Geometric Transformations

Geometric transformations are operations that can change the position, size, and orientation of geometric figures. They are fundamental in the field of geometry and have numerous applications in various fields, including computer graphics, image processing, and physics.

Types of Geometric Transformations

  1. Translation: A translation (or slide) moves every point of a figure the same distance in the same direction.

  2. Rotation: A rotation turns a figure around a fixed point, called the center of rotation. The amount of turn is specified by the angle of rotation.

  3. Reflection: A reflection (or flip) flips a figure over a line, called the line of reflection. Each point of the figure and its image are the same distance from the line of reflection.

  4. Dilation: A dilation (or scaling) enlarges or reduces a figure by a scale factor, relative to a fixed center point. The scale factor is the amount by which each dimension of the figure is lengthened or shortened.

Invariant Properties

Despite these transformations, certain properties of the figures remain unchanged, or invariant. These include:

  • Distance (Length): Under translation, rotation, and reflection, distances are preserved. This means that the length of a line segment, the perimeter of a polygon, or the circumference of a circle remains unchanged. However, dilation can change these distances.

  • Angle Measure: Under translation, rotation, and reflection, angle measures are preserved. This means that the measures of angles in a polygon or the measures of central angles in a circle remain unchanged. However, dilation does not affect angle measures.

  • Parallelism: Parallel lines remain parallel under all four types of transformations.

  • Collinearity: Points that are on the same line remain on the same line under all four types of transformations.