11.3 Projective transformations and homogeneous coordinates

Projective transformations are powerful tools in non-Euclidean geometry. They extend affine transformations to include projections, preserving collinearity and incidence while allowing for more flexible manipulations of geometric objects.

Homogeneous coordinates provide a unified way to represent points and lines in projective space. This system simplifies calculations and enables the representation of points and lines at infinity, making it essential for working with projective transformations.

Projective Transformations and Homogeneous Coordinates

Properties of projective transformations

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• Generalize affine transformations (translations, rotations, scaling, shearing) to include projections
• Preserve collinearity meaning if points lie on a line, they remain on a line after transformation
• Preserve incidence so if a point lies on a line or curve, it remains on the transformed line or curve
• Preserve cross-ratio, the ratio of ratios of distances between four collinear points
• Do not necessarily preserve parallelism (parallel lines may not remain parallel), angles, or distances
• Can be represented by invertible $4 \times 4$ matrices using homogeneous coordinates
• Composition of projective transformations corresponds to matrix multiplication of their representing matrices
• Set of all projective transformations forms a group under composition operation

Homogeneous coordinates in projective geometry

• Represent points and lines in projective space using 4-tuples
• Point $(x, y, z)$ in Euclidean space represented as $(x, y, z, 1)$ in homogeneous coordinates
• Point at infinity represented as $(x, y, z, 0)$ where at least one of $x$, $y$, or $z$ is non-zero
• Lines represented by 4-tuple $[a, b, c, d]$ satisfying equation $ax + by + cz + d = 0$
• Line at infinity represented by $[0, 0, 0, 1]$
• Incidence of point $(x, y, z, w)$ and line $[a, b, c, d]$ given by $ax + by + cz + dw = 0$
• Homogeneous coordinates allow representation of points and lines at infinity and simplify computations

Matrix representations for projective transformations

• Projective transformation represented by $4 \times 4$ matrix $M$
• Transformed point $P'$ obtained by multiplying $M$ with homogeneous coordinates of original point $P$: $P' = MP$
• Inverse of projective transformation matrix $M$ is another projective transformation matrix $M^{-1}$
• Composition of projective transformations $M_1$ and $M_2$ achieved by multiplying their matrices: $M_2M_1$
• Matrix representation simplifies computation and composition of projective transformations

Invariants under projective transformations

• Collinearity: Points on a line remain on a line after transformation
• Incidence: Point on a line or curve remains on the transformed line or curve
• Cross-ratio of four collinear points $A$, $B$, $C$, $D$ defined as $\frac{AC \cdot BD}{BC \cdot AD}$ remains invariant
• Degree of an algebraic curve (number of intersections with a line) remains invariant
• Invariants are properties that remain unchanged under projective transformations and are useful for analysis and recognition