# Transformation Rules

We’re working on rigid transformations in my Geometry classes. The basic transformation rules for translation and reflection over a vertical or horizontal line are straightforward; here, they’re written as functions, rather than the briefer vector notation.

• Translation of $$h$$ horizontally and $$k$$ vertically: $(x, y) \rightarrow (x + h, y + k)$
• Reflection over a vertical line $$y = k$$ or the x-axis (where $$k = 0$$): $(x, y) \rightarrow (x, 2k – y)$
• Reflection over a horizontal line $$x = h$$ or the y-axis (where $$h = 0$$): $(x, y) \rightarrow (2h – x, y)$

I’m only expecting my students to understand and apply the rules above, but I decided to work out the other cases for the sake of completeness. As I told them, the mathematics gets quite a bit more complicated:

• Reflection over the line $$y = mx + b$$: $(x, y) \rightarrow \left(\frac{(1-m^2)x+2m(y-b)}{m^2 + 1}, \frac{2(mx+b) – (1-m^2)y}{m^2 + 1}\right)$
• Counterclockwise rotation of $$\theta$$ around the point $$(h, k)$$: $(x, y) \rightarrow ( h + (x – h) \cos \theta – (y – k) \sin \theta, k + (x – h) \sin \theta + (y – k) \cos \theta)$