Contents
- Typical Types of Transformation of 2D Planes
- Affine Transformation
- Projective Transformation
- Perspecive Transformation
- References
Typical Types of Transformation of 2D Planes
Key differences about projective and affine transformations:
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projective: lines mapped to lines, but parallelism may not be kept;
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affine: collinearity and parallelism are both kept
Affine Transformation
A transformation that preserves lines and parallelism (maps parallel lines to parallel lines) is an affine transformation. There are two important particular cases of such transformations:
A nonproportional scaling transformation centered at the origin has the form
where a,b != 0 are the scaling factors (real numbers). The corresponding matrix in homogeneous coordinates is
A shear preserving horizontal lines has the form
where r is the shearing factor (see Figure 1). The corresponding matrix in homogeneous coordinates is
Figure 1: A shear with factor r=1/2.
Every affine transformation is obtained by composing a scaling transformation with an isometry, or a shear with a homothety and an isometry.
Projective Transformation
A transformation that maps lines to lines (but does not necessarily preserve parallelism) is a projective transformation. Any plane projective transformation can be expressed by an invertible 3×3 matrix in homogeneous coordinates:
Conversely, any invertible 3×3 matrix defines a projective transformation of the plane. Projective transformations (if not affine) are not defined on all of the plane, but only on the complement of a line (the missing line is “mapped to infinity”).
Figure above: In projective transformations (if not affine), a vanishing line in infinity can be warped to be a finite line.
Perspecive Transformation
The term perspecive transformation is also commonly seen. Perspective transformation projects a 3D geometric object into a 2D plane. It can be seen as a common example of projective transformation. Strictly speaking it gives a transformation from one plane to another, but if we identify the two planes by (for example) fixing a cartesian system in each, we get a projective transformation from the plane to itself, as shown in the figure below.
Figure above: A perspective transformation with center O, mapping the plane P to the plane Q. The transformation is not defined on the line L, where P intersects the plane parallel to Q and going throught O.
What is the main difference between perspectivity and general projectivity?
The distinctive property of a perspectivity is thatlines joining corresponding points are concurrent. The difference between a perspectivity and projectivity is made clear by considering the composition of two perspectivities.
The composition of two perspectivities is not in general a perspectivity. However, the composition is a projectivity because a perspectivity is a projectivity, and projectivities form a group (closed), so that the composition of two projectivities is a projectivity.
This can be illustrated by the figure below:
Figure above: A line projectivity. Points {a, b, c} are related to points {A, B, C} by a line-to-line perspectivity. Points {a’, b’, c’} are also related to points {A, B, C} by a perspectivity. However, points {a, b, c} are related to points {a’, b’, c’} by a projectivity; they are not related by a perspectivity because lines joining corresponding points are not concurrent. In fact the pairwise intersections result in three distinct points {p, q, r}.
References
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Multiview Geometry of Computer Vision