Halfturn and Glide Reflection Summary

Halfturn Summary

Definition: H_A, the halfturn with center A (also called the point reflection with center A) is the rotation A₁₈₀ with center A and angle 180 degrees.

A halfturn is its own inverse: A₁₈₀ A₁₈₀ = A₃₆₀ = I

A halfturn is the product of line reflections in perpendicular lines.

· If m and n are any two lines perpendicular at A, then H_A = R_mR_n.

· In particular, this means that also H_A = R_nR_m.

This is a special case of the general theorem about products of line reflections for lines intersecting at a point A. In general, the product is the rotation by twice the angle between the lines, so in this case, 2 * 90 = 180.

You can also see this fact directly from the same right angle figure we studied for the Carpenter theorem. The angle PQR is a right angle and distances AP = AQ = AR, so A is the circumcenter of the right triangle, so is the midpoint of PR.

Glide Reflection Summary

A Glide Reflection G is defined to be the composition G = TR of a line reflection R = R_k followed by a translation T = T_AB, where AB is parallel to m. (The notation T_AB means that T(A) = B. We can think of AB as the translation vector of T. We can choose A (and thus B) on line k if we wish.

T is the product of two line reflections T = R_mR_n. Since the direction of the translation is AB, parallel to k, then m and n are perpendicular to k.

So we can also write G = R_mR_nR_k. G is the product of 3 line reflections, two of which are perpendicular to the third. If we set A = intersection of n and k, then m passes through C, the midpoint of A and B.

· If we group G = (R_mR_n)R_k, we see that G = TR.

· But if we group G = R_m(R_nR_k), we see that G = R_mH_A.

Since line reflection in perpendicular lines commutes, we also have

G = R_mR_nR_k = R_mR_kR_n = R_kR_mR_n