Kata

There are exactly eight ways to map a square onto itself by rotation or reflection. We can take its mirror image (reflection in horizontal, vertical, or either of two 45-degree diagonal lines); rotate it (by 90 degrees clockwise, 90 degrees anti-clockwise, or 180 degrees); or just leave it alone.

These eight transformations together comprise the dihedral group Dih(4).

In this kata we're going to write a class Dih4 that represents this collection of rotations and reflections. An object (instance) of this class will have one of eight possible values, corresponding to the eight transformations.

The Dih4 class will need to include at least the following features:

• Names or constants for the possible values:

  IDENTITY
  ROTATE_90_ANTICLOCKWISE
  ROTATE_180
  ROTATE_90_CLOCKWISE
  REFLECT_VERTICAL
  REFLECT_FORWARD_DIAGONAL
  REFLECT_HORIZONTAL
  REFLECT_REVERSE_DIAGONAL
  

  (these are in the same order as in the picture above).

• A function then() that works out the effect of performing transformations successively. For example, if r is the 90-degree clockwise rotation and f is the reflection with vertical mirror line, r.then(f) should equal the reflection with reverse-diagonal mirror line.

• A function inv() that gives the inverse (opposite) of a transformation. For example, if r is the 90-degree clockwise rotation, then r.inv() should equal the 90-degree anticlockwise rotation.

• Functions is_rotation() and is_reflection() that test whether a transformation is a rotation or a reflection. (The identity transformation is considered to be a rotation (by 0 degrees) but not a reflection.)

• Tests for equality and inequality: an equals() method (Java) or == and != operators (C++).

See the example tests for more examples of how the syntax of Dih4 objects should go.

Note to C++ codewarriors: some of the test cases use const Dih4 objects -- plan accordingly.