2D and 3D Transformations
The rigid, similarity, affine, and projective hierarchy, with composition and inversion.
Key Terms
RigidSimilarityAffineProjectiveCompositionInverse Transform
Python Tools
NumPyOpenCVMatplotlib
Textbook Reference
Szeliski Ch.2Hartley & Zisserman Ch.2
Topics Covered
- 1Translation, rotation, scale, and shear as matrices
- 2The transformation hierarchy: rigid, similarity, affine, projective, and their invariants
- 33D rotation representations and rigid motion
- 4Composing transformations across coordinate frames
- 5Inverting transformations and interpreting the results geometrically
Learning Outcomes
- Place a given transformation in the rigid, similarity, affine, or projective class
- Compose and invert 2D and 3D transformations
- Identify which geometric properties each transformation class preserves
Lecture Infographics
1 image file
Lecture Infographics
Transformations Interactive Demo
Visualise 2D & 3D coordinate transforms with live matrices
Transformation Matrix
Original
Transformed
Translation — Adds (tx, ty) to every point — pure displacement, no distortion. 2 DOF.
Object Transformation Walkthrough
A house shape with labeled vertices — watch coordinates update live as each point transforms
Formula
Live Coordinates
| Pt | x | y | → | x′ | y′ |
|---|---|---|---|---|---|
| A | 0.0 | -52.0 | → | 0.0 | -52.0 |
| B | -40.0 | -8.0 | → | -40.0 | -8.0 |
| C | 40.0 | -8.0 | → | 40.0 | -8.0 |
| D | 40.0 | 44.0 | → | 40.0 | 44.0 |
| E | -40.0 | 44.0 | → | -40.0 | 44.0 |
Original position
Transformed position
Translate — Every vertex shifts by the same vector (tx, ty). The shape, size, and orientation are all preserved — pure displacement, 2 DOF.
Lecture PDFs
2D and 3D Transformations
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