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#region --- License ---

/*

Copyright (c) 2006 - 2008 The Open Toolkit library.

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.

*/

#endregion

using System;

using System.Runtime.InteropServices;

using System.Xml.Serialization;

namespace OpenTK.Math

{

    /// <summary>

    /// Represents a 3D vector using three double-precision floating-point numbers.

    /// </summary>

    [Obsolete("OpenTK.Math functions have been moved to the root OpenTK namespace (reason: XNA compatibility")]

    [Serializable]

    [StructLayout(LayoutKind.Sequential)]

    public struct Vector3d : IEquatable<Vector3d>

    {

        #region Fields

        /// <summary>

        /// The X component of the Vector3.

        /// </summary>

        public double X;

        /// <summary>

        /// The Y component of the Vector3.

        /// </summary>

        public double Y;

        /// <summary>

        /// The Z component of the Vector3.

        /// </summary>

        public double Z;

        #endregion

        #region Constructors

        /// <summary>

        /// Constructs a new Vector3.

        /// </summary>

        /// <param name="x">The x component of the Vector3.</param>

        /// <param name="y">The y component of the Vector3.</param>

        /// <param name="z">The z component of the Vector3.</param>

        public Vector3d(double x, double y, double z)

        {

            X = x;

            Y = y;

            Z = z;

        }

        /// <summary>

        /// Constructs a new instance from the given Vector2d.

        /// </summary>

        /// <param name="v">The Vector2d to copy components from.</param>

        public Vector3d(Vector2d v)

        {

            X = v.X;

            Y = v.Y;

            Z = 0.0f;

        }

        /// <summary>

        /// Constructs a new instance from the given Vector3d.

        /// </summary>

        /// <param name="v">The Vector3d to copy components from.</param>

        public Vector3d(Vector3d v)

        {

            X = v.X;

            Y = v.Y;

            Z = v.Z;

        }

        /// <summary>

        /// Constructs a new instance from the given Vector4d.

        /// </summary>

        /// <param name="v">The Vector4d to copy components from.</param>

        public Vector3d(Vector4d v)

        {

            X = v.X;

            Y = v.Y;

            Z = v.Z;

        }

        #endregion

        #region Public Members

        #region Instance

        #region public void Add()

        /// <summary>Add the Vector passed as parameter to this instance.</summary>

        /// <param name="right">Right operand. This parameter is only read from.</param>

        public void Add(Vector3d right)

        {

            this.X += right.X;

            this.Y += right.Y;

            this.Z += right.Z;

        }

        /// <summary>Add the Vector passed as parameter to this instance.</summary>

        /// <param name="right">Right operand. This parameter is only read from.</param>

        [CLSCompliant(false)]

        public void Add(ref Vector3d right)

        {

            this.X += right.X;

            this.Y += right.Y;

            this.Z += right.Z;

        }

        #endregion public void Add()

        #region public void Sub()

        /// <summary>Subtract the Vector passed as parameter from this instance.</summary>

        /// <param name="right">Right operand. This parameter is only read from.</param>

        public void Sub(Vector3d right)

        {

            this.X -= right.X;

            this.Y -= right.Y;

            this.Z -= right.Z;

        }

        /// <summary>Subtract the Vector passed as parameter from this instance.</summary>

        /// <param name="right">Right operand. This parameter is only read from.</param>

        [CLSCompliant(false)]

        public void Sub(ref Vector3d right)

        {

            this.X -= right.X;

            this.Y -= right.Y;

            this.Z -= right.Z;

        }

        #endregion public void Sub()

        #region public void Mult()

        /// <summary>Multiply this instance by a scalar.</summary>

        /// <param name="f">Scalar operand.</param>

        public void Mult(double f)

        {

            this.X *= f;

            this.Y *= f;

            this.Z *= f;

        }

        #endregion public void Mult()

        #region public void Div()

        /// <summary>Divide this instance by a scalar.</summary>

        /// <param name="f">Scalar operand.</param>

        public void Div(double f)

        {

            double mult = 1.0 / f;

            this.X *= mult;

            this.Y *= mult;

            this.Z *= mult;

        }

        #endregion public void Div()

        #region public double Length

        /// <summary>

        /// Gets the length (magnitude) of the vector.

        /// </summary>

        /// <see cref="LengthFast"/>

        /// <seealso cref="LengthSquared"/>

        public double Length

        {

            get

            {

                return (float)System.Math.Sqrt(X * X + Y * Y + Z * Z);

            }

        }

        #endregion

        #region public double LengthFast

        /// <summary>

        /// Gets an approximation of the vector length (magnitude).

        /// </summary>

        /// <remarks>

        /// This property uses an approximation of the square root function to calculate vector magnitude, with

        /// an upper error bound of 0.001.

        /// </remarks>

        /// <see cref="Length"/>

        /// <seealso cref="LengthSquared"/>

        public double LengthFast

        {

            get

            {

                return 1.0f / MathHelper.InverseSqrtFast(X * X + Y * Y + Z * Z);

            }

        }

        #endregion

        #region public double LengthSquared

        /// <summary>

        /// Gets the square of the vector length (magnitude).

        /// </summary>

        /// <remarks>

        /// This property avoids the costly square root operation required by the Length property. This makes it more suitable

        /// for comparisons.

        /// </remarks>

        /// <see cref="Length"/>

        /// <seealso cref="LengthFast"/>

        public double LengthSquared

        {

            get

            {

                return X * X + Y * Y + Z * Z;

            }

        }

        #endregion

        #region public void Normalize()

        /// <summary>

        /// Scales the Vector3d to unit length.

        /// </summary>

        public void Normalize()

        {

            double scale = 1.0f / this.Length;

            X *= scale;

            Y *= scale;

            Z *= scale;

        }

        #endregion

        #region public void NormalizeFast()

        /// <summary>

        /// Scales the Vector3d to approximately unit length.

        /// </summary>

        public void NormalizeFast()

        {

            double scale = Functions.InverseSqrtFast(X * X + Y * Y + Z * Z);

            X *= scale;

            Y *= scale;

            Z *= scale;

        }

        #endregion

        #region public void Scale()

        /// <summary>

        /// Scales the current Vector3d by the given amounts.

        /// </summary>

        /// <param name="sx">The scale of the X component.</param>

        /// <param name="sy">The scale of the Y component.</param>

        /// <param name="sz">The scale of the Z component.</param>

        public void Scale(double sx, double sy, double sz)

        {

            this.X = X * sx;

            this.Y = Y * sy;

            this.Z = Z * sz;

        }

        /// <summary>Scales this instance by the given parameter.</summary>

        /// <param name="scale">The scaling of the individual components.</param>

        public void Scale(Vector3d scale)

        {

            this.X *= scale.X;

            this.Y *= scale.Y;

            this.Z *= scale.Z;

        }

        /// <summary>Scales this instance by the given parameter.</summary>

        /// <param name="scale">The scaling of the individual components.</param>

        [CLSCompliant(false)]

        public void Scale(ref Vector3d scale)

        {

            this.X *= scale.X;

            this.Y *= scale.Y;

            this.Z *= scale.Z;

        }

        #endregion public void Scale()

        #endregion

        #region Static

        #region Fields

        /// <summary>

        /// Defines a unit-length Vector3d that points towards the X-axis.

        /// </summary>

        public static readonly Vector3d UnitX = new Vector3d(1, 0, 0);

        /// <summary>

        /// Defines a unit-length Vector3d that points towards the Y-axis.

        /// </summary>

        public static readonly Vector3d UnitY = new Vector3d(0, 1, 0);

        /// <summary>

        /// /// Defines a unit-length Vector3d that points towards the Z-axis.

        /// </summary>

        public static readonly Vector3d UnitZ = new Vector3d(0, 0, 1);

        /// <summary>

        /// Defines a zero-length Vector3.

        /// </summary>

        public static readonly Vector3d Zero = new Vector3d(0, 0, 0);

        /// <summary>

        /// Defines an instance with all components set to 1.

        /// </summary>

        public static readonly Vector3d One = new Vector3d(1, 1, 1);

        /// <summary>

        /// Defines the size of the Vector3d struct in bytes.

        /// </summary>

        public static readonly int SizeInBytes = Marshal.SizeOf(new Vector3d());

        #endregion

        #region Add

        /// <summary>

        /// Add two Vectors

        /// </summary>

        /// <param name="a">First operand</param>

        /// <param name="b">Second operand</param>

        /// <returns>Result of addition</returns>

        public static Vector3d Add(Vector3d a, Vector3d b)

        {

            a.X += b.X;

            a.Y += b.Y;

            a.Z += b.Z;

            return a;

        }

        /// <summary>

        /// Add two Vectors

        /// </summary>

        /// <param name="a">First operand</param>

        /// <param name="b">Second operand</param>

        /// <param name="result">Result of addition</param>

        public static void Add(ref Vector3d a, ref Vector3d b, out Vector3d result)

        {

            result.X = a.X + b.X;

            result.Y = a.Y + b.Y;

            result.Z = a.Z + b.Z;

        }

        #endregion

        #region Sub

        /// <summary>

        /// Subtract one Vector from another

        /// </summary>

        /// <param name="a">First operand</param>

        /// <param name="b">Second operand</param>

        /// <returns>Result of subtraction</returns>

        public static Vector3d Sub(Vector3d a, Vector3d b)

        {

            a.X -= b.X;

            a.Y -= b.Y;

            a.Z -= b.Z;

            return a;

        }

        /// <summary>

        /// Subtract one Vector from another

        /// </summary>

        /// <param name="a">First operand</param>

        /// <param name="b">Second operand</param>

        /// <param name="result">Result of subtraction</param>

        public static void Sub(ref Vector3d a, ref Vector3d b, out Vector3d result)

        {

            result.X = a.X - b.X;

            result.Y = a.Y - b.Y;

            result.Z = a.Z - b.Z;

        }

        #endregion

        #region Mult

        /// <summary>

        /// Multiply a vector and a scalar

        /// </summary>

        /// <param name="a">Vector operand</param>

        /// <param name="f">Scalar operand</param>

        /// <returns>Result of the multiplication</returns>

        public static Vector3d Mult(Vector3d a, double f)

        {

            a.X *= f;

            a.Y *= f;

            a.Z *= f;

            return a;

        }

        /// <summary>

        /// Multiply a vector and a scalar

        /// </summary>

        /// <param name="a">Vector operand</param>

        /// <param name="f">Scalar operand</param>

        /// <param name="result">Result of the multiplication</param>

        public static void Mult(ref Vector3d a, double f, out Vector3d result)

        {

            result.X = a.X * f;

            result.Y = a.Y * f;

            result.Z = a.Z * f;

        }

        #endregion

        #region Div

        /// <summary>

        /// Divide a vector by a scalar

        /// </summary>

        /// <param name="a">Vector operand</param>

        /// <param name="f">Scalar operand</param>

        /// <returns>Result of the division</returns>

        public static Vector3d Div(Vector3d a, double f)

        {

            double mult = 1.0f / f;

            a.X *= mult;

            a.Y *= mult;

            a.Z *= mult;

            return a;

        }

        /// <summary>

        /// Divide a vector by a scalar

        /// </summary>

        /// <param name="a">Vector operand</param>

        /// <param name="f">Scalar operand</param>

        /// <param name="result">Result of the division</param>

        public static void Div(ref Vector3d a, double f, out Vector3d result)

        {

            double mult = 1.0f / f;

            result.X = a.X * mult;

            result.Y = a.Y * mult;

            result.Z = a.Z * mult;

        }

        #endregion

        #region ComponentMin

        /// <summary>

        /// Calculate the component-wise minimum of two vectors

        /// </summary>

        /// <param name="a">First operand</param>

        /// <param name="b">Second operand</param>

        /// <returns>The component-wise minimum</returns>

        public static Vector3d ComponentMin(Vector3d a, Vector3d b)

        {

            a.X = a.X < b.X ? a.X : b.X;

            a.Y = a.Y < b.Y ? a.Y : b.Y;

            a.Z = a.Z < b.Z ? a.Z : b.Z;

            return a;

        }

        /// <summary>

        /// Calculate the component-wise minimum of two vectors

        /// </summary>

        /// <param name="a">First operand</param>

        /// <param name="b">Second operand</param>

        /// <param name="result">The component-wise minimum</param>

        public static void ComponentMin(ref Vector3d a, ref Vector3d b, out Vector3d result)

        {

            result.X = a.X < b.X ? a.X : b.X;

            result.Y = a.Y < b.Y ? a.Y : b.Y;

            result.Z = a.Z < b.Z ? a.Z : b.Z;

        }

        #endregion

        #region ComponentMax

        /// <summary>

        /// Calculate the component-wise maximum of two vectors

        /// </summary>

        /// <param name="a">First operand</param>

        /// <param name="b">Second operand</param>

        /// <returns>The component-wise maximum</returns>

        public static Vector3d ComponentMax(Vector3d a, Vector3d b)

        {

            a.X = a.X > b.X ? a.X : b.X;

            a.Y = a.Y > b.Y ? a.Y : b.Y;

            a.Z = a.Z > b.Z ? a.Z : b.Z;

            return a;

        }

        /// <summary>

        /// Calculate the component-wise maximum of two vectors

        /// </summary>

        /// <param name="a">First operand</param>

        /// <param name="b">Second operand</param>

        /// <param name="result">The component-wise maximum</param>

        public static void ComponentMax(ref Vector3d a, ref Vector3d b, out Vector3d result)

        {

            result.X = a.X > b.X ? a.X : b.X;

            result.Y = a.Y > b.Y ? a.Y : b.Y;

            result.Z = a.Z > b.Z ? a.Z : b.Z;

        }

        #endregion

        #region Min

        /// <summary>

        /// Returns the Vector3d with the minimum magnitude

        /// </summary>

        /// <param name="left">Left operand</param>

        /// <param name="right">Right operand</param>

        /// <returns>The minimum Vector3</returns>

        public static Vector3d Min(Vector3d left, Vector3d right)

        {

            return left.LengthSquared < right.LengthSquared ? left : right;

        }

        #endregion

        #region Max

        /// <summary>

        /// Returns the Vector3d with the minimum magnitude

        /// </summary>

        /// <param name="left">Left operand</param>

        /// <param name="right">Right operand</param>

        /// <returns>The minimum Vector3</returns>

        public static Vector3d Max(Vector3d left, Vector3d right)

        {

            return left.LengthSquared >= right.LengthSquared ? left : right;

        }

        #endregion

        #region Clamp

        /// <summary>

        /// Clamp a vector to the given minimum and maximum vectors

        /// </summary>

        /// <param name="vec">Input vector</param>

        /// <param name="min">Minimum vector</param>

        /// <param name="max">Maximum vector</param>

        /// <returns>The clamped vector</returns>

        public static Vector3d Clamp(Vector3d vec, Vector3d min, Vector3d max)

        {

            vec.X = vec.X < min.X ? min.X : vec.X > max.X ? max.X : vec.X;

            vec.Y = vec.Y < min.Y ? min.Y : vec.Y > max.Y ? max.Y : vec.Y;

            vec.Z = vec.Z < min.Z ? min.Z : vec.Z > max.Z ? max.Z : vec.Z;

            return vec;

        }

        /// <summary>

        /// Clamp a vector to the given minimum and maximum vectors

        /// </summary>

        /// <param name="vec">Input vector</param>

        /// <param name="min">Minimum vector</param>

        /// <param name="max">Maximum vector</param>

        /// <param name="result">The clamped vector</param>

        public static void Clamp(ref Vector3d vec, ref Vector3d min, ref Vector3d max, out Vector3d result)

        {

            result.X = vec.X < min.X ? min.X : vec.X > max.X ? max.X : vec.X;

            result.Y = vec.Y < min.Y ? min.Y : vec.Y > max.Y ? max.Y : vec.Y;

            result.Z = vec.Z < min.Z ? min.Z : vec.Z > max.Z ? max.Z : vec.Z;

        }

        #endregion

        #region Normalize

        /// <summary>

        /// Scale a vector to unit length

        /// </summary>

        /// <param name="vec">The input vector</param>

        /// <returns>The normalized vector</returns>

        public static Vector3d Normalize(Vector3d vec)

        {

            double scale = 1.0f / vec.Length;

            vec.X *= scale;

            vec.Y *= scale;

            vec.Z *= scale;

            return vec;

        }

        /// <summary>

        /// Scale a vector to unit length

        /// </summary>

        /// <param name="vec">The input vector</param>

        /// <param name="result">The normalized vector</param>

        public static void Normalize(ref Vector3d vec, out Vector3d result)

        {

            double scale = 1.0f / vec.Length;

            result.X = vec.X * scale;

            result.Y = vec.Y * scale;

            result.Z = vec.Z * scale;

        }

        #endregion

        #region NormalizeFast

        /// <summary>

        /// Scale a vector to approximately unit length

        /// </summary>

        /// <param name="vec">The input vector</param>

        /// <returns>The normalized vector</returns>

        public static Vector3d NormalizeFast(Vector3d vec)

        {

            double scale = Functions.InverseSqrtFast(vec.X * vec.X + vec.Y * vec.Y + vec.Z * vec.Z);

            vec.X *= scale;

            vec.Y *= scale;

            vec.Z *= scale;

            return vec;

        }

        /// <summary>

        /// Scale a vector to approximately unit length

        /// </summary>

        /// <param name="vec">The input vector</param>

        /// <param name="result">The normalized vector</param>

        public static void NormalizeFast(ref Vector3d vec, out Vector3d result)

        {

            double scale = Functions.InverseSqrtFast(vec.X * vec.X + vec.Y * vec.Y + vec.Z * vec.Z);

            result.X = vec.X * scale;

            result.Y = vec.Y * scale;

            result.Z = vec.Z * scale;

        }

        #endregion

        #region Dot

        /// <summary>

        /// Calculate the dot (scalar) product of two vectors

        /// </summary>

        /// <param name="left">First operand</param>

        /// <param name="right">Second operand</param>

        /// <returns>The dot product of the two inputs</returns>

        public static double Dot(Vector3d left, Vector3d right)

        {

            return left.X * right.X + left.Y * right.Y + left.Z * right.Z;

        }

        /// <summary>

        /// Calculate the dot (scalar) product of two vectors

        /// </summary>

        /// <param name="left">First operand</param>

        /// <param name="right">Second operand</param>

        /// <param name="result">The dot product of the two inputs</param>

        public static void Dot(ref Vector3d left, ref Vector3d right, out double result)

        {

            result = left.X * right.X + left.Y * right.Y + left.Z * right.Z;

        }

        #endregion

        #region Cross

        /// <summary>

        /// Caclulate the cross (vector) product of two vectors

        /// </summary>

        /// <param name="left">First operand</param>

        /// <param name="right">Second operand</param>

        /// <returns>The cross product of the two inputs</returns>

        public static Vector3d Cross(Vector3d left, Vector3d right)

        {

            return new Vector3d(left.Y * right.Z - left.Z * right.Y,

                                left.Z * right.X - left.X * right.Z,

                                left.X * right.Y - left.Y * right.X);

        }

        /// <summary>

        /// Caclulate the cross (vector) product of two vectors

        /// </summary>

        /// <param name="left">First operand</param>

        /// <param name="right">Second operand</param>

        /// <returns>The cross product of the two inputs</returns>

        /// <param name="result">The cross product of the two inputs</param>

        public static void Cross(ref Vector3d left, ref Vector3d right, out Vector3d result)

        {

            result.X = left.Y * right.Z - left.Z * right.Y;

            result.Y = left.Z * right.X - left.X * right.Z;

            result.Z = left.X * right.Y - left.Y * right.X;

        }

        #endregion

        #region Lerp

        /// <summary>

        /// Returns a new Vector that is the linear blend of the 2 given Vectors

        /// </summary>

        /// <param name="a">First input vector</param>

        /// <param name="b">Second input vector</param>

        /// <param name="blend">The blend factor. a when blend=0, b when blend=1.</param>

        /// <returns>a when blend=0, b when blend=1, and a linear combination otherwise</returns>

        public static Vector3d Lerp(Vector3d a, Vector3d b, double blend)

        {

            a.X = blend * (b.X - a.X) + a.X;

            a.Y = blend * (b.Y - a.Y) + a.Y;

            a.Z = blend * (b.Z - a.Z) + a.Z;

            return a;

        }

        /// <summary>

        /// Returns a new Vector that is the linear blend of the 2 given Vectors

        /// </summary>

        /// <param name="a">First input vector</param>

        /// <param name="b">Second input vector</param>

        /// <param name="blend">The blend factor. a when blend=0, b when blend=1.</param>

        /// <param name="result">a when blend=0, b when blend=1, and a linear combination otherwise</param>

        public static void Lerp(ref Vector3d a, ref Vector3d b, double blend, out Vector3d result)

        {

            result.X = blend * (b.X - a.X) + a.X;

            result.Y = blend * (b.Y - a.Y) + a.Y;

            result.Z = blend * (b.Z - a.Z) + a.Z;

        }

        #endregion

        #region Barycentric

        /// <summary>

        /// Interpolate 3 Vectors using Barycentric coordinates

        /// </summary>

        /// <param name="a">First input Vector</param>

        /// <param name="b">Second input Vector</param>

        /// <param name="c">Third input Vector</param>

        /// <param name="u">First Barycentric Coordinate</param>

        /// <param name="v">Second Barycentric Coordinate</param>

        /// <returns>a when u=v=0, b when u=1,v=0, c when u=0,v=1, and a linear combination of a,b,c otherwise</returns>

        public static Vector3d BaryCentric(Vector3d a, Vector3d b, Vector3d c, double u, double v)

        {

            return a + u * (b - a) + v * (c - a);

        }

        /// <summary>Interpolate 3 Vectors using Barycentric coordinates</summary>

        /// <param name="a">First input Vector.</param>

        /// <param name="b">Second input Vector.</param>

        /// <param name="c">Third input Vector.</param>

        /// <param name="u">First Barycentric Coordinate.</param>

        /// <param name="v">Second Barycentric Coordinate.</param>

        /// <param name="result">Output Vector. a when u=v=0, b when u=1,v=0, c when u=0,v=1, and a linear combination of a,b,c otherwise</param>

        public static void BaryCentric(ref Vector3d a, ref Vector3d b, ref Vector3d c, float u, float v, out Vector3d result)

        {

            result = a; // copy

            Vector3d temp = b; // copy

            temp.Sub(ref a);

            temp.Mult(u);

            result.Add(ref temp);

            temp = c; // copy

            temp.Sub(ref a);

            temp.Mult(v);

            result.Add(ref temp);

        }

        #endregion

        #region Transform

        /// <summary>Transform a direction vector by the given Matrix

        /// Assumes the matrix has a bottom row of (0,0,0,1), that is the translation part is ignored.

        /// </summary>

        /// <param name="vec">The vector to transform</param>

        /// <param name="mat">The desired transformation</param>

        /// <returns>The transformed vector</returns>

        public static Vector3d TransformVector(Vector3d vec, Matrix4d mat)

        {

            return new Vector3d(

                Vector3d.Dot(vec, new Vector3d(mat.Column0)),

                Vector3d.Dot(vec, new Vector3d(mat.Column1)),

                Vector3d.Dot(vec, new Vector3d(mat.Column2)));

        }

        /// <summary>Transform a direction vector by the given Matrix

        /// Assumes the matrix has a bottom row of (0,0,0,1), that is the translation part is ignored.

        /// </summary>

        /// <param name="vec">The vector to transform</param>

        /// <param name="mat">The desired transformation</param>

        /// <param name="result">The transformed vector</param>

        public static void TransformVector(ref Vector3d vec, ref Matrix4d mat, out Vector3d result)

        {

            result.X = vec.X * mat.Row0.X +

                       vec.Y * mat.Row1.X +

                       vec.Z * mat.Row2.X;

            result.Y = vec.X * mat.Row0.Y +

                       vec.Y * mat.Row1.Y +

                       vec.Z * mat.Row2.Y;