7 The plane at infinity, TT^, is a fixed plane under the projective transforma-

tion H if, and only if H is an affinity.

The proof is the analogue of the derivation of result 2.17(p48). It is worth clarifying two points:

(i) The plane 77^ is, in general, only fixed as a set under an affinity; it is not fixed pointwise.

(ii) Under a particular affinity (for example a Euclidean motion) there may be planes in addition to n^ which are fixed. However, only iv^ is fixed under any affinity.

These points are illustrated in more detail by the following example.

Example 3.8. Consider the Euclidean transformation represented by the matrix

H„ = R 0 0T 1 cos 9 — sin 9 0 0 sin 9 cos 9 0 0 0 0 1 0 0 0 0 1 (3.20)

This is a rotation by 9 about the Z-axis with a zero translation (it is a planar screw motion, see section 3.4.1). Geometrically it is evident that the family of XY-planes or- thogonal to the rotation axis are simply rotated about the Z-axis by this transformation. This means that there is a pencil of fixed planes orthogonal to the z-axis. The planes are fixed as sets, but not pointwise as any (finite) point (not on the axis) is rotated in horizontal circles by this Euclidean action. Algebraically, the fixed planes of H are the eigenvectors of H (refer to section 2.9). In this case the eigenvalues are {e w n-%e _1,1}

3.6 The absolute conic 81 and the corresponding eigenvectors of Ej are

E, = i 0

V o y

V o /

V i /

( M / M _o

V o y

The eigenvectors Ei and E2 do not correspond to real planes, and will not be discussed

further here. The eigenvectors E3 and E4 are degenerate. Thus there is a pencil of

fixed planes which is spanned by these eigenvectors. The axis of this pencil is the line of intersection of the the planes (perpendicular to the Z-axis) with TTQO, and the pencil

includes ir^. A The example also illustrates the connection between the geometry of the projective

plane, IP2, and projective 3-space, P3. A plane TT intersects TT^ in a line which is

the line at infinity, 1^, of the plane IT. A projective transformation of P3 induces a

subordinate plane projective transformation on ir.

Affine properties of a reconstruction. In later chapters on reconstruction, for exam- ple chapter 10, it will be seen that the projective coordinates of the (Euclidean) scene can be reconstructed from multiple views. Once -K^ is identified in projective 3-space, i.e. its projective coordinates are known, it is then possible to determine affine prop- erties of the reconstruction such as whether geometric entities are parallel - they are parallel if they intersect on -K^.

A more algorithmic approach is to transform P3 so that the identified TT^ is moved

to its canonical position at iv^ = (0, 0. 0,1)T. After this mapping we then have the

situation that the Euclidean scene, where n^ has the coordinates (0, 0, 0,1)T, and our

reconstruction are related by a projective transformation that fixes -K^ at (0, 0, 0,1)T. It

follows from result 3.7 that the scene and reconstruction are related by an affine trans- formation. Thus affine properties can now be measured directly from the coordinates of the entities.

3.6 The absolute conic

The absolute conic, fi^, is a (point) conic on n^. In a metric frame TV^ = (0, 0, 0,1)T,

and points on Q^ satisfy

X2 + X2 + X2

X4 '

(3.21) Note that two equations are required to define Q^.

For directions on ir^ (i.e. points with x4 = 0 ) the defining equation can be written

( X1, X2, X3) I ( X1, X 2 , X3)T = 0

so that QQO corresponds to a conic C with matrix C = I. It is thus a conic of purely imaginary points on -n^.

must fix the plane at infinity, and hence must be affine. Such a transformation is of the form

A t 0T 1

HA =

Restricting to the plane at infinity, the absolute conic is represented by the matrix I3 x 3,

and since it is fixed by HA, one has A_ TI A_ 1 = I (up to scale), and taking inverses gives

AAT = I. This means that A is orthogonal, hence a scaled rotation, or scaled rotation

with reflection. This completes the proof. D Even though fi^ does not have any real points, it shares the properties of any conic -

such as that a line intersects a conic in two points; the pole-polar relationship etc. Here are a few particular properties of O.^:

(i) QQO is only fixed as a set by a general similarity; it is not fixed pointwise. This means that under a similarity a point on Q.^ may travel to another point on fi^, but it is not mapped to a point off the conic.

(ii) All circles intersect 0.^ in two points. Suppose the support plane of the circle is TV. Then n intersects 7?^ in a line, and this line intersects Q.^ in two points. These two points are the circular points of TT.

(iii) All spheres intersect ir^ in fi^.

Metric properties. Once 0.^, (and its support plane 77^) have been identified in

projective 3-space then metric properties, such as angles and relative lengths, can be measured.

Consider two lines with directions (3-vectors) d i and d2. The angle between these

directions in a Euclidean world frame is given by (d! d2)

cos6» = _ v 1 ; = . (3.22)

/(dldTXdJd^

This may be written as

(d]"fiood2) . . .

cos 0 = = (3.23) ^(d]"fic od1)(d2 rno cd2)

where dx and d2 are the points of intersection of the lines with the plane 71-00 containing

3.7 The absolute dual quadric 83

di *d

d2

a b Fig. 3.8. Orthogonality and fioo- (a) On -KX orthogonal directions d^ d2 are conjugate with respect

to flo<j. (b) A plane normal direction d and the intersection line 1 of the plane with TT^ are in pole-polar relation with respect to fioo-

The expression (3.23) reduces to (3.22) in a Euclidean world frame where Q.^ — I. However, the expression is valid in any projective coordinate frame as may be verified from the transformation properties of points and conies (see (iv)(b) on page 63).

There is no simple formula for the angle between two planes computed from the directions of their surface normals.

Orthogonality and polarity. We now give a geometric representation of orthogo- nality in a projective space based on the absolute conic. The main device will be the pole-polar relationship between a point and line induced by a conic.

An immediate consequence of (3.23) is that two directions di and d2 are orthogonal

if dJfiocd-2 = 0. Thus orthogonality is encoded by conjugacy with respect to 0.^. The great advantage of this is that conjugacy is a projective relation, so that in a projective frame (obtained by a projective transformation of Euclidean 3-space) directions can be identified as orthogonal if they are conjugate with respect to Q^ in that frame (in general the matrix of Q.^ is not I in a projective frame). The geometric representation of orthogonality is shown in figure 3.8.

This representation is helpful when considering orthogonality between rays in a camera, for example in determining the normal to a plane through the camera cen- tre (see section 8.6(p213)). If image points are conjugate with respect to the image of floo then the corresponding rays are orthogonal.

Again, a more algorithmic approach is to projectively transform the coordinates so that fioo is mapped to its canonical position (3.21), and then metric properties can be determined directly from coordinates.

3.7 The absolute dual quadric

Recall that 0.^, is defined by two equations - it is a conic on the plane at infinity. The dual of the absolute conic ftoo is a degenerate dual quadric in 3-space called the absolute dual quadric, and denoted Q^. Geometrically Q^ consists of the planes tangent to ftoo, so that ftoo is the "rim" of Q^. This is called a rim quadric. Think of the set of planes tangent to an ellipsoid, and then squash the ellipsoid to a pancake.

the line in which the plane (vT, k)J meets the plane at infinity. This line is tangent to

the absolute conic if and only if vTI v = 0. Thus, the envelope of Q^, is made up of

just those planes tangent to the absolute conic.

Since this is an important fact, we consider it from another angle. Consider the ab- solute conic as the limit of a series of squashed ellipsoids, namely quadrics represented by the matrix Q = diag(l, 1, l,k). As k —> oo, these quadrics become increasingly close to the plane at infinity, and in the limit the only points they contain are the points (Xi, X2, X3, 0)T with x\ + xl + X3 = 0, that is points on the absolute conic. However,

the dual of Q is the quadric Q* = Q_1 = diag(l, 1,1, A;-1), which in the limit becomes

the absolute dual quadric Q^ = diag(l, 1,1, 0).

The dual quadric Q^ is a degenerate quadric and has 8 degrees of freedom (a symmet- ric matrix has 10 independent elements, but the irrelevant scale and zero determinant condition each reduce the degrees of freedom by 1). It is a geometric representation of the 8 degrees of freedom that are required to specify metric properties in a projective coordinate frame. Qj^ has a significant advantage over fi^ in algebraic manipulations because both ir^ and fl^ are contained in a single geometric object (unlike !)„, which requires two equations (3.21) in order to specify it). In the following we give its three most important properties.