Gauss Numbers: A complex number is a Gauss number if it lies on the (imaginary) unit circle: . A Gauss number satisfies . The equivalence between vectors and complex numbers used in the worksheet is based on these two definitions.

Reflection: For any Gauss number z, , which can be checked algebraically. If θ> π /2, the transformation z² no longer corresponds to minimum rotation. Whether or not the rotation is, in fact, "by 2θ", depends on the problem. Here are two examples:
A. A swinging door, like the Saloon doors in movies. Suppose they swing equally far in both directions. Now, it can't rotate through the wall. From above, let O be the hinge position, d the unit vector representing the door, and x the unit vector facing the center position. If θ is the angle from d to x (from the door back into the open doorway), the motion corresponding to "reflection about d" will, in fact, always take place by the signed angle 2θ.
B. Reflection of light. We may take the surface normal N in whatever direction we like. The reflected ray does not cross the threshold. "Reflection about N" will always correspond to minimum rotation: http://www.geogebratube.org/material/show/id/111895

Reflect vector s (example 1): We can just use the unit vectors of b2 and s, and rotate using (4). Why didn't I? The circle is defined by vector r. Consider its equation:
And point S on the curve:
Point O translates the curve, R scales and rotates. For these transformations, is constant. It only needs to be updated when S is manipulated directly. Likewise, changes only when B, on the unit circle, is moved.