Definition
Geometric Transformations are operations that move or change a geometric figure (the pre-image) to produce a new figure (the image). Isometries (or rigid motions) are transformations that preserve distance and angle measure, resulting in an image congruent to the pre-image.
Why It Matters
Transformation theory allows us to identify the ‘invariants’ in a changing world; by understanding what stays the same when a shape is slid, flipped, or rotated, we gain the foundation for modern computer graphics, robotic control, and the laws of physical symmetry.
Core Concepts
- Translation (Slide): Moves every point of a figure the same distance in the same direction. Defined by a vector .
- How to read: “The vector v.”
- Meaning: A rigid motion — distances and angles are preserved; the figure slides without rotating or flipping.
- Reflection (Flip): Flips a figure over a line called the axis of reflection. Each point and its image are equidistant from this line.
- Rotation (Turn): Turns a figure about a fixed point called the center of rotation through a specific angle.
- Dilation (Resizing): A non-isometric transformation that changes the size of a figure but not its shape. It is defined by a center and a scale factor (). Dilations result in similar rather than congruent figures.
- How to read: “The scale factor k.”
- Meaning: enlarges, shrinks; shape (angle measures) stays the same but side lengths scale uniformly.
- Symmetry: A property of a figure that remains unchanged under a specific transformation (e.g., reflectional symmetry or rotational symmetry).
- Mapping Notation: A transformation is a function that maps every point in the plane to a unique image point.
- How to read: “The transformation T maps point P to point P prime.”
- Meaning: Every pre-image point is sent to exactly one image point — a function on the set of geometric points.