Theorem 4.11

Theorem 4.11: If T is the translation

T(x, y) = (x - x₀, y - y₀)

U(x, y) = (x + x₀, y + y₀)

is the inverse of T.

Proof: If we compose the two translations, we get

T(U(x, y))

= T(x + x₀, y + y₀)

=((x + x₀) - x₀, (y + y₀) - y₀)

= (x, y)

If we compose them in the other order, we get

U(T(x, y))

= U(x - x₀, y - y₀)

=((x - x₀) + x₀, (y - y₀) + y₀)

= (x, y)

so U acts like an inverse for T.