Suppose H is a glide reflection What is the nature of H o HS

Suppose H is a glide reflection. What is the nature of H o H?

Solution

The Glide Reflection is an isometry because it is defined as the composition of two isometries.

We know that the composition of two isometries is an isometry. (as proved below)

Proof: Let H₁ and H₂ be two isometries.

If H₁(A) = A\' and H₁(B) = B\', Then by isometry H₁, we have AB = A\'B\'

If H₂(A) = A\'\' and H₂(B) = B\'\', Then by isometry H₂, we have AB = A\'\'B\'\'

Now, the composition of the above two isometries, we have

H₂ º H₁ = H₂(H₁(A)) = H₂(A\') = A\'\'\'. and

H₂ º H₁ = H₂(H₁(A)) = H₂(B\') = B\'\'\'.

Therefore, under composition AB = A\'\'\'B\'\'\' and distance is preserved.

Hence, the nature of H º H will be a glide reflection.