1X * xf2.R.Col1.X + tmp1Y * xf2.R.Col1.Y;
                    cp.LocalPoint.Y = tmp1X * xf2.R.Col2.X + tmp1Y * xf2.R.Col2.Y;
                    cp.Id = clipPoints2[i].ID;

                    if (flip)
                    {
                        // Swap features
                        ContactFeature cf = cp.Id.Features;
                        cp.Id.Features.IndexA = cf.IndexB;
                        cp.Id.Features.IndexB = cf.IndexA;
                        cp.Id.Features.TypeA = cf.TypeB;
                        cp.Id.Features.TypeB = cf.TypeA;
                    }

                    manifold.Points[pointCount] = cp;

                    ++pointCount;
                }
            }

            manifold.PointCount = pointCount;
        }

        /// <summary>
        /// Compute contact points for edge versus circle.
        /// This accounts for edge connectivity.
        /// </summary>
        /// <param name="manifold">The manifold.</param>
        /// <param name="edgeA">The edge A.</param>
        /// <param name="transformA">The transform A.</param>
        /// <param name="circleB">The circle B.</param>
        /// <param name="transformB">The transform B.</param>
        public static void CollideEdgeAndCircle(ref Manifold manifold,
                                                EdgeShape edgeA, ref Transform transformA,
                                                CircleShape circleB, ref Transform transformB)
        {
            manifold.PointCount = 0;

            // Compute circle in frame of edge
            Vector2 Q = MathUtils.MultiplyT(ref transformA, MathUtils.Multiply(ref transformB, ref circleB._position));

            Vector2 A = edgeA.Vertex1, B = edgeA.Vertex2;
            Vector2 e = B - A;

            // Barycentric coordinates
            float u = Vector2.Dot(e, B - Q);
            float v = Vector2.Dot(e, Q - A);

            float radius = edgeA.Radius + circleB.Radius;

            ContactFeature cf;
            cf.IndexB = 0;
            cf.TypeB = (byte)ContactFeatureType.Vertex;

            Vector2 P, d;

            // Region A
            if (v <= 0.0f)
            {
                P = A;
                d = Q - P;
                float dd;
                Vector2.Dot(ref d, ref d, out dd);
                if (dd > radius * radius)
                {
                    return;
                }

                // Is there an edge connected to A?
                if (edgeA.HasVertex0)
                {
                    Vector2 A1 = edgeA.Vertex0;
                    Vector2 B1 = A;
                    Vector2 e1 = B1 - A1;
                    float u1 = Vector2.Dot(e1, B1 - Q);

                    // Is the circle in Region AB of the previous edge?
                    if (u1 > 0.0f)
                    {
                        return;
                    }
                }

                cf.IndexA = 0;
                cf.TypeA = (byte)ContactFeatureType.Vertex;
                manifold.PointCount = 1;
                manifold.Type = ManifoldType.Circles;
                manifold.LocalNormal = Vector2.Zero;
                manifold.LocalPoint = P;
                ManifoldPoint mp = new ManifoldPoint();
                mp.Id.Key = 0;
                mp.Id.Features = cf;
                mp.LocalPoint = circleB.Position;
                manifold.Points[0] = mp;
                return;
            }

            // Region B
            if (u <= 0.0f)
            {
                P = B;
                d = Q - P;
                float dd;
                Vector2.Dot(ref d, ref d, out dd);
                if (dd > radius * radius)
                {
                    return;
                }

                // Is there an edge connected to B?
                if (edgeA.HasVertex3)
                {
                    Vector2 B2 = edgeA.Vertex3;
                    Vector2 A2 = B;
                    Vector2 e2 = B2 - A2;
                    float v2 = Vector2.Dot(e2, Q - A2);

                    // Is the circle in Region AB of the next edge?
                    if (v2 > 0.0f)
                    {
                        return;
                    }
                }

                cf.IndexA = 1;
                cf.TypeA = (byte)ContactFeatureType.Vertex;
                manifold.PointCount = 1;
                manifold.Type = ManifoldType.Circles;
                manifold.LocalNormal = Vector2.Zero;
                manifold.LocalPoint = P;
                ManifoldPoint mp = new ManifoldPoint();
                mp.Id.Key = 0;
                mp.Id.Features = cf;
                mp.LocalPoint = circleB.Position;
                manifold.Points[0] = mp;
                return;
            }

            // Region AB
            float den;
            Vector2.Dot(ref e, ref e, out den);
            Debug.Assert(den > 0.0f);
            P = (1.0f / den) * (u * A + v * B);
            d = Q - P;
            float dd2;
            Vector2.Dot(ref d, ref d, out dd2);
            if (dd2 > radius * radius)
            {
                return;
            }

            Vector2 n = new Vector2(-e.Y, e.X);
            if (Vector2.Dot(n, Q - A) < 0.0f)
            {
                n = new Vector2(-n.X, -n.Y);
            }
            n.Normalize();

            cf.IndexA = 0;
            cf.TypeA = (byte)ContactFeatureType.Face;
            manifold.PointCount = 1;
            manifold.Type = ManifoldType.FaceA;
            manifold.LocalNormal = n;
            manifold.LocalPoint = A;
            ManifoldPoint mp2 = new ManifoldPoint();
            mp2.Id.Key = 0;
            mp2.Id.Features = cf;
            mp2.LocalPoint = circleB.Position;
            manifold.Points[0] = mp2;
        }

        /// <summary>
        /// Collides and edge and a polygon, taking into account edge adjacency.
        /// </summary>
        /// <param name="manifold">The manifold.</param>
        /// <param name="edgeA">The edge A.</param>
        /// <param name="xfA">The xf A.</param>
        /// <param name="polygonB">The polygon B.</param>
        /// <param name="xfB">The xf B.</param>
        public static void CollideEdgeAndPolygon(ref Manifold manifold,
                                                 EdgeShape edgeA, ref Transform xfA,
                                                 PolygonShape polygonB, ref Transform xfB)
        {
            MathUtils.MultiplyT(ref xfA, ref xfB, out _xf);

            // Edge geometry
            _edgeA.V0 = edgeA.Vertex0;
            _edgeA.V1 = edgeA.Vertex1;
            _edgeA.V2 = edgeA.Vertex2;
            _edgeA.V3 = edgeA.Vertex3;
            Vector2 e = _edgeA.V2 - _edgeA.V1;

            // Normal points outwards in CCW order.
            _edgeA.Normal = new Vector2(e.Y, -e.X);
            _edgeA.Normal.Normalize();
            _edgeA.HasVertex0 = edgeA.HasVertex0;
            _edgeA.HasVertex3 = edgeA.HasVertex3;

            // Proxy for edge
            _proxyA.Vertices[0] = _edgeA.V1;
            _proxyA.Vertices[1] = _edgeA.V2;
            _proxyA.Normals[0] = _edgeA.Normal;
            _proxyA.Normals[1] = -_edgeA.Normal;
            _proxyA.Centroid = 0.5f * (_edgeA.V1 + _edgeA.V2);
            _proxyA.Count = 2;

            // Proxy for polygon
            _proxyB.Count = polygonB.Vertices.Count;
            _proxyB.Centroid = MathUtils.Multiply(ref _xf, ref polygonB.MassData.Centroid);
            for (int i = 0; i < polygonB.Vertices.Count; ++i)
            {
                _proxyB.Vertices[i] = MathUtils.Multiply(ref _xf, polygonB.Vertices[i]);
                _proxyB.Normals[i] = MathUtils.Multiply(ref _xf.R, polygonB.Normals[i]);
            }

            _radius = 2.0f * Settings.PolygonRadius;

            _limit11 = Vector2.Zero;
            _limit12 = Vector2.Zero;
            _limit21 = Vector2.Zero;
            _limit22 = Vector2.Zero;

            //Collide(ref manifold); inline start
            manifold.PointCount = 0;

            //ComputeAdjacency(); inline start
            Vector2 v0 = _edgeA.V0;
            Vector2 v1 = _edgeA.V1;
            Vector2 v2 = _edgeA.V2;
            Vector2 v3 = _edgeA.V3;

            // Determine allowable the normal regions based on adjacency.
            // Note: it may be possible that no normal is admissable.
            Vector2 centerB = _proxyB.Centroid;
            if (_edgeA.HasVertex0)
            {
                Vector2 e0 = v1 - v0;
                Vector2 e1 = v2 - v1;
                Vector2 n0 = new Vector2(e0.Y, -e0.X);
                Vector2 n1 = new Vector2(e1.Y, -e1.X);
                n0.Normalize();
                n1.Normalize();

                bool convex = MathUtils.Cross(n0, n1) >= 0.0f;
                bool front0 = Vector2.Dot(n0, centerB - v0) >= 0.0f;
                bool front1 = Vector2.Dot(n1, centerB - v1) >= 0.0f;

                if (convex)
                {
                    if (front0 || front1)
                    {
                        _limit11 = n1;
                        _limit12 = n0;
                    }
                    else
                    {
                        _limit11 = -n1;
                        _limit12 = -n0;
                    }
                }
                else
                {
                    if (front0 && front1)
                    {
                        _limit11 = n0;
                        _limit12 = n1;
                    }
                    else
                    {
                        _limit11 = -n0;
                        _limit12 = -n1;
                    }
                }
            }
            else
            {
                _limit11 = Vector2.Zero;
                _limit12 = Vector2.Zero;
            }

            if (_edgeA.HasVertex3)
            {
                Vector2 e1 = v2 - v1;
                Vector2 e2 = v3 - v2;
                Vector2 n1 = new Vector2(e1.Y, -e1.X);
                Vector2 n2 = new Vector2(e2.Y, -e2.X);
                n1.Normalize();
                n2.Normalize();

                bool convex = MathUtils.Cross(n1, n2) >= 0.0f;
                bool front1 = Vector2.Dot(n1, centerB - v1) >= 0.0f;
                bool front2 = Vector2.Dot(n2, centerB - v2) >= 0.0f;

                if (convex)
                {
                    if (front1 || front2)
                    {
                        _limit21 = n2;
                        _limit22 = n1;
                    }
                    else
                    {
                        _limit21 = -n2;
                        _limit22 = -n1;
                    }
                }
                else
                {
                    if (front1 && front2)
                    {
                        _limit21 = n1;
                        _limit22 = n2;
                    }
                    else
                    {
                        _limit21 = -n1;
                        _limit22 = -n2;
                    }
                }
            }
            else
            {
                _limit21 = Vector2.Zero;
                _limit22 = Vector2.Zero;
            }

            //ComputeAdjacency(); inline end

            //EPAxis edgeAxis = ComputeEdgeSeparation(); inline start
            EPAxis edgeAxis = ComputeEdgeSeparation();

            // If no valid normal can be found than this edge should not collide.
            // This can happen on the middle edge of a 3-edge zig-zag chain.
            if (edgeAxis.Type == EPAxisType.Unknown)
            {
                return;
            }

            if (edgeAxis.Separation > _radius)
            {
                return;
            }

            EPAxis polygonAxis = ComputePolygonSeparation();
            if (polygonAxis.Type != EPAxisType.Unknown && polygonAxis.Separation > _radius)
            {
                return;
            }

            // Use hysteresis for jitter reduction.
            const float k_relativeTol = 0.98f;
            const float k_absoluteTol = 0.001f;

            EPAxis primaryAxis;
            if (polygonAxis.Type == EPAxisType.Unknown)
            {
                primaryAxis = edgeAxis;
            }
            else if (polygonAxis.Separation > k_relativeTol * edgeAxis.Separation + k_absoluteTol)
            {
                primaryAxis = polygonAxis;
            }
            else
            {
                primaryAxis = edgeAxis;
            }

            EPProxy proxy1;
            EPProxy proxy2;
            FixedArray2<ClipVertex> incidentEdge = new FixedArray2<ClipVertex>();
            if (primaryAxis.Type == EPAxisType.EdgeA)
            {
                proxy1 = _proxyA;
                proxy2 = _proxyB;
                manifold.Type = ManifoldType.FaceA;
            }
            else
            {
                proxy1 = _proxyB;
                proxy2 = _proxyA;
                manifold.Type = ManifoldType.FaceB;
            }

            int edge1 = primaryAxis.Index;

            FindIncidentEdge(ref incidentEdge, proxy1, primaryAxis.Index, proxy2);
            int count1 = proxy1.Count;

            int iv1 = edge1;
            int iv2 = edge1 + 1 < count1 ? edge1 + 1 : 0;

            Vector2 v11 = proxy1.Vertices[iv1];
            Vector2 v12 = proxy1.Vertices[iv2];

            Vector2 tangent = v12 - v11;
            tangent.Normalize();

            Vector2 normal = MathUtils.Cross(tangent, 1.0f);
            Vector2 planePoint = 0.5f * (v11 + v12);

            // Face offset.
            float frontOffset = Vector2.Dot(normal, v11);

            // Side offsets, extended by polytope skin thickness.
            float sideOffset1 = -Vector2.Dot(tangent, v11) + _radius;
            float sideOffset2 = Vector2.Dot(tangent, v12) + _radius;

            // Clip incident edge against extruded edge1 side edges.
            FixedArray2<ClipVertex> clipPoints1;
            FixedArray2<ClipVertex> clipPoints2;
            int np;

            // Clip to box side 1
            np = ClipSegmentToLine(out clipPoints1, ref incidentEdge, -tangent, sideOffset1, iv1);

            if (np < Settings.MaxManifoldPoints)
            {
                return;
            }

            // Clip to negative box side 1
            np = ClipSegmentToLine(out clipPoints2, ref clipPoints1, tangent, sideOffset2, iv2);

            if (np < Settings.MaxManifoldPoints)
            {
                return;
            }

            // Now clipPoints2 contains the clipped points.
            if (primaryAxis.Type == EPAxisType.EdgeA)
            {
                manifold.LocalNormal = normal;
                manifold.LocalPoint = planePoint;
            }
            else
            {
                manifold.LocalNormal = MathUtils.MultiplyT(ref _xf.R, ref normal);
                manifold.LocalPoint = MathUtils.MultiplyT(ref _xf, ref planePoint);
            }

            int pointCount = 0;
            for (int i1 = 0; i1 < Settings.MaxManifoldPoints; ++i1)
            {
                float separation = Vector2.Dot(normal, clipPoints2[i1].V) - frontOffset;

                if (separation <= _radius)
                {
                    ManifoldPoint cp = manifold.Points[pointCount];

                    if (primaryAxis.Type == EPAxisType.EdgeA)
                    {
                        cp.LocalPoint = MathUtils.MultiplyT(ref _xf, clipPoints2[i1].V);
                        cp.Id = clipPoints2[i1].ID;
                    }
                    else
                    {
                        cp.LocalPoint = clipPoints2[i1].V;
                        cp.Id.Features.TypeA = clipPoints2[i1].ID.Features.TypeB;
                        cp.Id.Features.TypeB = clipPoints2[i1].ID.Features.TypeA;
                        cp.Id.Features.IndexA = clipPoints2[i1].ID.Features.IndexB;
                        cp.Id.Features.IndexB = clipPoints2[i1].ID.Features.IndexA;
                    }

                    manifold.Points[pointCount] = cp;

                    ++pointCount;
                }
            }

            manifold.PointCount = pointCount;

            //Collide(ref manifold); inline end
        }

        private static EPAxis ComputeEdgeSeparation()
        {
            // EdgeA separation
            EPAxis bestAxis;
            bestAxis.Type = EPAxisType.Unknown;
            bestAxis.Index = -1;
            bestAxis.Separation = -Settings.MaxFloat;
            _tmpNormals[0] = _edgeA.Normal;
            _tmpNormals[1] = -_edgeA.Normal;

            for (int i = 0; i < 2; ++i)
            {
                Vector2 n = _tmpNormals[i];

                // Adjacency
                bool valid1 = MathUtils.Cross(n, _limit11) >= -Settings.AngularSlop &&
                              MathUtils.Cross(_limit12, n) >= -Settings.AngularSlop;
                bool valid2 = MathUtils.Cross(n, _limit21) >= -Settings.AngularSlop &&
                              MathUtils.Cross(_limit22, n) >= -Settings.AngularSlop;

                if (valid1 == false || valid2 == false)
                {
                    continue;
                }

                EPAxis axis;
                axis.Type = EPAxisType.EdgeA;
                axis.Index = i;
                axis.Separation = Settings.MaxFloat;

                for (int j = 0; j < _proxyB.Count; ++j)
                {
                    float s = Vector2.Dot(n, _proxyB.Vertices[j] - _edgeA.V1);
                    if (s < axis.Separation)
                    {
                        axis.Separation = s;
                    }
                }

                if (axis.Separation > _radius)
                {
                    return axis;
                }

                if (axis.Separation > bestAxis.Separation)
                {
                    bestAxis = axis;
                }
            }

            return bestAxis;
        }

        private static EPAxis ComputePolygonSeparation()
        {
            EPAxis axis;
            axis.Type = EPAxisType.Unknown;
            axis.Index = -1;
            axis.Separation = -Settings.MaxFloat;
            for (int i = 0; i < _proxyB.Count; ++i)
            {
                Vector2 n = -_proxyB.Normals[i];

                // Adjacency
                bool valid1 = MathUtils.Cross(n, _limit11) >= -Settings.AngularSlop &&
                              MathUtils.Cross(_limit12, n) >= -Settings.AngularSlop;
                bool valid2 = MathUtils.Cross(n, _limit21) >= -Settings.AngularSlop &&
                              MathUtils.Cross(_limit22, n) >= -Settings.AngularSlop;

                if (valid1 == false && valid2 == false)
                {
                    continue;
                }

                float s1 = Vector2.Dot(n, _proxyB.Vertices[i] - _edgeA.V1);
                float s2 = Vector2.Dot(n, _proxyB.Vertices[i] - _edgeA.V2);
                float s = Math.Min(s1, s2);

                if (s > _radius)
                {
                    axis.Type = EPAxisType.EdgeB;
                    axis.Index = i;
                    axis.Separation = s;
                }

                if (s > axis.Separation)
                {
                    axis.Type = EPAxisType.EdgeB;
                    axis.Index = i;
                    axis.Separation = s;
                }
            }

            return axis;
        }

        private static void FindIncidentEdge(ref FixedArray2<ClipVertex> c, EPProxy proxy1, int edge1, EPProxy proxy2)
        {
            int count2 = proxy2.Count;

            Debug.Assert(0 <= edge1 && edge1 < proxy1.Count);

            // Get the normal of the reference edge in proxy2's frame.
            Vector2 normal1 = proxy1.Normals[edge1];

            // Find the incident edge on proxy2.
            int index = 0;
            float minDot = float.MaxValue;
            for (int i = 0; i < count2; ++i)
            {
                float dot = Vector2.Dot(normal1, proxy2.Normals[i]);
                if (dot < minDot)
                {
                    minDot = dot;
                    index = i;
                }
            }

            // Build the clip vertices for the incident edge.
            int i1 = index;
            int i2 = i1 + 1 < count2 ? i1 + 1 : 0;

            ClipVertex cTemp = new ClipVertex();
            cTemp.V = proxy2.Vertices[i1];
            cTemp.ID.Features.IndexA = (byte)edge1;
            cTemp.ID.Features.IndexB = (byte)i1;
            cTemp.ID.Features.TypeA = (byte)ContactFeatureType.Face;
            cTemp.ID.Features.TypeB = (byte)ContactFeatureType.Vertex;
            c[0] = cTemp;

            cTemp.V = proxy2.Vertices[i2];
            cTemp.ID.Features.IndexA = (byte)edge1;
            cTemp.ID.Features.IndexB = (byte)i2;
            cTemp.ID.Features.TypeA = (byte)ContactFeatureType.Face;
            cTemp.ID.Features.TypeB = (byte)ContactFeatureType.Vertex;
            c[1] = cTemp;
        }

        /// <summary>
        /// Clipping for contact manifolds.
        /// </summary>
        /// <param name="vOut">The v out.</param>
        /// <param name="vIn">The v in.</param>
        /// <param name="normal">The normal.</param>
        /// <param name="offset">The offset.</param>
        /// <param name="vertexIndexA">The vertex index A.</param>
        /// <returns></returns>
        private static int ClipSegmentToLine(out FixedArray2<ClipVertex> vOut, ref FixedArray2<ClipVertex> vIn,
                                             Vector2 normal, float offset, int vertexIndexA)
        {
            vOut = new FixedArray2<ClipVertex>();

            ClipVertex v0 = vIn[0];
            ClipVertex v1 = vIn[1];

            // Start with no output points
            int numOut = 0;

            // Calculate the distance of end points to the line
            float distance0 = normal.X * v0.V.X + normal.Y * v0.V.Y - offset;
            float distance1 = normal.X * v1.V.X + normal.Y * v1.V.Y - offset;

            // If the points are behind the plane
            if (distance0 <= 0.0f) vOut[numOut++] = v0;
            if (distance1 <= 0.0f) vOut[numOut++] = v1;

            // If the points are on different sides of the plane
            if (distance0 * distance1 < 0.0f)
            {
                // Find intersection point of edge and plane
                float interp = distance0 / (distance0 - distance1);

                ClipVertex cv = vOut[numOut];

                cv.V.X = v0.V.X + interp * (v1.V.X - v0.V.X);
                cv.V.Y = v0.V.Y + interp * (v1.V.Y - v0.V.Y);

                // VertexA is hitting edgeB.
                cv.ID.Features.IndexA = (byte)vertexIndexA;
                cv.ID.Features.IndexB = v0.ID.Features.IndexB;
                cv.ID.Features.TypeA = (byte)ContactFeatureType.Vertex;
                cv.ID.Features.TypeB = (byte)ContactFeatureType.Face;

                vOut[numOut] = cv;

                ++numOut;
            }

            return numOut;
        }

        /// <summary>
        /// Find the separation between poly1 and poly2 for a give edge normal on poly1.
        /// </summary>
        /// <param name="poly1">The poly1.</param>
        /// <param name="xf1">The XF1.</param>
        /// <param name="edge1">The edge1.</param>
        /// <param name="poly2">The poly2.</param>
        /// <param name="xf2">The XF2.</param>
        /// <returns></returns>
        private static float EdgeSeparation(PolygonShape poly1, ref Transform xf1, int edge1,
                                            PolygonShape poly2, ref Transform xf2)
        {
            int count2 = poly2.Vertices.Count;

            Debug.Assert(0 <= edge1 && edge1 < poly1.Vertices.Count);

            // Convert normal from poly1's frame into poly2's frame.
            Vector2 p1n = poly1.Normals[edge1];

            float normalWorldx = xf1.R.Col1.X * p1n.X + xf1.R.Col2.X * p1n.Y;
            float normalWorldy = xf1.R.Col1.Y * p1n.X + xf1.R.Col2.Y * p1n.Y;

            Vector2 normal = new Vector2(normalWorldx * xf2.R.Col1.X + normalWorldy * xf2.R.Col1.Y,
                                         normalWorldx * xf2.R.Col2.X + normalWorldy * xf2.R.Col2.Y);

            // Find support vertex on poly2 for -normal.
            int index = 0;
            float minDot = Settings.MaxFloat;

            for (int i = 0; i < count2; ++i)
            {
                float dot = Vector2.Dot(poly2.Vertices[i], normal);

                if (dot < minDot)
                {
                    minDot = dot;
                    index = i;
                }
            }

            Vector2 p1ve = poly1.Vertices[edge1];
            Vector2 p2vi = poly2.Vertices[index];

            return ((xf2.Position.X + xf2.R.Col1.X * p2vi.X + xf2.R.Col2.X * p2vi.Y) -
                    (xf1.Position.X + xf1.R.Col1.X * p1ve.X + xf1.R.Col2.X * p1ve.Y)) * normalWorldx +
                   ((xf2.Position.Y + xf2.R.Col1.Y * p2vi.X + xf2.R.Col2.Y * p2vi.Y) -
                    (xf1.Position.Y + xf1.R.Col1.Y * p1ve.X + xf1.R.Col2.Y * p1ve.Y)) * normalWorldy;
        }

        /// <summary>
        /// Find the max separation between poly1 and poly2 using edge normals from poly1.
        /// </summary>
        /// <param name="edgeIndex">Index of the edge.</param>
        /// <param name="poly1">The poly1.</param>
        /// <param name="xf1">The XF1.</param>
        /// <param name="poly2">The poly2.</param>
        /// <param name="xf2">The XF2.</param>
        /// <returns></returns>
        private static float FindMaxSeparation(out int edgeIndex,
                                               PolygonShape poly1, ref Transform xf1,
                                               PolygonShape poly2, ref Transform xf2)
        {
            int count1 = poly1.Vertices.Count;

            // Vector pointing from the centroid of poly1 to the centroid of poly2.
            float dx = (xf2.Position.X + xf2.R.Col1.X * poly2.MassData.Centroid.X +
                        xf2.R.Col2.X * poly2.MassData.Centroid.Y) -
                       (xf1.Position.X + xf1.R.Col1.X * poly1.MassData.Centroid.X +
                        xf1.R.Col2.X * poly1.MassData.Centroid.Y);
            float dy = (xf2.Position.Y + xf2.R.Col1.Y * poly2.MassData.Centroid.X +
                        xf2.R.Col2.Y * poly2.MassData.Centroid.Y) -
                       (xf1.Position.Y + xf1.R.Col1.Y * poly1.MassData.Centroid.X +
                        xf1.R.Col2.Y * poly1.MassData.Centroid.Y);
            Vector2 dLocal1 = new Vector2(dx * xf1.R.Col1.X + dy * xf1.R.Col1.Y, dx * xf1.R.Col2.X + dy * xf1.R.Col2.Y);

            // Find edge normal on poly1 that has the largest projection onto d.
            int edge = 0;
            float maxDot = -Settings.MaxFloat;
            for (int i = 0; i < count1; ++i)
            {
                float dot = Vector2.Dot(poly1.Normals[i], dLocal1);
                if (dot > maxDot)
                {
                    maxDot = dot;
                    edge = i;
                }
            }

            // Get the separation for the edge normal.
            float s = EdgeSeparation(poly1, ref xf1, edge, poly2, ref xf2);

            // Check the separation for the previous edge normal.
            int prevEdge = edge - 1 >= 0 ? edge - 1 : count1 - 1;
            float sPrev = EdgeSeparation(poly1, ref xf1, prevEdge, poly2, ref xf2);

            // Check the separation for the next edge normal.
            int nextEdge = edge + 1 < count1 ? edge + 1 : 0;
            float sNext = EdgeSeparation(poly1, ref xf1, nextEdge, poly2, ref xf2);

            // Find the best edge and the search direction.
            int bestEdge;
            float bestSeparation;
            int increment;
            if (sPrev > s && sPrev > sNext)
            {
                increment = -1;
                bestEdge = prevEdge;
                bestSeparation = sPrev;
            }
            else if (sNext > s)
            {
                increment = 1;
                bestEdge = nextEdge;
                bestSeparation = sNext;
            }
            else
            {
                edgeIndex = edge;
                return s;
            }

            // Perform a local search for the best edge normal.
            for (; ; )
            {
                if (increment == -1)
                    edge = bestEdge - 1 >= 0 ? bestEdge - 1 : count1 - 1;
                else
                    edge = bestEdge + 1 < count1 ? bestEdge + 1 : 0;

                s = EdgeSeparation(poly1, ref xf1, edge, poly2, ref xf2);

                if (s > bestSeparation)
                {
                    bestEdge = edge;
                    bestSeparation = s;
                }
                else
                {
                    break;
                }
            }

            edgeIndex = bestEdge;
            return bestSeparation;
        }

        private static void FindIncidentEdge(out FixedArray2<ClipVertex> c,
                                             PolygonShape poly1, ref Transform xf1, int edge1,
                                             PolygonShape poly2, ref Transform xf2)
        {
            c = new FixedArray2<ClipVertex>();

            int count2 = poly2.Vertices.Count;

            Debug.Assert(0 <= edge1 && edge1 < poly1.Vertices.Count);

            // Get the normal of the reference edge in poly2's frame.
            Vector2 v = poly1.Normals[edge1];
            float tmpx = xf1.R.Col1.X * v.X + xf1.R.Col2.X * v.Y;
            float tmpy = xf1.R.Col1.Y * v.X + xf1.R.Col2.Y * v.Y;
            Vector2 normal1 = new Vector2(tmpx * xf2.R.Col1.X + tmpy * xf2.R.Col1.Y,
                                          tmpx * xf2.R.Col2.X + tmpy * xf2.R.Col2.Y);

            // Find the incident edge on poly2.
            int index = 0;
            float minDot = Settings.MaxFloat;
            for (int i = 0; i < count2; ++i)
            {
                float dot = Vector2.Dot(normal1, poly2.Normals[i]);
                if (dot < minDot)
                {
                    minDot = dot;
                    index = i;
                }
            }

            // Build the clip vertices for the incident edge.
            int i1 = index;
            int i2 = i1 + 1 < count2 ? i1 + 1 : 0;

            ClipVertex cv0 = c[0];

            Vector2 v1 = poly2.Vertices[i1];
            cv0.V.X = xf2.Position.X + xf2.R.Col1.X * v1.X + xf2.R.Col2.X * v1.Y;
            cv0.V.Y = xf2.Position.Y + xf2.R.Col1.Y * v1.X + xf2.R.Col2.Y * v1.Y;
            cv0.ID.Features.IndexA = (byte)edge1;
            cv0.ID.Features.IndexB = (byte)i1;
            cv0.ID.Features.TypeA = (byte)ContactFeatureType.Face;
            cv0.ID.Features.TypeB = (byte)ContactFeatureType.Vertex;

            c[0] = cv0;

            ClipVertex cv1 = c[1];
            Vector2 v2 = poly2.Vertices[i2];
            cv1.V.X = xf2.Position.X + xf2.R.Col1.X * v2.X + xf2.R.Col2.X * v2.Y;
            cv1.V.Y = xf2.Position.Y + xf2.R.Col1.Y * v2.X + xf2.R.Col2.Y * v2.Y;
            cv1.ID.Features.IndexA = (byte)edge1;
            cv1.ID.Features.IndexB = (byte)i2;
            cv1.ID.Features.TypeA = (byte)ContactFeatureType.Face;
            cv1.ID.Features.TypeB = (byte)ContactFeatureType.Vertex;

            c[1] = cv1;
        }
    }
}