Thread: Composition of Reflections

Hello again,

How would one find a pair of lines m and n such that r_n ° r_m is the translation that is the given transformation?

a) the translation that maps the origin onto (5,0)
b) rotation of 180 degrees about a point (x_0, y_0)
c) rotation of 40 degrees about origin
d) translation T with T(x,y) = (x+3, y+4)

I understand that this has to do with the two-reflection theorem for rotations and translations, but I can't seem to find any examples so that I can fully understand the concept. I came across these in an old textbook and was wondering if anyone would be able to explain how to do problems like this. I appreciate it!

Well, once you know the theorems, it's not difficult. For a), there are two translations (preserving orientation) that map (0, 0) to (5, 0): a shift 5 units to the right and a rotation by 180 degrees around (2.5, 0). The shift can be obtained as the composition of two reflections over any two vertical lines 2.5 units apart (first over the left one, then over the right one). The rotation is the composition of reflections over any two perpendicular lines that intersect at (2.5, 0).

Take d). The origin is mapped to (3, 4), so the distance between the two points is 5. Therefore, one needs any two parallel lines 2.5 units apart with the slope -3/4 (because -3/4 * 4/3 = -1; this is the condition that the lines with these slopes are perpendicular).