NOTES: [list] [*][b]Gauss Numbers:[/b] A complex number is a Gauss number if it lies on the (imaginary) unit circle: [math]{\small z= \cos θ + i \sin θ}[/math]. A Gauss number satisfies [math] {\small |z| = \sqrt{z\overline{z}} = 1}[/math]. The equivalence between vectors and complex numbers used in the worksheet is based on these two definitions. [*][b]Reflection:[/b] For any Gauss number z, [math] { \small z^2 = \cos(2θ) + i \sin(2θ) }[/math], 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. [i]A swinging door[/i], 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, [b]d[/b] the unit vector representing the door, and [b]x[/b] the unit vector facing the center position. If θ is the angle from [b]d[/b] to [b]x[/b] (from the door back into the open doorway), the motion corresponding to "reflection about [b]d[/b]" will, in fact, always take place by the [i]signed [/i]angle 2θ. B. [i]Reflection of light.[/i] We may take the surface normal [b]N[/b] in whatever direction we like. The reflected ray [i]does not[/i] cross the threshold. "Reflection about N" will always correspond to minimum rotation: [url]http://www.geogebratube.org/material/show/id/111895[/url] [*][i]Reflect vector s (example 1):[/i] 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: [math] \;\; {\small{\bf x}(t) = O + (\cos t + i \sin t) {\bf r},\;\;\; -π ≤ t ≤ π }[/math] And point S on the curve: [math] \;\; {\small S = {\bf x}(t_s) = O + (\cos t_s + i \sin t_s) {\bf r}}[/math] [math] \;\;\;\;\; {\small =O + ρ_s{\bf r}}[/math] Point O translates the curve, R scales and rotates. For these transformations, [math] ρ_s[/math] is [i]constant[/i]. It only needs to be updated when S is manipulated directly. Likewise, [math] z_δ[/math] changes only when B, on the unit circle, is moved. [/list]