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 \((h, k)\): \[(x, y) \rightarrow ( h + (x – h) \cos \theta – (y – k) \sin \theta, k + (x – h) \sin \theta + (y – k) \cos \theta) \]

 

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