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2D roation with exp(I phi)

A general rotation in two dimensions can be described by writing it in components parallel and
orthogonal to the vector to be rotated:

x' = x cos(phi) + x I sin(phi) = x [cos(phi) + I sin(phi)] = x exp(I phi)
= exp(-I phi/2) x exp (I phi/2),

where I is the unit magnitude bivector of the rotation plane. As demonstrated
elsewhere,
xI = x.I (if x is in the I plane) will always be orthogonal to x. The last equality
follows from II = -1, xI = -Ix, cos(phi/2)cos(phi/2) - sin(phi/2)sin(phi/2) = cos(phi) and
2sin(phi/2)cos(phi/2) = sin(phi). This "double sided" expression corresponds both to the
descriptions of rotations by two reflections or by
rotors.

You can change the bright red points interactively by dragging them with the mouse.

The basic idea for this applet stems from the "DEMOrotdefinition" in Dorst, Mann and Bouma's
geometric algebra MATLAB tutorial GABLE.

[ GA with Cinderella ]

Soli Deo Gloria. Created with Cinderella by E. Hitzer (Fukui).