Plane.cs

// Copyright (c) .NET Foundation and Contributors (https://dotnetfoundation.org/ & https://stride3d.net) and Silicon Studio Corp. (https://www.siliconstudio.co.jp)
// Distributed under the MIT license. See the LICENSE.md file in the project root for more information.
//
// -----------------------------------------------------------------------------
// Original code from SlimMath project. http://code.google.com/p/slimmath/
// Greetings to SlimDX Group. Original code published with the following license:
// -----------------------------------------------------------------------------
/*
* Copyright (c) 2007-2011 SlimDX Group
* 
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
* 
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
* 
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
* THE SOFTWARE.
*/
using System;
using System.Globalization;
using System.Runtime.InteropServices;

namespace Stride.Core.Mathematics
{
    /// <summary>
    /// Represents a plane in three dimensional space.
    /// </summary>
    [DataContract]
    [StructLayout(LayoutKind.Sequential, Pack = 4)]
    public struct Plane : IEquatable<Plane>, IFormattable, IIntersectableWithRay
    {
        /// <summary>
        /// The normal vector of the plane.
        /// </summary>
        public Vector3 Normal;

        /// <summary>
        /// The distance of the plane along its normal from the origin.
        /// </summary>
        public float D;

        /// <summary>
        /// Initializes a new instance of the <see cref="Stride.Core.Mathematics.Plane"/> struct.
        /// </summary>
        /// <param name="value">The value that will be assigned to all components.</param>
        public Plane(float value)
        {
            Normal.X = Normal.Y = Normal.Z = D = value;
        }

        /// <summary>
        /// Initializes a new instance of the <see cref="Stride.Core.Mathematics.Plane"/> struct.
        /// </summary>
        /// <param name="a">The X component of the normal.</param>
        /// <param name="b">The Y component of the normal.</param>
        /// <param name="c">The Z component of the normal.</param>
        /// <param name="d">The distance of the plane along its normal from the origin.</param>
        public Plane(float a, float b, float c, float d)
        {
            Normal.X = a;
            Normal.Y = b;
            Normal.Z = c;
            D = d;
        }

        /// <summary>
        /// Initializes a new instance of the <see cref="Plane"/> struct.
        /// </summary>
        /// <param name="point">Any point that lies along the plane.</param>
        /// <param name="normal">The normal vector to the plane.</param>
        public Plane(Vector3 point, Vector3 normal)
        {
            Normal = normal;
            Vector3.Dot(ref normal, ref point, out D);
        }

        /// <summary>
        /// Initializes a new instance of the <see cref="Stride.Core.Mathematics.Plane"/> struct.
        /// </summary>
        /// <param name="value">The normal of the plane.</param>
        /// <param name="d">The distance of the plane along its normal from the origin</param>
        public Plane(Vector3 value, float d)
        {
            Normal = value;
            D = d;
        }

        /// <summary>
        /// Initializes a new instance of the <see cref="Stride.Core.Mathematics.Plane"/> struct.
        /// </summary>
        /// <param name="point1">First point of a triangle defining the plane.</param>
        /// <param name="point2">Second point of a triangle defining the plane.</param>
        /// <param name="point3">Third point of a triangle defining the plane.</param>
        public Plane(Vector3 point1, Vector3 point2, Vector3 point3)
        {
            float x1 = point2.X - point1.X;
            float y1 = point2.Y - point1.Y;
            float z1 = point2.Z - point1.Z;
            float x2 = point3.X - point1.X;
            float y2 = point3.Y - point1.Y;
            float z2 = point3.Z - point1.Z;
            float yz = (y1 * z2) - (z1 * y2);
            float xz = (z1 * x2) - (x1 * z2);
            float xy = (x1 * y2) - (y1 * x2);
            float invPyth = 1.0f / MathF.Sqrt((yz * yz) + (xz * xz) + (xy * xy));

            Normal.X = yz * invPyth;
            Normal.Y = xz * invPyth;
            Normal.Z = xy * invPyth;
            D = -((Normal.X * point1.X) + (Normal.Y * point1.Y) + (Normal.Z * point1.Z));
        }

        /// <summary>
        /// Initializes a new instance of the <see cref="Stride.Core.Mathematics.Plane"/> struct.
        /// </summary>
        /// <param name="values">The values to assign to the A, B, C, and D components of the plane. This must be an array with four elements.</param>
        /// <exception cref="ArgumentNullException">Thrown when <paramref name="values"/> is <c>null</c>.</exception>
        /// <exception cref="ArgumentOutOfRangeException">Thrown when <paramref name="values"/> contains more or less than four elements.</exception>
        public Plane(float[] values)
        {
            if (values == null)
                throw new ArgumentNullException("values");
            if (values.Length != 4)
                throw new ArgumentOutOfRangeException("values", "There must be four and only four input values for Plane.");

            Normal.X = values[0];
            Normal.Y = values[1];
            Normal.Z = values[2];
            D = values[3];
        }

        /// <summary>
        /// Gets or sets the component at the specified index.
        /// </summary>
        /// <value>The value of the A, B, C, or D component, depending on the index.</value>
        /// <param name="index">The index of the component to access. Use 0 for the A component, 1 for the B component, 2 for the C component, and 3 for the D component.</param>
        /// <returns>The value of the component at the specified index.</returns>
        /// <exception cref="System.ArgumentOutOfRangeException">Thrown when the <paramref name="index"/> is out of the range [0, 3].</exception>
        public float this[int index]
        {
            get
            {
                switch (index)
                {
                    case 0: return Normal.X;
                    case 1: return Normal.Y;
                    case 2: return Normal.Z;
                    case 3: return D;
                }

                throw new ArgumentOutOfRangeException("index", "Indices for Plane run from 0 to 3, inclusive.");
            }

            set
            {
                switch (index)
                {
                    case 0: Normal.X = value; break;
                    case 1: Normal.Y = value; break;
                    case 2: Normal.Z = value; break;
                    case 3: D = value; break;
                    default: throw new ArgumentOutOfRangeException("index", "Indices for Plane run from 0 to 3, inclusive.");
                }
            }
        }

        /// <summary>
        /// Negates a plane by negating all its coefficients, which result in a plane in opposite direction.
        /// </summary>
        public void Negate()
        {
            Normal.X = -Normal.X;
            Normal.Y = -Normal.Y;
            Normal.Z = -Normal.Z;
            D = -D;
        }

        /// <summary>
        /// Changes the coefficients of the normal vector of the plane to make it of unit length.
        /// </summary>
        public void Normalize()
        {
            float magnitude = 1.0f / MathF.Sqrt((Normal.X * Normal.X) + (Normal.Y * Normal.Y) + (Normal.Z * Normal.Z));

            Normal.X *= magnitude;
            Normal.Y *= magnitude;
            Normal.Z *= magnitude;
            D *= magnitude;
        }

        /// <summary>
        /// Creates an array containing the elements of the plane.
        /// </summary>
        /// <returns>A four-element array containing the components of the plane.</returns>
        public float[] ToArray()
        {
            return new float[] { Normal.X, Normal.Y, Normal.Z, D };
        }

        /// <summary>
        /// Determines if there is an intersection between the current object and a point.
        /// </summary>
        /// <param name="point">The point to test.</param>
        /// <returns>Whether the two objects intersected.</returns>
        public PlaneIntersectionType Intersects(ref readonly Vector3 point)
        {
            return CollisionHelper.PlaneIntersectsPoint(ref this, in point);
        }

        /// <summary>
        /// Determines if there is an intersection between the current object and a <see cref="Stride.Core.Mathematics.Ray"/>.
        /// </summary>
        /// <param name="ray">The ray to test.</param>
        /// <returns>Whether the two objects intersected.</returns>
        public bool Intersects(ref readonly Ray ray)
        {
            float distance;
            return CollisionHelper.RayIntersectsPlane(in ray, ref this, out distance);
        }

        /// <summary>
        /// Determines if there is an intersection between the current object and a <see cref="Stride.Core.Mathematics.Ray"/>.
        /// </summary>
        /// <param name="ray">The ray to test.</param>
        /// <param name="distance">When the method completes, contains the distance of the intersection,
        /// or 0 if there was no intersection.</param>
        /// <returns>Whether the two objects intersected.</returns>
        public bool Intersects(ref readonly Ray ray, out float distance)
        {
            return CollisionHelper.RayIntersectsPlane(in ray, ref this, out distance);
        }

        /// <summary>
        /// Determines if there is an intersection between the current object and a <see cref="Stride.Core.Mathematics.Ray"/>.
        /// </summary>
        /// <param name="ray">The ray to test.</param>
        /// <param name="point">When the method completes, contains the point of intersection,
        /// or <see cref="Stride.Core.Mathematics.Vector3.Zero"/> if there was no intersection.</param>
        /// <returns>Whether the two objects intersected.</returns>
        public bool Intersects(ref readonly Ray ray, out Vector3 point)
        {
            return CollisionHelper.RayIntersectsPlane(in ray, ref this, out point);
        }

        /// <summary>
        /// Determines if there is an intersection between the current object and a <see cref="Stride.Core.Mathematics.Plane"/>.
        /// </summary>
        /// <param name="plane">The plane to test.</param>
        /// <returns>Whether the two objects intersected.</returns>
        public bool Intersects(ref readonly Plane plane)
        {
            return CollisionHelper.PlaneIntersectsPlane(ref this, in plane);
        }

        /// <summary>
        /// Determines if there is an intersection between the current object and a <see cref="Stride.Core.Mathematics.Plane"/>.
        /// </summary>
        /// <param name="plane">The plane to test.</param>
        /// <param name="line">When the method completes, contains the line of intersection
        /// as a <see cref="Stride.Core.Mathematics.Ray"/>, or a zero ray if there was no intersection.</param>
        /// <returns>Whether the two objects intersected.</returns>
        public bool Intersects(ref readonly Plane plane, out Ray line)
        {
            return CollisionHelper.PlaneIntersectsPlane(ref this, in plane, out line);
        }

        /// <summary>
        /// Determines if there is an intersection between the current object and a triangle.
        /// </summary>
        /// <param name="vertex1">The first vertex of the triangle to test.</param>
        /// <param name="vertex2">The second vertex of the triagnle to test.</param>
        /// <param name="vertex3">The third vertex of the triangle to test.</param>
        /// <returns>Whether the two objects intersected.</returns>
        public PlaneIntersectionType Intersects(ref readonly Vector3 vertex1, ref readonly Vector3 vertex2, ref readonly Vector3 vertex3)
        {
            return CollisionHelper.PlaneIntersectsTriangle(ref this, in vertex1, in vertex2, in vertex3);
        }

        /// <summary>
        /// Determines if there is an intersection between the current object and a <see cref="Stride.Core.Mathematics.BoundingBox"/>.
        /// </summary>
        /// <param name="box">The box to test.</param>
        /// <returns>Whether the two objects intersected.</returns>
        public PlaneIntersectionType Intersects(ref readonly BoundingBox box)
        {
            return CollisionHelper.PlaneIntersectsBox(ref this, in box);
        }

        /// <summary>
        /// Determines if there is an intersection between the current object and a <see cref="Stride.Core.Mathematics.BoundingSphere"/>.
        /// </summary>
        /// <param name="sphere">The sphere to test.</param>
        /// <returns>Whether the two objects intersected.</returns>
        public PlaneIntersectionType Intersects(ref readonly BoundingSphere sphere)
        {
            return CollisionHelper.PlaneIntersectsSphere(ref this, in sphere);
        }

        /// <summary>
        /// Scales the plane by the given scaling factor.
        /// </summary>
        /// <param name="value">The plane to scale.</param>
        /// <param name="scale">The amount by which to scale the plane.</param>
        /// <param name="result">When the method completes, contains the scaled plane.</param>
        public static void Multiply(ref readonly Plane value, float scale, out Plane result)
        {
            result.Normal.X = value.Normal.X * scale;
            result.Normal.Y = value.Normal.Y * scale;
            result.Normal.Z = value.Normal.Z * scale;
            result.D = value.D * scale;
        }

        /// <summary>
        /// Scales the plane by the given scaling factor.
        /// </summary>
        /// <param name="value">The plane to scale.</param>
        /// <param name="scale">The amount by which to scale the plane.</param>
        /// <returns>The scaled plane.</returns>
        public static Plane Multiply(Plane value, float scale)
        {
            return new Plane(value.Normal.X * scale, value.Normal.Y * scale, value.Normal.Z * scale, value.D * scale);
        }

        /// <summary>
        /// Calculates the dot product of the specified vector and plane.
        /// </summary>
        /// <param name="left">The source plane.</param>
        /// <param name="right">The source vector.</param>
        /// <param name="result">When the method completes, contains the dot product of the specified plane and vector.</param>
        public static void Dot(ref readonly Plane left, ref readonly Vector4 right, out float result)
        {
            result = (left.Normal.X * right.X) + (left.Normal.Y * right.Y) + (left.Normal.Z * right.Z) + (left.D * right.W);
        }

        /// <summary>
        /// Calculates the dot product of the specified vector and plane.
        /// </summary>
        /// <param name="left">The source plane.</param>
        /// <param name="right">The source vector.</param>
        /// <returns>The dot product of the specified plane and vector.</returns>
        public static float Dot(Plane left, Vector4 right)
        {
            return (left.Normal.X * right.X) + (left.Normal.Y * right.Y) + (left.Normal.Z * right.Z) + (left.D * right.W);
        }

        /// <summary>
        /// Calculates the dot product of a specified vector and the normal of the plane plus the distance value of the plane.
        /// </summary>
        /// <param name="left">The source plane.</param>
        /// <param name="right">The source vector.</param>
        /// <param name="result">When the method completes, contains the dot product of a specified vector and the normal of the Plane plus the distance value of the plane.</param>
        public static void DotCoordinate(ref readonly Plane left, ref readonly Vector3 right, out float result)
        {
            result = (left.Normal.X * right.X) + (left.Normal.Y * right.Y) + (left.Normal.Z * right.Z) + left.D;
        }

        /// <summary>
        /// Calculates the dot product of a specified vector and the normal of the plane plus the distance value of the plane.
        /// </summary>
        /// <param name="left">The source plane.</param>
        /// <param name="right">The source vector.</param>
        /// <returns>The dot product of a specified vector and the normal of the Plane plus the distance value of the plane.</returns>
        public static float DotCoordinate(Plane left, Vector3 right)
        {
            return (left.Normal.X * right.X) + (left.Normal.Y * right.Y) + (left.Normal.Z * right.Z) + left.D;
        }

        /// <summary>
        /// Calculates the dot product of the specified vector and the normal of the plane.
        /// </summary>
        /// <param name="left">The source plane.</param>
        /// <param name="right">The source vector.</param>
        /// <param name="result">When the method completes, contains the dot product of the specified vector and the normal of the plane.</param>
        public static void DotNormal(ref readonly Plane left, ref readonly Vector3 right, out float result)
        {
            result = (left.Normal.X * right.X) + (left.Normal.Y * right.Y) + (left.Normal.Z * right.Z);
        }

        /// <summary>
        /// Calculates the dot product of the specified vector and the normal of the plane.
        /// </summary>
        /// <param name="left">The source plane.</param>
        /// <param name="right">The source vector.</param>
        /// <returns>The dot product of the specified vector and the normal of the plane.</returns>
        public static float DotNormal(Plane left, Vector3 right)
        {
            return (left.Normal.X * right.X) + (left.Normal.Y * right.Y) + (left.Normal.Z * right.Z);
        }

        /// <summary>
        /// Projects a point onto a plane.
        /// </summary>
        /// <param name="plane">The plane to project the point to.</param>
        /// <param name="point">The point to project.</param>
        /// <param name="result">The projected point.</param>
        public static void Project(ref readonly Plane plane, ref readonly Vector3 point, out Vector3 result)
        {
            float distance;
            DotCoordinate(in plane, in point, out distance);

            // compute: point - distance * plane.Normal
            Vector3.Multiply(in plane.Normal, distance, out result);
            Vector3.Subtract(in point, ref result, out result);
        }

        /// <summary>
        /// Projects a point onto a plane.
        /// </summary>
        /// <param name="plane">The plane to project the point to.</param>
        /// <param name="point">The point to project.</param>
        /// <returns>The projected point.</returns>
        public static Vector3 Project(Plane plane, Vector3 point)
        {
            Vector3 result;
            Project(ref plane, ref point, out result);
            return result;
        }

        /// <summary>
        /// Creates a plane of unit length.
        /// </summary>
        /// <param name="normalX">The X component of the normal.</param>
        /// <param name="normalY">The Y component of the normal.</param>
        /// <param name="normalZ">The Z component of the normal.</param>
        /// <param name="planeD">The distance of the plane along its normal from the origin.</param>
        /// <param name="result">When the method completes, contains the normalized plane.</param>
        public static void Normalize(float normalX, float normalY, float normalZ, float planeD, out Plane result)
        {
            float magnitude = 1.0f / MathF.Sqrt((normalX * normalX) + (normalY * normalY) + (normalZ * normalZ));

            result.Normal.X = normalX * magnitude;
            result.Normal.Y = normalY * magnitude;
            result.Normal.Z = normalZ * magnitude;
            result.D = planeD * magnitude;
        }

        /// <summary>
        /// Changes the coefficients of the normal vector of the plane to make it of unit length.
        /// </summary>
        /// <param name="plane">The source plane.</param>
        /// <param name="result">When the method completes, contains the normalized plane.</param>
        public static void Normalize(ref readonly Plane plane, out Plane result)
        {
            float magnitude = 1.0f / MathF.Sqrt((plane.Normal.X * plane.Normal.X) + (plane.Normal.Y * plane.Normal.Y) + (plane.Normal.Z * plane.Normal.Z));

            result.Normal.X = plane.Normal.X * magnitude;
            result.Normal.Y = plane.Normal.Y * magnitude;
            result.Normal.Z = plane.Normal.Z * magnitude;
            result.D = plane.D * magnitude;
        }

        /// <summary>
        /// Changes the coefficients of the normal vector of the plane to make it of unit length.
        /// </summary>
        /// <param name="plane">The source plane.</param>
        /// <returns>The normalized plane.</returns>
        public static Plane Normalize(Plane plane)
        {
            float magnitude = 1.0f / MathF.Sqrt((plane.Normal.X * plane.Normal.X) + (plane.Normal.Y * plane.Normal.Y) + (plane.Normal.Z * plane.Normal.Z));
            return new Plane(plane.Normal.X * magnitude, plane.Normal.Y * magnitude, plane.Normal.Z * magnitude, plane.D * magnitude);
        }

        /// <summary>
        /// Negates a plane by negating all its coefficients, which result in a plane in opposite direction.
        /// </summary>
        /// <param name="plane">The source plane.</param>
        /// <param name="result">When the method completes, contains the flipped plane.</param>
        public static void Negate(ref readonly Plane plane, out Plane result)
        {
            result.Normal.X = -plane.Normal.X;
            result.Normal.Y = -plane.Normal.Y;
            result.Normal.Z = -plane.Normal.Z;
            result.D = -plane.D;
        }

        /// <summary>
        /// Negates a plane by negating all its coefficients, which result in a plane in opposite direction.
        /// </summary>
        /// <param name="plane">The source plane.</param>
        /// <returns>The flipped plane.</returns>
        public static Plane Negate(Plane plane)
        {
            float magnitude = 1.0f / MathF.Sqrt((plane.Normal.X * plane.Normal.X) + (plane.Normal.Y * plane.Normal.Y) + (plane.Normal.Z * plane.Normal.Z));
            return new Plane(plane.Normal.X * magnitude, plane.Normal.Y * magnitude, plane.Normal.Z * magnitude, plane.D * magnitude);
        }

        /// <summary>
        /// Transforms a normalized plane by a quaternion rotation.
        /// </summary>
        /// <param name="plane">The normalized source plane.</param>
        /// <param name="rotation">The quaternion rotation.</param>
        /// <param name="result">When the method completes, contains the transformed plane.</param>
        public static void Transform(ref readonly Plane plane, ref readonly Quaternion rotation, out Plane result)
        {
            float x2 = rotation.X + rotation.X;
            float y2 = rotation.Y + rotation.Y;
            float z2 = rotation.Z + rotation.Z;
            float wx = rotation.W * x2;
            float wy = rotation.W * y2;
            float wz = rotation.W * z2;
            float xx = rotation.X * x2;
            float xy = rotation.X * y2;
            float xz = rotation.X * z2;
            float yy = rotation.Y * y2;
            float yz = rotation.Y * z2;
            float zz = rotation.Z * z2;

            float x = plane.Normal.X;
            float y = plane.Normal.Y;
            float z = plane.Normal.Z;

            result.Normal.X = ((x * ((1.0f - yy) - zz)) + (y * (xy - wz))) + (z * (xz + wy));
            result.Normal.Y = ((x * (xy + wz)) + (y * ((1.0f - xx) - zz))) + (z * (yz - wx));
            result.Normal.Z = ((x * (xz - wy)) + (y * (yz + wx))) + (z * ((1.0f - xx) - yy));
            result.D = plane.D;
        }

        /// <summary>
        /// Transforms a normalized plane by a quaternion rotation.
        /// </summary>
        /// <param name="plane">The normalized source plane.</param>
        /// <param name="rotation">The quaternion rotation.</param>
        /// <returns>The transformed plane.</returns>
        public static Plane Transform(Plane plane, Quaternion rotation)
        {
            Plane result;
            float x2 = rotation.X + rotation.X;
            float y2 = rotation.Y + rotation.Y;
            float z2 = rotation.Z + rotation.Z;
            float wx = rotation.W * x2;
            float wy = rotation.W * y2;
            float wz = rotation.W * z2;
            float xx = rotation.X * x2;
            float xy = rotation.X * y2;
            float xz = rotation.X * z2;
            float yy = rotation.Y * y2;
            float yz = rotation.Y * z2;
            float zz = rotation.Z * z2;

            float x = plane.Normal.X;
            float y = plane.Normal.Y;
            float z = plane.Normal.Z;

            result.Normal.X = ((x * ((1.0f - yy) - zz)) + (y * (xy - wz))) + (z * (xz + wy));
            result.Normal.Y = ((x * (xy + wz)) + (y * ((1.0f - xx) - zz))) + (z * (yz - wx));
            result.Normal.Z = ((x * (xz - wy)) + (y * (yz + wx))) + (z * ((1.0f - xx) - yy));
            result.D = plane.D;

            return result;
        }

        /// <summary>
        /// Transforms an array of normalized planes by a quaternion rotation.
        /// </summary>
        /// <param name="planes">The array of normalized planes to transform.</param>
        /// <param name="rotation">The quaternion rotation.</param>
        /// <exception cref="ArgumentNullException">Thrown when <paramref name="planes"/> is <c>null</c>.</exception>
        public static void Transform(Plane[] planes, ref readonly Quaternion rotation)
        {
            if (planes == null)
                throw new ArgumentNullException("planes");

            float x2 = rotation.X + rotation.X;
            float y2 = rotation.Y + rotation.Y;
            float z2 = rotation.Z + rotation.Z;
            float wx = rotation.W * x2;
            float wy = rotation.W * y2;
            float wz = rotation.W * z2;
            float xx = rotation.X * x2;
            float xy = rotation.X * y2;
            float xz = rotation.X * z2;
            float yy = rotation.Y * y2;
            float yz = rotation.Y * z2;
            float zz = rotation.Z * z2;

            for (int i = 0; i < planes.Length; ++i)
            {
                float x = planes[i].Normal.X;
                float y = planes[i].Normal.Y;
                float z = planes[i].Normal.Z;

                /*
                 * Note:
                 * Factor common arithmetic out of loop.
                */
                planes[i].Normal.X = ((x * ((1.0f - yy) - zz)) + (y * (xy - wz))) + (z * (xz + wy));
                planes[i].Normal.Y = ((x * (xy + wz)) + (y * ((1.0f - xx) - zz))) + (z * (yz - wx));
                planes[i].Normal.Z = ((x * (xz - wy)) + (y * (yz + wx))) + (z * ((1.0f - xx) - yy));
            }
        }

        /// <summary>
        /// Transforms a normalized plane by a matrix.
        /// </summary>
        /// <param name="plane">The normalized source plane.</param>
        /// <param name="transformation">The transformation matrix.</param>
        /// <param name="result">When the method completes, contains the transformed plane.</param>
        public static void Transform(ref readonly Plane plane, ref readonly Matrix transformation, out Plane result)
        {
            float x = plane.Normal.X;
            float y = plane.Normal.Y;
            float z = plane.Normal.Z;
            float d = plane.D;

            Matrix inverse;
            Matrix.Invert(in transformation, out inverse);

            result.Normal.X = (((x * inverse.M11) + (y * inverse.M12)) + (z * inverse.M13)) + (d * inverse.M14);
            result.Normal.Y = (((x * inverse.M21) + (y * inverse.M22)) + (z * inverse.M23)) + (d * inverse.M24);
            result.Normal.Z = (((x * inverse.M31) + (y * inverse.M32)) + (z * inverse.M33)) + (d * inverse.M34);
            result.D = (((x * inverse.M41) + (y * inverse.M42)) + (z * inverse.M43)) + (d * inverse.M44);
        }

        /// <summary>
        /// Transforms a normalized plane by a matrix.
        /// </summary>
        /// <param name="plane">The normalized source plane.</param>
        /// <param name="transformation">The transformation matrix.</param>
        /// <returns>When the method completes, contains the transformed plane.</returns>
        public static Plane Transform(Plane plane, Matrix transformation)
        {
            Plane result;
            float x = plane.Normal.X;
            float y = plane.Normal.Y;
            float z = plane.Normal.Z;
            float d = plane.D;

            transformation.Invert();
            result.Normal.X = (((x * transformation.M11) + (y * transformation.M12)) + (z * transformation.M13)) + (d * transformation.M14);
            result.Normal.Y = (((x * transformation.M21) + (y * transformation.M22)) + (z * transformation.M23)) + (d * transformation.M24);
            result.Normal.Z = (((x * transformation.M31) + (y * transformation.M32)) + (z * transformation.M33)) + (d * transformation.M34);
            result.D = (((x * transformation.M41) + (y * transformation.M42)) + (z * transformation.M43)) + (d * transformation.M44);

            return result;
        }

        /// <summary>
        /// Transforms an array of normalized planes by a matrix.
        /// </summary>
        /// <param name="planes">The array of normalized planes to transform.</param>
        /// <param name="transformation">The transformation matrix.</param>
        /// <exception cref="ArgumentNullException">Thrown when <paramref name="planes"/> is <c>null</c>.</exception>
        public static void Transform(Plane[] planes, ref readonly Matrix transformation)
        {
            if (planes == null)
                throw new ArgumentNullException("planes");

            Matrix inverse;
            Matrix.Invert(in transformation, out inverse);

            for (int i = 0; i < planes.Length; ++i)
            {
                Transform(ref planes[i], in transformation, out planes[i]);
            }
        }

        /// <summary>
        /// Scales a plane by the given value.
        /// </summary>
        /// <param name="scale">The amount by which to scale the plane.</param>
        /// <param name="plane">The plane to scale.</param>
        /// <returns>The scaled plane.</returns>
        public static Plane operator *(float scale, Plane plane)
        {
            return new Plane(plane.Normal.X * scale, plane.Normal.Y * scale, plane.Normal.Z * scale, plane.D * scale);
        }

        /// <summary>
        /// Scales a plane by the given value.
        /// </summary>
        /// <param name="plane">The plane to scale.</param>
        /// <param name="scale">The amount by which to scale the plane.</param>
        /// <returns>The scaled plane.</returns>
        public static Plane operator *(Plane plane, float scale)
        {
            return new Plane(plane.Normal.X * scale, plane.Normal.Y * scale, plane.Normal.Z * scale, plane.D * scale);
        }

        /// <summary>
        /// Negates a plane by negating all its coefficients, which result in a plane in opposite direction.
        /// </summary>
        /// <returns>The negated plane.</returns>
        public static Plane operator -(Plane plane)
        {
            return new Plane(-plane.Normal.X, -plane.Normal.Y, -plane.Normal.Z, -plane.D);
        }

        /// <summary>
        /// Tests for equality between two objects.
        /// </summary>
        /// <param name="left">The first value to compare.</param>
        /// <param name="right">The second value to compare.</param>
        /// <returns><c>true</c> if <paramref name="left"/> has the same value as <paramref name="right"/>; otherwise, <c>false</c>.</returns>
        public static bool operator ==(Plane left, Plane right)
        {
            return left.Equals(right);
        }

        /// <summary>
        /// Tests for inequality between two objects.
        /// </summary>
        /// <param name="left">The first value to compare.</param>
        /// <param name="right">The second value to compare.</param>
        /// <returns><c>true</c> if <paramref name="left"/> has a different value than <paramref name="right"/>; otherwise, <c>false</c>.</returns>
        public static bool operator !=(Plane left, Plane right)
        {
            return !left.Equals(right);
        }

        /// <summary>
        /// Returns a <see cref="string"/> that represents this instance.
        /// </summary>
        /// <returns>
        /// A <see cref="string"/> that represents this instance.
        /// </returns>
        public override string ToString()
        {
            return string.Format(CultureInfo.CurrentCulture, "A:{0} B:{1} C:{2} D:{3}", Normal.X, Normal.Y, Normal.Z, D);
        }

        /// <summary>
        /// Returns a <see cref="string"/> that represents this instance.
        /// </summary>
        /// <param name="format">The format.</param>
        /// <returns>
        /// A <see cref="string"/> that represents this instance.
        /// </returns>
        public string ToString(string format)
        {
            return string.Format(CultureInfo.CurrentCulture, "A:{0} B:{1} C:{2} D:{3}", Normal.X.ToString(format, CultureInfo.CurrentCulture),
                Normal.Y.ToString(format, CultureInfo.CurrentCulture), Normal.Z.ToString(format, CultureInfo.CurrentCulture), D.ToString(format, CultureInfo.CurrentCulture));
        }

        /// <summary>
        /// Returns a <see cref="string"/> that represents this instance.
        /// </summary>
        /// <param name="formatProvider">The format provider.</param>
        /// <returns>
        /// A <see cref="string"/> that represents this instance.
        /// </returns>
        public string ToString(IFormatProvider formatProvider)
        {
            return string.Format(formatProvider, "A:{0} B:{1} C:{2} D:{3}", Normal.X, Normal.Y, Normal.Z, D);
        }

        /// <summary>
        /// Returns a <see cref="string"/> that represents this instance.
        /// </summary>
        /// <param name="format">The format.</param>
        /// <param name="formatProvider">The format provider.</param>
        /// <returns>
        /// A <see cref="string"/> that represents this instance.
        /// </returns>
        public string ToString(string format, IFormatProvider formatProvider)
        {
            return string.Format(formatProvider, "A:{0} B:{1} C:{2} D:{3}", Normal.X.ToString(format, formatProvider),
                Normal.Y.ToString(format, formatProvider), Normal.Z.ToString(format, formatProvider), D.ToString(format, formatProvider));
        }

        /// <summary>
        /// Returns a hash code for this instance.
        /// </summary>
        /// <returns>
        /// A hash code for this instance, suitable for use in hashing algorithms and data structures like a hash table.
        /// </returns>
        public override int GetHashCode()
        {
            return Normal.GetHashCode() + D.GetHashCode();
        }

        /// <summary>
        /// Determines whether the specified <see cref="Stride.Core.Mathematics.Vector4"/> is equal to this instance.
        /// </summary>
        /// <param name="value">The <see cref="Stride.Core.Mathematics.Vector4"/> to compare with this instance.</param>
        /// <returns>
        /// <c>true</c> if the specified <see cref="Stride.Core.Mathematics.Vector4"/> is equal to this instance; otherwise, <c>false</c>.
        /// </returns>
        public bool Equals(Plane value)
        {
            return Normal == value.Normal && D == value.D;
        }

        /// <summary>
        /// Determines whether the specified <see cref="object"/> is equal to this instance.
        /// </summary>
        /// <param name="value">The <see cref="object"/> to compare with this instance.</param>
        /// <returns>
        /// <c>true</c> if the specified <see cref="object"/> is equal to this instance; otherwise, <c>false</c>.
        /// </returns>
        public override bool Equals(object value)
        {
            if (value == null)
                return false;

            if (value.GetType() != GetType())
                return false;

            return Equals((Plane)value);
        }

#if SlimDX1xInterop
        /// <summary>
        /// Performs an implicit conversion from <see cref="Stride.Core.Mathematics.Plane"/> to <see cref="SlimDX.Plane"/>.
        /// </summary>
        /// <param name="value">The value.</param>
        /// <returns>The result of the conversion.</returns>
        public static implicit operator SlimDX.Plane(Plane value)
        {
            return new SlimDX.Plane(value.Normal, value.D);
        }

        /// <summary>
        /// Performs an implicit conversion from <see cref="SlimDX.Plane"/> to <see cref="Stride.Core.Mathematics.Plane"/>.
        /// </summary>
        /// <param name="value">The value.</param>
        /// <returns>The result of the conversion.</returns>
        public static implicit operator Plane(SlimDX.Plane value)
        {
            return new Plane(value.Normal, value.D);
        }
#endif

#if XnaInterop
        /// <summary>
        /// Performs an implicit conversion from <see cref="Stride.Core.Mathematics.Plane"/> to <see cref="Microsoft.Xna.Framework.Plane"/>.
        /// </summary>
        /// <param name="value">The value.</param>
        /// <returns>The result of the conversion.</returns>
        public static implicit operator Microsoft.Xna.Framework.Plane(Plane value)
        {
            return new Microsoft.Xna.Framework.Plane(value.Normal, value.D);
        }

        /// <summary>
        /// Performs an implicit conversion from <see cref="Microsoft.Xna.Framework.Plane"/> to <see cref="Stride.Core.Mathematics.Plane"/>.
        /// </summary>
        /// <param name="value">The value.</param>
        /// <returns>The result of the conversion.</returns>
        public static implicit operator Plane(Microsoft.Xna.Framework.Plane value)
        {
            return new Plane(value.Normal, value.D);
        }
#endif
    }
}