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