# Complex Algebra: Rotation

- Author:
- Ryan Hirst

NOTES:

**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.