/// <summary>
/// Direct-Matrix-Slerp: for the sake of completeness, I have included the following expression
/// for Spherical-Linear-Interpolation without using quaternions. This is much faster then converting
/// both matrices into quaternions in order to do a quaternion slerp and then converting the slerped
/// quaternion back into a matrix.
/// This is a high-precision calculation. Given two orthonormal 3x3 matrices this function calculates
/// the shortest possible interpolation-path between the two rotations. The interpolation curve forms
/// a great arc on the rotation sphere (geodesic). Not only does Slerp follow a great arc it follows
/// the shortest great arc. Furthermore Slerp has constant angular velocity. All in all Slerp is the
/// optimal interpolation curve between two rotations.
/// STABILITY PROBLEM: There are two singularities at angle=0 and angle=PI. At 0 the interpolation-axis
/// is arbitrary, which means any axis will produce the same result because we have no rotation. Thats
/// why I'm using (1,0,0). At PI the rotations point away from each other and the interpolation-axis
/// is unpredictable. In this case I'm also using the axis (1,0,0). If the angle is ~0 or ~PI, then we
/// have to normalize a very small vector and this can cause numerical instability. The quaternion-slerp
/// has exactly the same problems. Ivo
/// </summary>
/// <param name="m"></param>
/// <param name="n"></param>
/// <param name="t"></param>
/// <example>Matrix33 slerp=Matrix33::CreateSlerp( m,n,0.333f );</example>
public void SetSlerp(Matrix34 m, Matrix34 n, float t)
{
// calculate delta-rotation between m and n (=39 flops)
Matrix33 d = new Matrix33(), i = new Matrix33();
d.M00 = m.M00 * n.M00 + m.M10 * n.M10 + m.M20 * n.M20; d.M01 = m.M00 * n.M01 + m.M10 * n.M11 + m.M20 * n.M21; d.M02 = m.M00 * n.M02 + m.M10 * n.M12 + m.M20 * n.M22;
d.M10 = m.M01 * n.M00 + m.M11 * n.M10 + m.M21 * n.M20; d.M11 = m.M01 * n.M01 + m.M11 * n.M11 + m.M21 * n.M21; d.M12 = m.M01 * n.M02 + m.M11 * n.M12 + m.M21 * n.M22;
d.M20 = d.M01 * d.M12 - d.M02 * d.M11; d.M21 = d.M02 * d.M10 - d.M00 * d.M12; d.M22 = d.M00 * d.M11 - d.M01 * d.M10;
// extract angle and axis
double cosine = MathHelpers.Clamp((d.M00 + d.M11 + d.M22 - 1.0) * 0.5, -1.0, +1.0);
double angle = Math.Atan2(Math.Sqrt(1.0 - cosine * cosine), cosine);
var axis = new Vec3(d.M21 - d.M12, d.M02 - d.M20, d.M10 - d.M01);
double l = Math.Sqrt(axis | axis); if (l > 0.00001)
{
axis /= (float)l;
}
else
{
axis = new Vec3(1, 0, 0);
}
i.SetRotationAA((float)angle * t, axis); // angle interpolation and calculation of new delta-matrix (=26 flops)
// final concatenation (=39 flops)
M00 = m.M00 * i.M00 + m.M01 * i.M10 + m.M02 * i.M20; M01 = m.M00 * i.M01 + m.M01 * i.M11 + m.M02 * i.M21; M02 = m.M00 * i.M02 + m.M01 * i.M12 + m.M02 * i.M22;
M10 = m.M10 * i.M00 + m.M11 * i.M10 + m.M12 * i.M20; M11 = m.M10 * i.M01 + m.M11 * i.M11 + m.M12 * i.M21; M12 = m.M10 * i.M02 + m.M11 * i.M12 + m.M12 * i.M22;
M20 = M01 * M12 - M02 * M11; M21 = M02 * M10 - M00 * M12; M22 = M00 * M11 - M01 * M10;
M03 = m.M03 * (1 - t) + n.M03 * t;
M13 = m.M13 * (1 - t) + n.M13 * t;
M23 = m.M23 * (1 - t) + n.M23 * t;
}
public static Quat FromMatrix33(Matrix33 m)
{
float s, p, tr = m.M00 + m.M11 + m.M22;
//check the diagonal
if (tr > (float)0.0)
{
s = (float)Math.Sqrt(tr + 1.0f); p = 0.5f / s;
return(new Quat(s * 0.5f, (m.M21 - m.M12) * p, (m.M02 - m.M20) * p, (m.M10 - m.M01) * p));
}
//diagonal is negative. now we have to find the biggest element on the diagonal
//check if "M00" is the biggest element
if ((m.M00 >= m.M11) && (m.M00 >= m.M22))
{
s = (float)Math.Sqrt(m.M00 - m.M11 - m.M22 + 1.0f); p = 0.5f / s;
return(new Quat((m.M21 - m.M12) * p, s * 0.5f, (m.M10 + m.M01) * p, (m.M20 + m.M02) * p));
}
//check if "M11" is the biggest element
if ((m.M11 >= m.M00) && (m.M11 >= m.M22))
{
s = (float)Math.Sqrt(m.M11 - m.M22 - m.M00 + 1.0f); p = 0.5f / s;
return(new Quat((m.M02 - m.M20) * p, (m.M01 + m.M10) * p, s * 0.5f, (m.M21 + m.M12) * p));
}
//check if "M22" is the biggest element
if ((m.M22 >= m.M00) && (m.M22 >= m.M11))
{
s = (float)Math.Sqrt(m.M22 - m.M00 - m.M11 + 1.0f); p = 0.5f / s;
return(new Quat((m.M10 - m.M01) * p, (m.M02 + m.M20) * p, (m.M12 + m.M21) * p, s * 0.5f));
}
return(Quat.Identity); // if it ends here, then we have no valid rotation matrix
}
public static Matrix33 CreateScale(Vec3 s)
{
var matrix = new Matrix33();
matrix.SetScale(s);
return(matrix);
}
public static Matrix33 CreateFromVectors(Vec3 vx, Vec3 vy, Vec3 vz)
{
var matrix = new Matrix33();
matrix.SetFromVectors(vx, vy, vz);
return(matrix);
}
public static Matrix33 CreateRotationXYZ(Vec3 rad)
{
var matrix = new Matrix33();
matrix.SetRotationXYZ(rad);
return(matrix);
}
public static Matrix33 CreateRotationZ(float rad)
{
var matrix = new Matrix33();
matrix.SetRotationZ(rad);
return(matrix);
}
public static Matrix33 CreateRotationAA(float c, float s, Vec3 axis)
{
var matrix = new Matrix33();
matrix.SetRotationAA(c, s, axis);
return(matrix);
}
public static Matrix33 CreateIdentity()
{
var matrix = new Matrix33();
matrix.SetIdentity();
return(matrix);
}
public static Matrix33 CreateRotationAA(Vec3 rot)
{
var matrix = new Matrix33();
matrix.SetRotationAA(rot);
return(matrix);
}
public static Matrix33 operator *(Matrix33 left, Matrix33 right)
{
var m = new Matrix33();
m.M00 = left.M00 * right.M00 + left.M01 * right.M10 + left.M02 * right.M20;
m.M01 = left.M00 * right.M01 + left.M01 * right.M11 + left.M02 * right.M21;
m.M02 = left.M00 * right.M02 + left.M01 * right.M12 + left.M02 * right.M22;
m.M10 = left.M10 * right.M00 + left.M11 * right.M10 + left.M12 * right.M20;
m.M11 = left.M10 * right.M01 + left.M11 * right.M11 + left.M12 * right.M21;
m.M12 = left.M10 * right.M02 + left.M11 * right.M12 + left.M12 * right.M22;
m.M20 = left.M20 * right.M00 + left.M21 * right.M10 + left.M22 * right.M20;
m.M21 = left.M20 * right.M01 + left.M21 * right.M11 + left.M22 * right.M21;
m.M22 = left.M20 * right.M02 + left.M21 * right.M12 + left.M22 * right.M22;
return m;
}
public static Matrix33 operator *(Matrix33 left, Matrix33 right)
{
var m = new Matrix33();
m.M00 = left.M00 * right.M00 + left.M01 * right.M10 + left.M02 * right.M20;
m.M01 = left.M00 * right.M01 + left.M01 * right.M11 + left.M02 * right.M21;
m.M02 = left.M00 * right.M02 + left.M01 * right.M12 + left.M02 * right.M22;
m.M10 = left.M10 * right.M00 + left.M11 * right.M10 + left.M12 * right.M20;
m.M11 = left.M10 * right.M01 + left.M11 * right.M11 + left.M12 * right.M21;
m.M12 = left.M10 * right.M02 + left.M11 * right.M12 + left.M12 * right.M22;
m.M20 = left.M20 * right.M00 + left.M21 * right.M10 + left.M22 * right.M20;
m.M21 = left.M20 * right.M01 + left.M21 * right.M11 + left.M22 * right.M21;
m.M22 = left.M20 * right.M02 + left.M21 * right.M12 + left.M22 * right.M22;
return(m);
}
public static Matrix33 CreateScale(Vec3 s)
{
var matrix = new Matrix33();
matrix.SetScale(s);
return matrix;
}
public static Matrix33 CreateRotationZ(float rad)
{
var matrix = new Matrix33();
matrix.SetRotationZ(rad);
return matrix;
}
public static Matrix33 CreateRotationXYZ(Vec3 rad)
{
var matrix = new Matrix33();
matrix.SetRotationXYZ(rad);
return matrix;
}
public static Matrix33 CreateRotationAA(Vec3 rot)
{
var matrix = new Matrix33();
matrix.SetRotationAA(rot);
return matrix;
}
public static Matrix33 CreateRotationAA(float c, float s, Vec3 axis)
{
var matrix = new Matrix33();
matrix.SetRotationAA(c, s, axis);
return matrix;
}
public static Matrix33 CreateIdentity()
{
var matrix = new Matrix33();
matrix.SetIdentity();
return matrix;
}
/*!
* Create a rotation matrix around an arbitrary axis (Eulers Theorem).
* The axis is specified as an normalized Vector3. The angle is assumed to be in radians.
* This function also assumes a translation-vector and stores it in the right column.
*
* Example:
* Matrix34 m34;
* Vector3 axis=GetNormalized( Vector3(-1.0f,-0.3f,0.0f) );
* m34.SetRotationAA( 3.14314f, axis, Vector3(5,5,5) );
*/
public void SetRotationAA(float rad, Vec3 axis, Vec3 t = default(Vec3))
{
this = new Matrix34(Matrix33.CreateRotationAA(rad, axis));
SetTranslation(t);
}
public Quat(Matrix33 matrix)
{
this = FromMatrix33(matrix);
}
/// <summary>
/// Direct-Matrix-Slerp: for the sake of completeness, I have included the following expression
/// for Spherical-Linear-Interpolation without using quaternions. This is much faster then converting
/// both matrices into quaternions in order to do a quaternion slerp and then converting the slerped
/// quaternion back into a matrix.
/// This is a high-precision calculation. Given two orthonormal 3x3 matrices this function calculates
/// the shortest possible interpolation-path between the two rotations. The interpolation curve forms
/// a great arc on the rotation sphere (geodesic). Not only does Slerp follow a great arc it follows
/// the shortest great arc. Furthermore Slerp has constant angular velocity. All in all Slerp is the
/// optimal interpolation curve between two rotations.
/// STABILITY PROBLEM: There are two singularities at angle=0 and angle=PI. At 0 the interpolation-axis
/// is arbitrary, which means any axis will produce the same result because we have no rotation. Thats
/// why I'm using (1,0,0). At PI the rotations point away from each other and the interpolation-axis
/// is unpredictable. In this case I'm also using the axis (1,0,0). If the angle is ~0 or ~PI, then we
/// have to normalize a very small vector and this can cause numerical instability. The quaternion-slerp
/// has exactly the same problems. Ivo
/// </summary>
/// <param name="m"></param>
/// <param name="n"></param>
/// <param name="t"></param>
/// <example>Matrix33 slerp=Matrix33::CreateSlerp( m,n,0.333f );</example>
public void SetSlerp(Matrix34 m, Matrix34 n, float t)
{
// calculate delta-rotation between m and n (=39 flops)
Matrix33 d = new Matrix33(), i = new Matrix33();
d.M00 = m.M00 * n.M00 + m.M10 * n.M10 + m.M20 * n.M20; d.M01 = m.M00 * n.M01 + m.M10 * n.M11 + m.M20 * n.M21; d.M02 = m.M00 * n.M02 + m.M10 * n.M12 + m.M20 * n.M22;
d.M10 = m.M01 * n.M00 + m.M11 * n.M10 + m.M21 * n.M20; d.M11 = m.M01 * n.M01 + m.M11 * n.M11 + m.M21 * n.M21; d.M12 = m.M01 * n.M02 + m.M11 * n.M12 + m.M21 * n.M22;
d.M20 = d.M01 * d.M12 - d.M02 * d.M11; d.M21 = d.M02 * d.M10 - d.M00 * d.M12; d.M22 = d.M00 * d.M11 - d.M01 * d.M10;
// extract angle and axis
double cosine = MathHelpers.Clamp((d.M00 + d.M11 + d.M22 - 1.0) * 0.5, -1.0, +1.0);
double angle = Math.Atan2(Math.Sqrt(1.0 - cosine * cosine), cosine);
var axis = new Vec3(d.M21 - d.M12, d.M02 - d.M20, d.M10 - d.M01);
double l = Math.Sqrt(axis | axis); if (l > 0.00001) axis /= (float)l; else axis = new Vec3(1, 0, 0);
i.SetRotationAA((float)angle * t, axis); // angle interpolation and calculation of new delta-matrix (=26 flops)
// final concatenation (=39 flops)
M00 = m.M00 * i.M00 + m.M01 * i.M10 + m.M02 * i.M20; M01 = m.M00 * i.M01 + m.M01 * i.M11 + m.M02 * i.M21; M02 = m.M00 * i.M02 + m.M01 * i.M12 + m.M02 * i.M22;
M10 = m.M10 * i.M00 + m.M11 * i.M10 + m.M12 * i.M20; M11 = m.M10 * i.M01 + m.M11 * i.M11 + m.M12 * i.M21; M12 = m.M10 * i.M02 + m.M11 * i.M12 + m.M12 * i.M22;
M20 = M01 * M12 - M02 * M11; M21 = M02 * M10 - M00 * M12; M22 = M00 * M11 - M01 * M10;
M03 = m.M03 * (1 - t) + n.M03 * t;
M13 = m.M13 * (1 - t) + n.M13 * t;
M23 = m.M23 * (1 - t) + n.M23 * t;
}
public Matrix34(Matrix33 m33)
: this()
{
}
public void SetRotation33(Matrix33 m33)
{
M00 = m33.M00; M01 = m33.M01; M02 = m33.M02;
M10 = m33.M10; M11 = m33.M11; M12 = m33.M12;
M20 = m33.M20; M21 = m33.M21; M22 = m33.M22;
}
public void SetRotationAA(Vec3 rot, Vec3 t = default(Vec3))
{
this = new Matrix34(Matrix33.CreateRotationAA(rot));
SetTranslation(t);
}
public void SetScale(Vec3 s, Vec3 t = default(Vec3))
{
this = new Matrix34(Matrix33.CreateScale(s));
SetTranslation(t);
}
public void SetRotationAA(float c, float s, Vec3 axis, Vec3 t = default(Vec3))
{
this = new Matrix34(Matrix33.CreateRotationAA(c, s, axis));
M03 = t.X; M13 = t.Y; M23 = t.Z;
}
/*!
*
* Convert three Euler angle to mat33 (rotation order:XYZ)
* The Euler angles are assumed to be in radians.
* The translation-vector is set to zero.
*
* Example 1:
* Matrix34 m34;
* m34.SetRotationXYZ( Ang3(0.5f,0.2f,0.9f), translation );
*
* Example 2:
* Matrix34 m34=Matrix34::CreateRotationXYZ( Ang3(0.5f,0.2f,0.9f), translation );
*/
public void SetRotationXYZ(Vec3 rad, Vec3 t = default(Vec3))
{
this = new Matrix34(Matrix33.CreateRotationXYZ(rad));
SetTranslation(t);
}
public void SetRotation33(Matrix33 m33)
{
M00 = m33.M00; M01 = m33.M01; M02 = m33.M02;
M10 = m33.M10; M11 = m33.M11; M12 = m33.M12;
M20 = m33.M20; M21 = m33.M21; M22 = m33.M22;
}
public static Quat FromMatrix33(Matrix33 m)
{
float s, p, tr = m.M00 + m.M11 + m.M22;
//check the diagonal
if (tr > (float)0.0)
{
s = (float)Math.Sqrt(tr + 1.0f); p = 0.5f / s;
return new Quat(s * 0.5f, (m.M21 - m.M12) * p, (m.M02 - m.M20) * p, (m.M10 - m.M01) * p);
}
//diagonal is negative. now we have to find the biggest element on the diagonal
//check if "M00" is the biggest element
if ((m.M00 >= m.M11) && (m.M00 >= m.M22))
{
s = (float)Math.Sqrt(m.M00 - m.M11 - m.M22 + 1.0f); p = 0.5f / s;
return new Quat((m.M21 - m.M12) * p, s * 0.5f, (m.M10 + m.M01) * p, (m.M20 + m.M02) * p);
}
//check if "M11" is the biggest element
if ((m.M11 >= m.M00) && (m.M11 >= m.M22))
{
s = (float)Math.Sqrt(m.M11 - m.M22 - m.M00 + 1.0f); p = 0.5f / s;
return new Quat((m.M02 - m.M20) * p, (m.M01 + m.M10) * p, s * 0.5f, (m.M21 + m.M12) * p);
}
//check if "M22" is the biggest element
if ((m.M22 >= m.M00) && (m.M22 >= m.M11))
{
s = (float)Math.Sqrt(m.M22 - m.M00 - m.M11 + 1.0f); p = 0.5f / s;
return new Quat((m.M10 - m.M01) * p, (m.M02 + m.M20) * p, (m.M12 + m.M21) * p, s * 0.5f);
}
return Quat.Identity; // if it ends here, then we have no valid rotation matrix
}
/// <summary>
/// x-YAW
/// y-PITCH (negative=looking down / positive=looking up)
/// z-ROLL
/// Note: If we are looking along the z-axis, its not possible to specify the x and z-angle.
/// </summary>
/// <param name="m"></param>
/// <returns></returns>
public static Angles3 CreateAnglesYPR(Matrix3x4 m)
{
Matrix33 m33 = new Matrix33(m);
return(CCamera.CreateAnglesYPR(m33));
}
public Quat(Matrix33 matrix)
{
this = FromMatrix33(matrix);
}
public static Matrix33 CreateFromVectors(Vec3 vx, Vec3 vy, Vec3 vz)
{
var matrix = new Matrix33();
matrix.SetFromVectors(vx, vy, vz);
return matrix;
}
public Matrix34(Matrix33 m33)
: this()
{
}
