A geometric reflection flips a shape over a line of reflection. The coordinates of any reflected point are determined by these standard transformation rules:

• Reflection across the x-axis: $(x, y) ightarrow (x, -y)$
• Reflection across the y-axis: $(x, y) ightarrow (-x, y)$
• Reflection across the line $y = x$: $(x, y) ightarrow (y, x)$
• Reflection across the line $y = -x$: $(x, y) ightarrow (-y, -x)$
• Reflection across the origin: $(x, y) ightarrow (-x, -y)$

Reflections have distinct mathematical properties that govern how geometric shapes behave under the transformation:

• Isometry (Distance-Preserving): The distance between any two points in the original shape is identical to the distance between their corresponding reflected points.
• Angle-Preserving: The angles of a polygon remain unchanged. The original shape and the reflected shape are congruent.
• Orientation Reversal: The orientation of the shape is reversed. For example, a clockwise triangle will become counter-clockwise when reflected.
• Involutive (Double Reflection): Reflecting a shape twice across the same line returns it to its original position: $R(R(P)) = P$.