Maths - Affine Conversion Euler To Matrix

Maths - Affine Conversion - Euler/Vector To Matrix

Definition of terms:

It you want to consider the rotation only then the code for this is shown here.

Alternatively if we want to consider the possibility that there is also a translation from the centre, and that the rotation may not be about the centre but may be about some arbitrary point, then we need to extend the notation as follows:

A 3D vector has to be added to the euler angles so that together they can specify the position and orientation.
The matrix has to be increased from 3x3 to 4x4 so that it can specify the position and orientation. (as explained here).

For the derivation of how to rotate about a point see this page.

void setRotate(Point3d A, // rotate about a line centred on A
   double theta,
   double phi,
   double alpha)
   { double cosAlpha, sinAlpha, cosPhi, sinPhi,
   cosTheta, sinTheta, cosPhi2, sinPhi2,
   cosTheta2, sinTheta2, pi, c, a1,a2,a3;
   if (A == null) {
     a1=a2=a3=0;
   } else {
     a1 = A.x; a2 = A.y; a3 = A.z;
   }
   cosPhi = Math.cos(phi); sinPhi = Math.sin(phi);
   cosPhi2 = cosPhi * cosPhi; sinPhi2 = sinPhi * sinPhi;
   cosTheta = Math.cos(theta);
   sinTheta = Math.sin(theta);
   cosTheta2 = cosTheta * cosTheta;
   sinTheta2 = sinTheta * sinTheta;
   cosAlpha = Math.cos(alpha);
   sinAlpha = Math.sin(alpha);
   c = 1.0 - cosAlpha;
   m00 = cosTheta2 * (cosAlpha * cosPhi2 + sinPhi2)
   + cosAlpha * sinTheta2;
   m10 = sinAlpha * cosPhi + c * sinPhi2 * cosTheta * sinTheta;
   m20 = sinPhi * (cosPhi * cosTheta * c - sinAlpha * sinTheta);
   m01 = sinPhi2 * cosTheta * sinTheta * c - sinAlpha * cosPhi;
   m11 = sinTheta2 * (cosAlpha * cosPhi2 + sinPhi2)
   + cosAlpha * cosTheta2;
   m21 = sinPhi * (cosPhi * sinTheta * c + sinAlpha * cosTheta);
   m02 = sinPhi * (cosPhi * cosTheta * c + sinAlpha * sinTheta);
   m12 = sinPhi * (cosPhi * sinTheta * c - sinAlpha * cosTheta);
   m22 = cosAlpha * sinPhi2 + cosPhi2;
   m03 = a1 - a1 * m00 - a2 * m01 - a3 * m02;
   m13 = a2 - a1 * m10 - a2 * m11 - a3 * m12;
   m23 = a3 - a1 * m20 - a2 * m21 - a3 * m22;
   }

Rotation about a point - using Euler

/** rotate about a line centred on A */
void setRotate(sfvec3f A,double theta,double phi,double alpha){
   double cosAlpha, sinAlpha, cosPhi, sinPhi,
   cosTheta, sinTheta, cosPhi2, sinPhi2,
   cosTheta2, sinTheta2, pi, c, a1,a2,a3;
   if (A == null) {
     a1=a2=a3=0;
   } else {
     a1 = A.x; a2 = A.y; a3 = A.z;
   }
   cosPhi = Math.cos(phi); sinPhi = Math.sin(phi);
   cosPhi2 = cosPhi * cosPhi; sinPhi2 = sinPhi * sinPhi;
   cosTheta = Math.cos(theta);
   sinTheta = Math.sin(theta);
   cosTheta2 = cosTheta * cosTheta;
   sinTheta2 = sinTheta * sinTheta;
   cosAlpha = Math.cos(alpha);
   sinAlpha = Math.sin(alpha);
   c = 1.0 - cosAlpha;
   m00 = cosTheta2 * (cosAlpha * cosPhi2 + sinPhi2) + cosAlpha * sinTheta2;
   m10 = sinAlpha * cosPhi + c * sinPhi2 * cosTheta * sinTheta;
   m20 = sinPhi * (cosPhi * cosTheta * c - sinAlpha * sinTheta);
   m01 = sinPhi2 * cosTheta * sinTheta * c - sinAlpha * cosPhi;
   m11 = sinTheta2 * (cosAlpha * cosPhi2 + sinPhi2) + cosAlpha * cosTheta2;
   m21 = sinPhi * (cosPhi * sinTheta * c + sinAlpha * cosTheta);
   m02 = sinPhi * (cosPhi * cosTheta * c + sinAlpha * sinTheta);
   m12 = sinPhi * (cosPhi * sinTheta * c - sinAlpha * cosTheta);
   m22 = cosAlpha * sinPhi2 + cosPhi2;
   m03 = a1 - a1 * m00 - a2 * m01 - a3 * m02;
   m13 = a2 - a1 * m10 - a2 * m11 - a3 * m12;
   m23 = a3 - a1 * m20 - a2 * m21 - a3 * m22;
   m30 = m31 = m32 = 0.0;
   m33 = 1.0;
}

Rotation about a point - using Axis Angle

public void setRotate(sfrotation rot,sfvec3f A) {

    double c = Math.cos(rot.angle);
    double s = Math.sin(rot.angle);
    double t = 1.0 - c;
    m00 = c + rot.x*rot.x*t;
    m11 = c + rot.y*rot.y*t;
    m22 = c + rot.z*rot.z*t;


    double tmp1 = rot.x*rot.y*t;
    double tmp2 = rot.z*s;
    m10 = tmp1 + tmp2;
    m01 = tmp1 - tmp2;


    tmp1 = rot.x*rot.z*t;
    tmp2 = rot.y*s;
    m20 = tmp1 - tmp2;
    m02 = tmp1 + tmp2;
	
    tmp1 = rot.y*rot.z*t;
    tmp2 = rot.x*s;
    m21 = tmp1 + tmp2;
    m12 = tmp1 - tmp2;


   double a1,a2,a3;
   if (A == null) {
     a1=a2=a3=0;
   } else {
     a1 = A.x; a2 = A.y; a3 = A.z;
   }

    m03 = a1 - a1 * m00 - a2 * m01 - a3 * m02;
    m13 = a2 - a1 * m10 - a2 * m11 - a3 * m12;
    m23 = a3 - a1 * m20 - a2 * m21 - a3 * m22;
    m30 = m31 = m32 = 0.0;
    m33 = 1.0;
}

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This is a version of basic designed for building games, for example to rotate a cube you might do the following:
make object cube 1,100
for x=1 to 360
rotate object 1,x,x,0
next x

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