Intersecting lines [math]{\small l_0, l_1}[/math], with unit direction vectors [math]{\small u_0, u_1}[/math], and normals [math]{\small n_0, n_1}[/math]. [br][br]Let point [math]{\small X_0} [/math] on [math]{\small l_0} [/math] cast a ray in direction [math]{\small R_0}[/math]. What is the path of the ray?

NOTES:[br]The index m of the last bounce is based on the following observation {link prev}.[br][br]For [math]{\small {\bf u}_k} [/math]and [math]{\small {\bf n}_k}\, [/math] take [math] {\small k\, ({\rm mod} \;2)}[/math][br][br]Then the sequences are as follows:[br][list=1][br][*]Given the (k-1)st reflection vector, the next reflection has direction:[br] [math]\;\;\; {\small R_k = R_{k-1} - 2 (R_{k-1} \!\cdot\! {\bf n}_k) {\bf n}_k}[/math][br]Or, in complex numbers:**[br] [math] \;\;\; z_{Rk} = -\overline{z_{R_{k-1}}}(z_{n_{k-1}})^2 [/math][br][br][*]Let the kth bounce fall at point[br] [math]\;\;\; {\small X_k = A+ t_k {\bf u}_k}[/math] [br] Then the times are given by:[br] [math]\;\;\; t_k = t_{k-1}\frac{R_{k-1} \times {\bf u}_{k-1}}{R_{k-1} \times {\bf u}_k} [/math] [br][/list][br][br]Simplifying the recursions, I obtain the formulas shown in the worksheet. [br][br]But let me simplify a bit further...[br]_______[br][b]Light>Reflection[/b][br]Prev: [url]http://tube.geogebra.org/material/show/id/128176[/url][br]Next: [url]http://www.geogebratube.org/material/show/id/130551[/url][br][br]**[b]Rotation using Gauss numbers (Complex Algebra):[/b][br][url]http://tube.geogebra.org/material/show/id/115348[/url]