Plane.cs

// Copyright (c) 2010-2014 SharpDX - Alexandre Mutel
// 
// 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.
// -----------------------------------------------------------------------------
// 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.CompilerServices;
using System.Runtime.InteropServices;

namespace TheMaths
{
    /// <summary>
    /// Represents a plane in three dimensional space.
    /// </summary>
    [StructLayout(LayoutKind.Sequential, Pack = 4)]
    public struct Plane : IEquatable<Plane>, IFormattable
    {
        /// <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="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="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="T:SharpDX.Plane" /> class.
        /// </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)
        {
            this.Normal = normal;
            this.D = -Vector3.Dot(normal, point);
        }

        /// <summary>
        /// Initializes a new instance of the <see cref="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="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 / (float)(Math.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="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>
        /// Changes the coefficients of the normal vector of the plane to make it of unit length.
        /// </summary>
        public void Normalize()
        {
            float magnitude = 1.0f / (float)(Math.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 Vector3 point)
        {
            return Collision.PlaneIntersectsPoint(ref this, ref point);
        }

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

        /// <summary>
        /// Determines if there is an intersection between the current object and a <see cref="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 Ray ray, out float distance)
        {
            return Collision.RayIntersectsPlane(ref ray, ref this, out distance);
        }

        /// <summary>
        /// Determines if there is an intersection between the current object and a <see cref="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="Vector3.Zero"/> if there was no intersection.</param>
        /// <returns>Whether the two objects intersected.</returns>
        public bool Intersects(ref Ray ray, out Vector3 point)
        {
            return Collision.RayIntersectsPlane(ref ray, ref this, out point);
        }

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

        /// <summary>
        /// Determines if there is an intersection between the current object and a <see cref="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="Ray"/>, or a zero ray if there was no intersection.</param>
        /// <returns>Whether the two objects intersected.</returns>
        public bool Intersects(ref Plane plane, out Ray line)
        {
            return Collision.PlaneIntersectsPlane(ref this, ref 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 triangle 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 Vector3 vertex1, ref Vector3 vertex2, ref Vector3 vertex3)
        {
            return Collision.PlaneIntersectsTriangle(ref this, ref vertex1, ref vertex2, ref vertex3);
        }

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

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

        /// <summary>
        /// Builds a matrix that can be used to reflect vectors about a plane.
        /// </summary>
        /// <param name="plane">The plane for which the reflection occurs. This parameter is assumed to be normalized.</param>
        /// <param name="result">When the method completes, contains the reflection matrix.</param>
        public void Reflection(out Matrix result)
        {
            float x = this.Normal.X;
            float y = this.Normal.Y;
            float z = this.Normal.Z;
            float x2 = -2.0f * x;
            float y2 = -2.0f * y;
            float z2 = -2.0f * z;

            result.M11 = (x2 * x) + 1.0f;
            result.M12 = y2 * x;
            result.M13 = z2 * x;
            result.M14 = 0.0f;
            result.M21 = x2 * y;
            result.M22 = (y2 * y) + 1.0f;
            result.M23 = z2 * y;
            result.M24 = 0.0f;
            result.M31 = x2 * z;
            result.M32 = y2 * z;
            result.M33 = (z2 * z) + 1.0f;
            result.M34 = 0.0f;
            result.M41 = x2 * this.D;
            result.M42 = y2 * this.D;
            result.M43 = z2 * this.D;
            result.M44 = 1.0f;
        }

        /// <summary>
        /// Builds a matrix that can be used to reflect vectors about a plane.
        /// </summary>
        /// <returns>The reflection matrix.</returns>
        public Matrix Reflection()
        {
            Matrix result;
            Reflection(out result);
            return result;
        }

        /// <summary>
        /// Creates a matrix that flattens geometry into a shadow from this the plane onto which to project the geometry as a shadow. 
        /// This plane  is assumed to be normalized
        /// </summary>
        /// <param name="light">The light direction. If the W component is 0, the light is directional light; if the
        /// W component is 1, the light is a point light.</param>
        /// <param name="result">When the method completes, contains the shadow matrix.</param>
        public void Shadow(ref Vector4 light, out Matrix result)
        {
            float dot = (this.Normal.X * light.X) + (this.Normal.Y * light.Y) + (this.Normal.Z * light.Z) + (this.D * light.W);
            float x = -this.Normal.X;
            float y = -this.Normal.Y;
            float z = -this.Normal.Z;
            float d = -this.D;

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

        /// <summary>
        /// Creates a matrix that flattens geometry into a shadow from this the plane onto which to project the geometry as a shadow. 
        /// This plane  is assumed to be normalized
        /// </summary>
        /// <param name="light">The light direction. If the W component is 0, the light is directional light; if the
        /// W component is 1, the light is a point light.</param>
        /// <returns>The shadow matrix.</returns>
        public Matrix Shadow(Vector4 light)
        {
            Matrix result;
            Shadow(ref light, out result);
            return result;
        }

        /// <summary>
        /// Builds a Matrix3x3 that can be used to reflect vectors about a plane for which the reflection occurs. 
        /// This plane is assumed to be normalized
        /// </summary>
        /// <param name="result">When the method completes, contains the reflection Matrix3x3.</param>
        public void Reflection(out Matrix3x3 result)
        {
            float x = this.Normal.X;
            float y = this.Normal.Y;
            float z = this.Normal.Z;
            float x2 = -2.0f * x;
            float y2 = -2.0f * y;
            float z2 = -2.0f * z;

            result.M11 = (x2 * x) + 1.0f;
            result.M12 = y2 * x;
            result.M13 = z2 * x;
            result.M21 = x2 * y;
            result.M22 = (y2 * y) + 1.0f;
            result.M23 = z2 * y;
            result.M31 = x2 * z;
            result.M32 = y2 * z;
            result.M33 = (z2 * z) + 1.0f;
        }

        /// <summary>
        /// Builds a Matrix3x3 that can be used to reflect vectors about a plane for which the reflection occurs. 
        /// This plane is assumed to be normalized
        /// </summary>
        /// <returns>The reflection Matrix3x3.</returns>
        public Matrix3x3 Reflection3x3()
        {
            Matrix3x3 result;
            Reflection(out result);
            return result;
        }

        /// <summary>
        /// Creates a Matrix3x3 that flattens geometry into a shadow.
        /// </summary>
        /// <param name="light">The light direction. If the W component is 0, the light is directional light; if the
        /// W component is 1, the light is a point light.</param>
        /// <param name="plane">The plane onto which to project the geometry as a shadow. This parameter is assumed to be normalized.</param>
        /// <param name="result">When the method completes, contains the shadow Matrix3x3.</param>
        public static void Shadow(ref Vector4 light, ref Plane plane, out Matrix3x3 result)
        {
            float dot = (plane.Normal.X * light.X) + (plane.Normal.Y * light.Y) + (plane.Normal.Z * light.Z) + (plane.D * light.W);
            float x = -plane.Normal.X;
            float y = -plane.Normal.Y;
            float z = -plane.Normal.Z;
            float d = -plane.D;

            result.M11 = (x * light.X) + dot;
            result.M21 = y * light.X;
            result.M31 = z * light.X;
            result.M12 = x * light.Y;
            result.M22 = (y * light.Y) + dot;
            result.M32 = z * light.Y;
            result.M13 = x * light.Z;
            result.M23 = y * light.Z;
            result.M33 = (z * light.Z) + dot;
        }

        /// <summary>
        /// Creates a Matrix3x3 that flattens geometry into a shadow.
        /// </summary>
        /// <param name="light">The light direction. If the W component is 0, the light is directional light; if the
        /// W component is 1, the light is a point light.</param>
        /// <param name="plane">The plane onto which to project the geometry as a shadow. This parameter is assumed to be normalized.</param>
        /// <returns>The shadow Matrix3x3.</returns>
        public static Matrix3x3 Shadow(Vector4 light, Plane plane)
        {
            Matrix3x3 result;
            Shadow(ref light, ref plane, out result);
            return result;
        }


        /// <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 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 Plane left, ref 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 Plane left, ref 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 Plane left, ref 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>
        /// 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 Plane plane, out Plane result)
        {
            float magnitude = 1.0f / (float)(Math.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 / (float)(Math.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 Plane plane, ref 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 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 Plane plane, ref 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(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;

            Matrix.Invert(transformation, out transformation);
            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 Matrix transformation)
        {
            if (planes == null)
                throw new ArgumentNullException("planes");

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

            for (int i = 0; i < planes.Length; ++i)
            {
                Transform(ref planes[i], ref 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>
        /// 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>
        [MethodImpl((MethodImplOptions)0x100)] // MethodImplOptions.AggressiveInlining
        public static bool operator ==(Plane left, Plane right)
        {
            return left.Equals(ref 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>
        [MethodImpl((MethodImplOptions)0x100)] // MethodImplOptions.AggressiveInlining
        public static bool operator !=(Plane left, Plane right)
        {
            return !left.Equals(ref right);
        }

        /// <summary>
        /// Returns a <see cref="System.String"/> that represents this instance.
        /// </summary>
        /// <returns>
        /// A <see cref="System.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="System.String"/> that represents this instance.
        /// </summary>
        /// <param name="format">The format.</param>
        /// <returns>
        /// A <see cref="System.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="System.String"/> that represents this instance.
        /// </summary>
        /// <param name="formatProvider">The format provider.</param>
        /// <returns>
        /// A <see cref="System.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="System.String"/> that represents this instance.
        /// </summary>
        /// <param name="format">The format.</param>
        /// <param name="formatProvider">The format provider.</param>
        /// <returns>
        /// A <see cref="System.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()
        {
            unchecked
            {
                return (Normal.GetHashCode() * 397) ^ D.GetHashCode();
            }
        }

        /// <summary>
        /// Determines whether the specified <see cref="Vector4"/> is equal to this instance.
        /// </summary>
        /// <param name="value">The <see cref="Vector4"/> to compare with this instance.</param>
        /// <returns>
        /// <c>true</c> if the specified <see cref="Vector4"/> is equal to this instance; otherwise, <c>false</c>.
        /// </returns>
        [MethodImpl((MethodImplOptions)0x100)] // MethodImplOptions.AggressiveInlining
        public bool Equals(ref Plane value)
        {
            return Normal == value.Normal && D == value.D;
        }

        /// <summary>
        /// Determines whether the specified <see cref="Vector4"/> is equal to this instance.
        /// </summary>
        /// <param name="value">The <see cref="Vector4"/> to compare with this instance.</param>
        /// <returns>
        /// <c>true</c> if the specified <see cref="Vector4"/> is equal to this instance; otherwise, <c>false</c>.
        /// </returns>
        [MethodImpl((MethodImplOptions)0x100)] // MethodImplOptions.AggressiveInlining
        public bool Equals(Plane value)
        {
            return Equals(ref value);
        }

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

            var strongValue = (Plane)value;
            return Equals(ref strongValue);
        }
    }
}