Invariance in Transformations

Describing Changes in Orientation

The orientation (the 'up' direction or facing of a shape) behaves differently depending on the transformation. First, during a translation, orientation is entirely invariant, meaning the shape remains the same way up. Next, during a rotation, the orientation changes as the shape spins, but the order of the vertices (e.g., reading them clockwise) is fully preserved.

Finally, during a reflection, the orientation is specifically reversed (or flipped). If vertices $A, B, C$ are arranged clockwise on the object, they will be anti-clockwise on the image.

Tracking Composite Transformations

When an object undergoes composite transformations (multiple transformations applied in a row), you must track the changes step-by-step. First, apply the initial transformation to the object and note the position and orientation of the intermediate image. Then, apply the second transformation directly to this intermediate image to find the final position.

Observe the overall changes when combining reflections. Applying two reflections in parallel lines will result in a translation. Applying two reflections in lines intersecting at angle $\theta$ creates a rotation of $2\theta$ about the point of intersection.

Completing an even number of reflections will always restore the original clockwise or anti-clockwise vertex order.

Invariant Points and Lines

Sometimes, specific locations do not move at all during a transformation. An invariant point is a coordinate that maps exactly onto itself, remaining identical before and after. An invariant line is a line where every point maps to another point on that exact same line, keeping the line's overall position fixed.

A line of invariant points goes further; every single point on it remains fixed in its exact original spot.

Specific Invariant Features by Transformation

Different transformations have unique invariant features. For reflections, the mirror line is the only line of invariant points, while any line perpendicular to it is an invariant line. For a translation by a non-zero vector, there are NO invariant points whatsoever, though any line parallel to the movement direction is an invariant line.

For rotations, the center of rotation is the only invariant point (unless it is a full 360-degree turn). Additionally, for a 180° rotation, any straight line passing through the center of rotation is an invariant line.

Calculating Translations with Column Vectors

We describe a translation using a column vector, written mathematically as $\binom{x}{y}$. The top number represents the horizontal displacement (positive means move right, negative means move left), and the bottom number represents the vertical displacement (positive means move up, negative means move down). To find the new coordinates, you mathematically add the vector components to the original coordinates.

Worked Example: Translating a Coordinate

A point $P$ at $(2, -1)$ is translated by the column vector $\binom{-5}{5}$. Calculate the coordinates of the new image $P'$.

Step 1: Write down the mapping formula.

• To translate a point $(x, y)$ by vector $\binom{a}{b}$, the image is calculated as $(x+a, y+b)$.

Step 2: Identify the components.

• Original coordinate: $x = 2$, $y = -1$
• Horizontal displacement: $a = -5$
• Vertical displacement: $b = 5$

Step 3: Substitute the values into the formula.

• New $x$-coordinate $= 2 + (-5)$
• New $y$-coordinate $= -1 + 5$

Step 4: Calculate the final answer.

• $P'$ is at $(-3, 4)$.