Posted by dxli on Jun 07, 2013; 3:01pm
URL: https://forum.librecad.org/don-t-understand-how-rotation-is-done-tp5708244p5708246.html

2D rotation around origin can be down equivalently by:

1, angle;
2, unit complex number (vector);
3, matrix;

3D rotation around origin is trickier, but can be rotated equivalently by:

1, axis-angle;
2, unit quaternion; (vector)
3, matrix;

to rotate a complex by a unit complex number:

z= x + i y;
angleVector = cos t + i sin t;

rotate z by angleVector using multiplication of complex number:

(x + i y) ( cos t + i sin t) = x cos t - y sin t + i ( x sin t + y cos t)

to rotate around any point P, simply translate P to origin, do rotation around origin, translate back by -P, something like conjugate,

Translate(P)^-1 Rotate(angle) Translate(P)

forgot to answer on ellipse angle:

ellipse angle is the angle used in parametric equation of ellipse, namely,

x = a cos t
y = b sin t

and the angle t is called ellipse angle, which is related to the actual angle of the point by:

tan(angle)=x/y = a/b tan(t)