Module IV·Article II·~1 min read
Affine and Projective Transformations in Space
Quadric Surfaces
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Projective Geometry in Space
Projective Space
We add a “plane at infinity” to $\mathbb{R}^3$: $\mathbb{R}P^3$. Points are nonzero vectors $(x:y:z:w)$ in homogeneous coordinates (up to scale).
The intersection of two distinct planes is a line. In projective geometry: any two planes intersect (parallel planes intersect at infinity).
Projection and Perspective
Perspective projection is a projective transformation. A pinhole camera projects a 3D scene onto a sensor (plane): $(X, Y, Z) \mapsto (fX/Z, fY/Z)$, where $f$ is the focal length.
In homogeneous coordinates: camera matrix $3 \times 4$ ($K[R|t]$).
Camera calibration = finding the parameters of the projective transformation. This is the basis of 3D reconstruction and augmented reality.
Desargues' Theorem
Two triangles are perspective from a point (corresponding vertices are joined by lines passing through one point) if and only if they are perspective from a line (intersections of corresponding sides lie on one line).
This is a purely projective result—it does not use distances and angles.
Duality
In the projective plane: “point” and “line” are dual. Theorem remains true if we swap “point” and “line”. Desargues' Theorem is self-dual.
In $\mathbb{R}P^3$: “point” and “plane” are dual.
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