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PROJECTIVE GEOMETRY IN 3D
Hierarchy of transformations
⎥⎦
⎤⎢⎣
⎡vTvtAProjective
15dof
Affine12dof
Similarity7dof
Euclidean6dof
Intersection and tangency
Parallellism of planes,Volume ratios, centroids,The plane at infinity π∞
The absolute conic Ω∞
Volume
⎥⎦
⎤⎢⎣
⎡10tA
T
⎥⎦
⎤⎢⎣
⎡10tR
T
s
⎥⎦
⎤⎢⎣
⎡10tR
T
INVARIANTS
3D points
( )TT
1 ,,,1,,,X4
3
4
2
4
1 ZYXXX
XX
XX
=⎟⎟⎠
⎞⎜⎜⎝
⎛=
in R3
( )04 ≠X
( )TZYX ,,
in P3
XX' H= (4x4-1=15 DOF)
projective transformation
3D point
( )T4321 ,,,X XXXX=
X_4 = 0 ideal point
Planes
0ππππ 4321 =+++ ZYX
0ππππ 44332211 =+++ XXXX
0Xπ =T
Dual: points ↔ planes, lines ↔ lines
3D plane
0X~.n =+ d ( )T321 π,π,πn = ( )TZYX ,,X~ =14 =Xd=4π
Euclidean representation
n/d
XX' H=ππ' -TH=
Transformation
n . (X - X_0) = 0
inhomogeneous homogeneous
distance of the planefrom the origin
~
X_0
A unique plane: joint of three points; or joint of a line and a point in general position. Two distinct planes intersect in a unique line. Three distinct planes intersect in a unique point.
det [ X X_1 X_2 X_3] = 0
a homogeous point X =[X 1]^T
where X = [X Y Z]^T
3x4 matrix
~~
= determinant transpose below is row X=(X_1...X_4)^T
X in homogeneous coordinates in P^3
Parametrized points on a planeM is 4x3 matrixx homogeneous coordinates in P^2
The matrix is not unique! You have in the plane
If a plane is given
X_1, X_2, X_3 defines a 3D plane. The linear combinationX = q_1 X_1 + q_2 X_2 + q_3 X_3 is the 3D null-space of the matrix. see the equation above
3D LinesA line is joint of two 3D points at the intersection of two planes.
A line have four degrees of freedom in 3D. One way to justify it: 3 DOF of a points on the 3D line plus1 DOF for a rotation perpendicular to the 3D line. Several representations exist. We do only the null-space and span representation A line represented by the span of two vectors.
A 3D line is a one-parameter family, defined by two points.
P Q
2x4 matrices
Points, lines and planes
⎥⎦
⎤⎢⎣
⎡= TX
WM 0π =M
⎥⎦
⎤⎢⎣
⎡= Tπ
W*
M 0X =M
W
X
*Wπ
The nullspace of the 3x4 matrix M.
plane
point
line and point
line and plane
If X is on W, or W* is on the plane, the matrix M is only rank 2.
Quadrics and dual quadrics
Classification of quadrics
conerank 3
two planes, rank 2
<- null vectornull space
Hierarchy of transformations
⎥⎦
⎤⎢⎣
⎡vTvtAProjective
15dof
Affine12dof
Similarity7dof
Euclidean6dof
Intersection and tangency
Parallellism of planes,Volume ratios, centroids,The plane at infinity π∞
The absolute conic Ω∞
Volume
⎥⎦
⎤⎢⎣
⎡10tA
T
⎥⎦
⎤⎢⎣
⎡10tR
T
s
⎥⎦
⎤⎢⎣
⎡10tR
T
INVARIANTS
The plane at infinity
∞
−
∞−
∞ =
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎥⎦
⎤⎢⎣
⎡
−==′ π
1000
1t0
ππA
AH
TTA
The plane at infinity π∞ is a fixed plane under a affine transformation H. Not fixed pointwise.
1. canical position2. contains directions 3. two planes are parallel ⇔ line of intersection in π∞4. line // line (or a plane) ⇔ point of intersection in π∞
( )T1,0,0,0π =∞
( )T0,,,D 321 XXX=
3D AFFINE TRANSFORMATION
3 DOFcanonical
The absolute conic
The absolute conic Ω∞ is a fixed conic under thesimilarity transformation H . Not fixed pointwise.
04
23
22
21 =
⎭⎬⎫++
XXXX
The absolute conic Ω∞ is a point conic on π∞. In a metric frame:
( ) ( )T321321 ,,I,, XXXXXXor conic for directions:only imaginary points
1. Ω∞ is only fixed as a set not pointwise2. Circles intersect Ω∞ in two points3. Spheres intersect π∞ in Ω∞
3D SIMILARITY TRANSFORMATION
= 0 C=I
The conic C=Idoes not change.
Metric properties
Orthogonality and polarity
Will use it later in the course, for camera calibration.
Absolute dual quadric
The quadric Q becomes absolute conic k -> infinity.The dual of Q
holds if the above is correct
in an Euclidean frame
euclidean coordinates
invariant to transformations
Screw decomposition. EUCLIDEAN trans.R and t.
Euclidean translation and rotation is equivalent with rotationabout a screw axis (parallel to the rotation axis) and translationalong the screw axis.2D case: screw axis = perpendicular bisector;S making angle theta
Hierarchy of transformations
⎥⎦
⎤⎢⎣
⎡vTvtAProjective
15dof
Affine12dof
Similarity7dof
Euclidean6dof
Intersection and tangency
Parallellism of planes,Volume ratios, centroids,The plane at infinity π∞
The absolute conic Ω∞
Volume
⎥⎦
⎤⎢⎣
⎡10tA
T
⎥⎦
⎤⎢⎣
⎡10tR
T
s
⎥⎦
⎤⎢⎣
⎡10tR
T
INVARIANTS
3D Rotation of Points
counter-clockwise
3D elementary rotations
Rotation around the coordinates axes, counter-clockwise.
Rotation in 3D
The three-dimensional orthonormal matrices satisfy
R⊤R = RR
⊤ = I3 det(R) = 1
and are called rotation matrices. In group theory this special orthogonal group iscalledSO(3). Is an example of a Lie group.
If a rotation matrix changes its values with timet, it can be describes as
R(t) : R → SO(3) .
The derivate with respect to timet is
R(t)R⊤(t) + R(t)R⊤(t) = 0 or R(t)R⊤(t) = −(R(t)R⊤(t))⊤
which is a skew-symmetric matrix. Any3 × 3 skew-symmetric matrix can bewritten as a matrix derived from a vector
R(t)R⊤(t) = [ω(t)]× where ω(t) = [ω1(t) ω2(t) ω3(t)]⊤ and
[ω(t)]× =
0 −ω3(t) ω2(t)ω3(t) 0 −ω1(t)−ω2(t) ω1(t) 0
.
Gives the equationR(t) = [ω(t)]×R(t) .
If the rotation does not depend on time,[ω(t)]× = [ω]×θ, where[ω1 ω2 ω3]⊤
is aunit vector andθ in the angle of rotation. This is the axis-angle representation.The matrix has no rotation as the initial value, and we obtain
R = exp([ω]× θ)
a rotation withθ in the 3D space around a rotation axisω. The matrixR is arotation matrix since[ω]⊤× = −[ω]× and therefore
[exp([ω]× θ)]−1 = exp(−[ω]× θ) = exp([ω]⊤× θ) = [exp([ω]× θ)]⊤ .
Locally, the elements ofSO(3) depend only on the three parameters of the vectorω. This is thetangent space of SO(3) and is called the Lie algebra,so(3). The
tangent space isalways a vector space, but different points inSO(3) lead to dif-ferent planes inso(3). TheSO(3) is the unit sphere andso(3) are planes. Thetransformation betweenSO(3) andso(3) is the matrix exponential or the matrixlogarithm
exp : so(3) → SO(3) log : SO(3) → so(3) .
exp : from the rotation angle to the matrix [ω]×θ −→ R
log : from the matrix to the rotation angle R −→ [ω]×θ = log R
θ = arccos
(
traceR − 1
2
)
ω =1
2 sin θ
r32 − r23
r13 − r31
r21 − r12
||ω|| = 1.
We can see that
log R =
0 if θ = 0θ
2 sin θ(R − R
⊤) if θ 6= 0
The3× 3 matrixR = exp([ω]× θ), with ||ω|| = 1 and angle of rotationθ, has theangle of rotationθ in the range[−π, π] since otherwise an infinity ofθ-s resultfrom the inverse cosine function.This is the axis-angle represention of a 3D rotation.
Rodrigues formula
The rotation matrix isR = exp([ω]× θ) with ||ω|| = 1 and angle of rotationθ canbe developed
exp([ω]× θ) = I + θ[ω]× +θ2
2![ω]2× +
θ3
3![ω]3× + ...
but [ω]2× = ω ω⊤ − I while the fourth pover is
[ω]4× = [ω ω⊤ − I]2 = ω ω
⊤ω ω
⊤ − 2ω ω⊤ + I = −[ω ω
⊤ − I] = −[ω]2×
because
(ω ω⊤) (ω ω
⊤) =
ω2
1ω1ω2 ω1ω3
ω1ω2 ω2
2ω2ω3
ω1ω3 ω2ω3 ω2
3
ω2
1ω1ω2 ω1ω3
ω1ω2 ω2
2ω2ω3
ω1ω3 ω2ω3 ω2
3
= 1 ∗ ω ω⊤ .
2
Every new even power just changes the sign of[ω]2×.The odd powers follow the same rule since[ω]3× = −[ω]×.The exponential can be rewritten as
exp([ω]× θ) = I +
(
θ −θ3
3!+
θ5
5!− · · ·
)
[ω]× +
(
θ2
2!−
θ4
4!+
θ6
6!− · · ·
)
[ω]2× .
Since Taylor series of
sin θ = θ −θ3
3!+
θ5
5!− · · ·
cos θ = 1 −θ2
2!+
θ4
4!− · · ·
we obtainexp([ω]× θ) = I + [ω]× sin θ + [ω]2×(1 − cos θ) .
If we use the fact thatω ω⊤ − I is equivalent with[ω]2× we obtain the other form
of the Rodrigues formula
exp([ω]× θ) = I cosθ + [ω]× sin θ + ω ω⊤(1 − cos θ)
with the formulae for[ω]× andω ω⊤ can be obtained from above.
3