Non-Rigid Transformations- Definitions and Examples in Geometry
What Are Non-Rigid Transformations?
Non-rigid transformations change a figure's size or shape. Unlike rigid transformations, which only move a shape without altering it, non-rigid transformations stretch, shrink, or distort the original figure. The result looks different from the starting shape.
The key point: congruence is lost. After a non-rigid transformation, the image is similar to the original but never identical in both shape and size.
Types of Non-Rigid Transformations
Dilation 🔍
Dilation resizes a shape by a scale factor. The shape stays the same, but everything gets bigger or smaller. A scale factor greater than 1 enlarges the figure. A scale factor between 0 and 1 shrinks it.
Example: A triangle with vertices at (0,0), (2,0), and (0,2) dilated by a factor of 3 becomes a triangle with vertices at (0,0), (6,0), and (0,6). The shape is identical. The size tripled.
Stretching
Stretching pulls a shape in one direction. You pick a line called the axis of transformation, and points move away from or toward that line.
Horizontal stretches move points left or right. Vertical stretches move points up or down. The shape distorts in one direction only.
Example: A square stretched vertically by a factor of 2 becomes a rectangle twice as tall. The width stays the same.
Compression
Compression is the opposite of stretching. Points get pushed closer together along the chosen axis. The shape squishes in one direction.
Example: A rectangle compressed horizontally by a factor of 0.5 becomes half as wide. Height remains unchanged.
Non-Rigid vs Rigid Transformations
Here's the direct comparison:
| Feature | Rigid Transformations | Non-Rigid Transformations |
|---|---|---|
| Size | Preserved | Changes |
| Shape | Preserved | Preserved (similar) or distorted |
| Congruence | Original and image are congruent | Original and image are NOT congruent |
| Similarity | Always similar | Always similar (for uniform scaling) |
| Types | Translation, rotation, reflection | Dilation, stretch, compression |
Bottom line: rigid transformations keep the original intact in every measurable way. Non-rigid transformations give you something that looks related but measures differently.
How to Identify Non-Rigid Transformations
Ask these questions:
- Does the image look bigger or smaller than the original? → Dilation
- Does the image look stretched in one direction? → Stretch
- Does the image look squished or compressed? → Compression
- Are the angles preserved but side lengths changed? → Dilation
- Are the side lengths proportional to the original? → Dilation
The fastest way to check: measure one side of the original, then measure the corresponding side of the image. If the ratio is not 1:1, you're dealing with a non-rigid transformation.
Real-World Applications
Non-rigid transformations aren't just classroom material.
- Graphic design and digital imaging — Resizing photos uses dilation. Warping images for special effects uses stretching and compression.
- Architecture and engineering — Scale drawings apply dilation. Models are proportional copies of real structures.
- Medical imaging — MRI and CT scans use transformations to adjust and compare images at different scales.
- Animation — Character rigging and deformation rely on non-rigid transformations to make movement look natural.
Getting Started: How to Perform a Dilation
Step 1: Identify the center of dilation. This is the fixed point everything expands from or contracts toward.
Step 2: Find the scale factor. A number greater than 1 enlarges. A number between 0 and 1 shrinks.
Step 3: Multiply each coordinate of every vertex by the scale factor, relative to the center point.
Step 4: Plot the new points and connect them.
Example calculation: Take point (3, 4) with center (0,0) and scale factor 2. New point: (6, 8). That's it.
For stretches and compressions, pick your axis first. Then multiply the perpendicular distance from that axis by your scale factor. Points on the axis stay fixed.