Symmetry plays a crucial role in everyday life; many man-made objects possess symmetry, and this has been exploited in vision systems [55, 102, 120, 128], [5, 9]. It is therefore reasonable to ask what happens to symmetry under a general projective transformation.
Two points are said to possess symmetry with respect to a line, the axis of sym-metry, if the line is the perpendicular bisector of the line segment joining the two points. They are said to be symmetric with respect to a third point, the centre of symmetry, if the third point bisects the line joining the points [65]. Symmetry is therefore an inherently non-projective quality, since it depends on the invariance of
50 Symmetry under Projective Transformations
midpoints and, in the case of axial symmetry, angles3. It is nonetheless possible to identify some properties of symmetry that do not rely on the invariance of angles or the ratio of lengths and this will be done in Section 2.8.1. We will then see in Section 2.8.2 that these properties describe a particular type of projective trans-formation, namely a plane harmonic homology. Section 2.8.3 finally shows that a plane harmonic homology maintains its structure under arbitrary projective trans-formations, and therefore is the most appropriate description of symmetry under projective transformations.
2.8.1 Properties of Symmetry
Symmetry can be described in terms of the transformationH∈IR3×3which trans-forms one sidexinto its symmetric complementx′, we get the necessary condition
Hx=k′x′ Hx′=kx
=⇒ HHx′=kk′x′
=⇒ HH=kk′I3. (2.56)
It is always possible to scale H so that kk′ = 1. H is called an involution or automorphism.
Equation (2.56) is only the necessary condition for symmetry; additional restric-tions are needed in order to ensure thatHrepresents a symmetry transformation.
In the case of axial symmetry the transformationHobviously leaves the axis itself unchanged; in other words, the axis forms a set of fixed points or united points.
In the case of point symmetry, the centre of symmetry is left unchanged. It turns out on closer inspection that axial symmetry has another fixed point at infinity, in the direction perpendicular to the axis, while point-symmetry has a fixed line at infinity.
In the projective case, the condition above reduces to that of a fixed point and a line of fixed points in arbitrary position, as long as the point is not located on the line. Interestingly, this means that there is no intrinsic difference between axial symmetry and point symmetry in a projective space.
Finally, symmetry is characterised by the fact that the line segment joiningxand its symmetric complement x′ is bisected by the axis of symmetry (in the case of axial symmetry) or the point of symmetry (in the case of point symmetry).
3For an affine transformation the concept of skewed symmetry can be defined.
2.8.2 Homologies 51
The ratio of collinear lengths is, however, not a projective invariant; the closest approximation within a projective space would be a constraint on the crossratio which any pair of symmetric points forms with its midpoint and the point at infinity: the crossratio is always cr =−1.This is called harmonic separation (see Section 2.6.2).
To identify a transformation that could take the role of symmetry within a pro-jective space, we are looking for a transformation with a line of fixed points and an additional fixed point not on that line which fulfils Equation (2.56) and with the required crossratio of cr =−1. We will see in the next section that a plane harmonic homology has all these attributes.
2.8.2 Homologies
One condition on a transformation that could take the role of symmetry within a projective space was the existence of a line of fixed points and an additional fixed point not on that line. We can see from Equation (2.6) on Page 20 that any 2D projective transformation of a homogeneous vectorx ∈IR3 can be expressed as its multiplication with a matrix P ∈ IR3×3. This matrix will in general have 3 eigenvectorsxi∈IR3,xi6=0and corresponding eigenvaluesλi, such that
Pxi=λixi. (2.57)
Since homogeneous coordinates are invariant to overall scale this means that a general projective transformation will have at least 3 points which remain fixed under this particular transformation4. Depending on the multiplicity of the λi
there are 6 distinctive cases. These are discussed in more detail in [138].
Here, we are only interested in cases that produce a line of united points, that is an eigenvalue of geometric multiplicity 2. There are two and only two such cases, the first case with one degenerate eigenvalueλ0of algebraic and geometric multiplicity 2 and one simple eigenvalueλ2 and the second case of one degenerate eigenvalue λ0of algebraic multiplicity 3 and geometric multiplicity 2. These cases are custom-arily called theplane homology and thespecial plane homology respectively, and the corresponding set of united points is formed by a lineℓ∈IR3of united points and a single united pointv ∈IR3, also called the vertex. Of these two only the plane homology is of interest to us, since in the case of the special plane homology the line of united points and the single united point coincide,vTℓ= 0.
4It is possible that two of these points — or even all three — coincide. It is also possible that more than 3 such points exist. A simple example for the latter isP=I3, the identical transformation.
52 Symmetry under Projective Transformations
According to [138] any plane homologyH∈IR3×3can always be parameterised as H=I3+1−cr
cr · vℓT
vTℓ (2.58)
as long asvTℓ6= 0, that is the homology is not a special plane homology. Accord-ingly, any plane homology with crossratio cr =−1 can always be parametrised as
H=I3−2vℓT
vTℓ. (2.59)
This is called aplane harmonic homology. By construction, any plane harmonic homology has a line of united points ℓ and a single united point v as well as cr =−1. It also satisfies the necessary condition for symmetry (2.56), it is
HH=I3−4vℓT
This shows that Equation (2.59) really describes a transformation as outlined in Section 2.8.1. It also describes Euclidean symmetry: ℓ = (a, b, c)T andv = (a, b,0)Tdescribe axial symmetry;v= (x, y, z)Tandℓ= (0,0,1)Tdescribe point symmetry.
2.8.3 Symmetry under Projection
Under an arbitrary projective transformationP∈IR3×3withPv=veandP−Tℓ= eℓthe plane harmonic homologyHtransforms asHe =PHP−1. ThatHe is again of the form (2.59) can be seen from:
He=PHP−1
The Gaussian Sphere 53
(a)Symmetry with respect to a line.
(b) Symmetry under affine transformation.
(c)Symmetry under projective transformation.
Figure 2.17: Symmetry under transformations.
It follows (from the previous section) that any symmetry-transformation is of the form given in Equation (2.59), and (from Equation (2.61)) that all symmetry-transformations keep this form under an arbitrary projective transformation. Con-versely, it is always possible to find a projection matrixP such thatPHP−1 de-scribes a symmetry. A plane harmonic homology therefore dede-scribes the form of a symmetry-transformation under an arbitrary projective transformation. Fig-ure 2.17 gives examples for symmetry under different transformations.