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:

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;
}

metadata block
see also:

 

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

 

This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2017 Martin John Baker - All rights reserved - privacy policy.