Homogeneous and Projective Geometry
Points, lines, and planes in homogeneous coordinates, with incidence, duality, and points at infinity.
Key Terms
Python Tools
Textbook Reference
Topics Covered
- 1Homogeneous representation of points, lines, and planes
- 2Incidence, join, and intersection via dot and cross products
- 3Duality of points and lines in the projective plane
- 4Points and lines at infinity and the line at infinity
- 5Projective transformations acting on points and on lines
Learning Outcomes
- Convert between Euclidean and homogeneous representations
- Compute intersections and joins using cross products
- Explain points at infinity and how projective transformations move them
Lecture Infographics
1 image file
Lecture Infographics
Composite 3D Transformation Demo
Triangle vertices are transformed around an arbitrary pivot P: translate to origin, rotate and scale, then translate back.
Current stage
1. Translate pivot P to origin
Subtract P from each vertex so rotation and scale happen around the pivot, not the world origin.
Live parameters
(1.20, 0.80, 0.50)
0.00 deg
(1.00, 1.00, 1.00)
Process
Move pivot to origin
q = v - P
Apply local transform
r = S R q
Move back to world frame
v' = r + P
| Point | v - P | x' | y' | z' |
|---|---|---|---|---|
| A | (-0.85, -0.60, -0.25) | 0.35 | 0.20 | 0.25 |
| B | (0.95, -0.55, 0.10) | 2.15 | 0.25 | 0.60 |
| C | (-0.15, 1.10, -0.05) | 1.05 | 1.90 | 0.45 |
v' = T(P) S R T(-P) v, where v = (x, y, z, 1). Homogeneous coordinates let translation, rotation, and scale live inside one 4 x 4 matrix chain.
Lecture PDFs
Homogeneous and Projective Geometry
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