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2.1 Geometric primitives and transformations 33

y projective
similarity
translation

Euclidean affine
x

Figure 2.4 Basic set of 2D planar transformations.

3D quadrics. The 3D analog of a conic section is a quadric surface

T
¯ x Q¯x =0 (2.13)
(Hartley and Zisserman 2004, Chapter 2). Again, while quadric surfaces are useful in the
study of multi-view geometry and can also serve as useful modeling primitives (spheres,
ellipsoids, cylinders), we do not study them in great detail in this book.

2.1.2 2D transformations

Having defined our basic primitives, we can now turn our attention to how they can be trans-
formed. The simplest transformations occur in the 2D plane and are illustrated in Figure 2.4.

Translation. 2D translations can be written as x = x + t or

x = I t ¯ x (2.14)
where I is the (2 × 2) identity matrix or

I t

¯ x = ¯ x (2.15)
0 T 1
where 0 is the zero vector. Using a 2 × 3 matrix results in a more compact notation, whereas
using a full-rank 3 × 3 matrix (which can be obtained from the 2 × 3 matrix by appending a
T
[0 1] row) makes it possible to chain transformations using matrix multiplication. Note that
in any equation where an augmented vector such as ¯x appears on both sides, it can always be
replaced with a full homogeneous vector ˜x.

Rotation + translation. This transformation is also known as 2D rigid body motion or
the 2D Euclidean transformation (since Euclidean distances are preserved). It can be written
as x = Rx + t or

x = R t ¯ x (2.16)
where

cos θ − sin θ
R = (2.17)
sin θ cos θ
T
is an orthonormal rotation matrix with RR = I and |R| =1.