Projective 3D geometry. In this lecture we examine quadrics in 3D projective
space P. 3 . Among others, we study. • Quadrics. • Dual Quadrics. • Twisted
Cubics.
Multiple View Geometry in Computer Vision
Prasanna Sahoo Department of Mathematics University of Louisville 1
Projective 3D Geometry (Back to Chapter 2)
Lecture 7
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Projective 3D geometry In this lecture we examine quadrics in 3D projective space P3. Among others, we study • Quadrics • Dual Quadrics • Twisted Cubics • Plane at Infinity • Absolute Conic • Absolute Dual Conic 3
Quadrics in P3 A quadric is a surface in P3 and given by the following 2nd degree equation: 2 + a x2 + a x2 a11x2 + a x 22 2 33 3 44 4 1
+ a12x1x2 + a13x1x3 + a14x1x4 + a23x2x3 + a24x2x4 + a34x3x4 = 0.
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If we denote (x1, x2, x3, x4)T by x, then the last quadratic equation becomes
xT Q x = 0 where
Q=
a
11 a 12 2 a13 2 a 14
2
a12 2
a22
a13 2 a23 2
a23 2
a33
a24 2
a34 2
a14 2
a24 2 . a34 2
a44
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Properties of Quadrics Many properties of quadrics follows directly from those of conics. • The matrix Q represents a quadric. • The matrix Q is a 4 × 4 symmetric matrix. • Since multiplying Q by a non-zero scalar does not affect the equation xT Q x = 0, therefore Q is a homogeneous representation of a quadric. 6
• The quadric has 9 degrees of freedom. • Nine points in general position define a quadric. • If the matrix Q is singular, then the quadric is called a degenerate quadric. • The intersection of a plane Π with a quadric is a conic C. • Under point transformation x0 = H x, a point quadric transforms as Q0 = H−T Q H−1. 7
Dual Quadrics The quadric defined above is called a point quadrics, since it defines an equation on points. Because of duality principle one can define a quadric using an equation on planes. This dual quadric is represented by a 4 × 4 matrix denoted by Q?. 8
The dual quadric are equations on planes: The tangent planes Π to the point quadric Q satisfy
ΠTQ? Π = 0. Under point transformation x0 = H x, a dual quadric transforms as Q?0 = H Q? HT. Here Q? is the adjoint of Q, or Q−1 if Q is invertible.
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Classification of Point Quadrics Since Q is a symmetric matrix, it may be decomposed as Q = UT D U where U is a real orthogonal matrix and D is a real diagonal matrix.
By appropriate scaling of the rows of U, one may write Q = HT D H, where D is a diagonal matrix with entries equal to 0, 1, −1. 10
We may further ensure that the 0 entries of D appear last along the diagonal, and that the 1 entries appear first. Because of the relationship Q = HT D H, we may assume that Q is equivalent to D up to a projective transformation.
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Signature of the diagonal matrix D The signature σ, of the matrix D is defined as the number of 1 entries minus the number of -1 entries in D. Since Q is defined only up sign, we may assume that σ is non-negative. The projective type of a quadric is uniquely determined by its rank and signature. 12
Classification of Point Quadrics A quadric represented by a diagonal matrix diag(d1, d2, d3, d4) correspond to a set of the points satisfying an equation d1x2 + d2y 2 + d3z 2 + d4w2 = 0. One may set w = 1 to get an equation for the noninfinite points on the quadric. 13
Classification of Point Quadrics
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Ruled and Unruled Quadrics Quadrics can be divided into two groups - ruled and unruled quadrics. A ruled quadrics is one that contains a straight line.
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Unruled quadrics They are all projectively equivalent and have rank 4.
Sphere, ellipsoid, hyperboloid of two sheets and paraboloid 16
Ruled quadrics These surfaces are made up of
two sets of disjoint
straight lines, and that each line from one set meets each line from the other set.
Hyperboloid of one sheet 17
Degenerate ruled quadrics Degenerate quadrics have rank less than 4. Cone has rank 3, and two planes has rank 2. They are ruled.
Cone and two planes 18
Twisted Cubics A twisted cubic in the 3-dimensional projective space
P3 is defined to be a parametrized curve given by the equation 0 x 1
0 x 2 x 0 3
x04
a a12 a13 a14 1 11 a21 a22 a23 a24 t 2 a 31 a32 a33 a34t
=
a41 a42 a43 a44
t3
=
1
t A . 2 t
t3
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The twisted cubic may be considered as a 3-dimensional analogue of a 2D conic.
Various views of the twisted cubic (t3 , t2 , t)T .
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• If c is a non-singular twisted cubic, then c is not contained within any plane. • If c is a non-singular twisted cubic, then in general
c intersects a plane at three distinct points. • The twisted cubic c has 12 degrees of freedom (15 DOF for A minus 3 DOF for parametrization). • The twisted cubic c can be uniquely defined by 6 points (2 constraints per point on cubic). 21
Hierarchy of Transformations 2.3. CONCLUSION ambiguity
DOF
projective
15
19 transformation
WW W , & , WW , , [ W , ,
WW W W & , W , ,, , W , ( ( ( WW W, & , W , , W ( (, WW W W & , W , ,, , W ( ( , (
affine
metric
Euclidean
12
7
6
invariants
W W , ,
cross-ratio
W ,
W ,
(
relative distances along direction parallelism plane at infinity relative distances angles absolute conic
absolute distances
L
Table 2.1: Number of degrees of freedom, transformations and invariants corresponding to the different geometric strata (the coefficients form orthonormal matrices)
22 Projective
TP
metric
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&
&
W , ( ( W W W, W ( WW W, , WW , , ( ( ,
( W , W , , , (, ( W , (
parallelism plane at infinity relative distances angles absolute conic
Effects of Transformations Euclidean
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absolute distances
L
Table 2.1: Number of degrees of freedom, transformations and invariants corresponding to the different geometric strata (the coefficients form orthonormal matrices)
Projective
TP
Affine
TA
Metric (similarity)
TM Euclidean
TE Figure 2.5: Shapes which are equivalent to a cube for the different geometric ambiguities
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The Plane at Infinity In planar projective geometry the identification of the line at infinity `∞ allows one to measure the affine properties of the plane.
Similarly one can measure
the metric properties of the plane by identifying the circular points on `∞. In P3 the corresponding geometric entities are the plane at infinity Π∞ and the absolute conic Ω∞. 24
• The plane at infinity has the canonical position
Π∞ = (0, 0, 0, 1)T. • It contains the directions (or the ideal points at infinity) d = (x1, x2, x3, 0)T. • The plane at infinity Π∞ is a fixed plane under an affine transformation since
−T A Π0∞ = H−T A Π∞ = −t−T A−T
0 0 0 1 0
1
0
=
0 0
= Π∞.
1
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Result 2.1. The plane at infinity Π∞ is a fixed plane under a projective transformation H if and only if H is an affine transformation.
The identification of Π∞ allows one to recover the affine properties (parallellism, ratio of volumes).
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• Two planes are parallel if and only if their line of intersection is on Π∞. • Two lines are parallel to each other if the point intersection is on Π∞. • A line and a plane are parallel to each other if the point intersection is on Π∞. • Any plane intersects Π∞ at line `∞. 27
Absolute Conic The absolute conic Ω∞ is a point conic on Π∞. The points on Ω∞ satisfy the equations 2 + x2 = 0 x2 + x 1 2 3
and
x4 = 0.
For directions (or points at infinity) on Π∞ the above equations can be written as (x1, x2, x3) I (x1, x2, x3)T = 0 where I is a 3 × 3 identity matrix. 28
Result 2.2. The absolute conic Ω∞ is a fixed conic under the projective transformation H if and only if H is a similarity.
• This result shows that Ω∞ stays as Ω∞ under a similarity transformation.
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Some properties • Ω∞ is a complex 2D point conic on Π∞ plane. • Only HP and HA transform Ω∞. • Ω∞ has 5 degrees of freedom for HA. • All circles (in any plane) intersect Ω∞ in two points. • All spheres intersect Π∞ in Ω∞. 30
Why learn Ω∞? Angles between directions d1 and d2 or planes Π1 and
Π2 can be measured using the absolute conic Ω∞. • The angle θ between two directions d1 and d2 is given by cos(θ) =
dT 1 Ω∞ d2 r
dT 1 Ω∞ d1
dT 2 Ω∞ d2
.
• The directions are orthogonal iff dT 1 Ω∞ d2 = 0. 31
• The angle θ between two planes Π1 and Π2 is given by cos(θ) =
ΠT 1 Ω∞ Π2 r
ΠT 1 Ω∞ Π1
ΠT 2 Ω∞ Π2
.
• The planes are orthogonal iff ΠT 1 Ω∞ Π2 = 0.
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Absolute Dual Quadric The dual of the absolute conic Ω∞ is a degenerate dual quadric in P3 and is called the absolute dual quadric. It is denoted by Q?∞. Geometrically Q?∞ consists of a set of tangent planes to Ω∞, so that Ω∞ is the ‘rim’ of Q?∞. Think of the set of planes tangent to an ellipsoid, and then squash the ellipsoid to a pancake. 33
Illustration of Absolute Dual Quadric
The absolute qual quadric is a conic in Π∞ 34
• Absolute dual quadric Q?∞ has 8 degrees of freedom.
1 0 1 ? • In world space, Q∞ = 0 0 0 0 0
0 0 1 0
0 0 0 0
• Π∞ is always a tangent plane to Q?∞ • The angle between planes Π1 and Π2 is given by cos(θ) =
? Π ΠT Q ∞ 2 1 r
? Π ΠT Q ∞ 1 1
? Π Q ΠT ∞ 2 2
.
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