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Projective 3D geometry

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Projective 3D geometry. Singular Value Decomposition. Singular Value Decomposition. Homogeneous least-squares Span and null-space Closest rank r approximation Pseudo inverse. Projective 3D Geometry. Points, lines, planes and quadrics Transformations П ∞ , ω ∞ and Ω ∞. 3D points. - PowerPoint PPT Presentation

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Projective 3D geometry

Singular Value Decomposition

Tnnnmmmnm VΣUA

IUU T

021 n

IVV T

nm

XXVT XVΣ T XVUΣ T

TTTnnn VUVUVUA 222111

000

00

00

00

Σ

2

1

n

Singular Value Decomposition

• Homogeneous least-squares

• Span and null-space

• Closest rank r approximation

• Pseudo inverse

1X AXmin subject to nVX solution

0 ,, 0 ,,,,~

21 rdiag TVΣ~

UA~ TVUΣA

0000

0000

000

000

Σ 2

1

4321 UU;UU LL NS 4321 VV;VV RR NS

TUVΣA 0 ,, 0 ,,,, 112

11 rdiag

TVUΣA

Projective 3D Geometry

• Points, lines, planes and quadrics

• Transformations

• П∞, ω∞ and Ω ∞

3D points

TT

1 ,,,1,,,X4

3

4

2

4

1 ZYXX

X

X

X

X

X

in R3

04 X

TZYX ,,

in P3

XX' H (4x4-1=15 dof)

projective transformation

3D point

T4321 ,,,X XXXX

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 Xd4π

Euclidean representation

n/d

XX' Hππ' -TH

Transformation

Planes from points

X

X

X

3

2

1

T

T

T

2341DX T123124134234 ,,,π DDDD

0det

4342414

3332313

2322212

1312111

XXXX

XXXX

XXXX

XXXX

0πX 0πX 0,πX π 321 TTT andfromSolve

(solve as right nullspace of )π

T

T

T

3

2

1

X

X

X

0XXX Xdet 321

Or implicitly from coplanarity condition

124313422341 DXDXDX 01234124313422341 DXDXDXDX 13422341 DXDX

Points from planes

0X

π

π

π

3

2

1

T

T

T

xX M 321 XXXM

0π MT

I

pM

Tdcba ,,,π T

a

d

a

c

a

bp ,,

0Xπ 0Xπ 0,Xπ X 321 TTT andfromSolve

(solve as right nullspace of )X

T

T

T

3

2

1

π

π

π

Representing a plane by its span

Lines

T

T

B

AW μBλA

T

T

Q

PW* μQλP

22** 0WWWW TT

0001

1000W

0010

0100W*

Example: X-axis

(4dof)

two points A and B two planes P and Q

Points, lines and planes

TX

WM 0π M

W*

M 0X M

W

X

*W

π

Plücker matrices

jijiij ABBAl TT BAABL

Plücker matrix (4x4 skew-symmetric homogeneous matrix)

1. L has rank 22. 4dof3. generalization of

4. L independent of choice A and B5. Transformation

24* 0LW T

yxl

THLHL'

0001

0000

0000

1000

1000

0

0

0

1

0001

1

0

0

0

L T

Example: x-axis

Plücker matrices

TT QPPQL*

Dual Plücker matrix L*

-1TLHHL -'*

*12

*13

*14

*23

*42

*34344223141312 :::::::::: llllllllllll

XLπ *

LπX

Correspondence

Join and incidence

0XL*

(plane through point and line)

(point on line)

(intersection point of plane and line)

(line in plane)0Lπ

0π,L,L 21 (coplanar lines)

Plücker line coordinates

0B,AB,A,detˆ|

0Q,PQ,P,detˆ|

0BPAQBQAPˆ| TTTT

0| (Plücker internal constraint)

(two lines intersect)

(two lines intersect)

(two lines intersect)

(Q : 4x4 symmetric matrix)0QXX T

1. 9 d.o.f.

2. in general 9 points define quadric

3. det Q=0 ↔ degenerate quadric

4. pole – polar

6. transformation

Q

QXπ QMMC T MxX:π

-1-TQHHQ'

0πQπ * T

-1* QQ 1. relation to quadric (non-degenerate)

2. transformation THHQQ' **

Rank Sign. Diagonal Equation Realization

4 4 (1,1,1,1) X2+ Y2+ Z2+1=0 No real points

2 (1,1,1,-1) X2+ Y2+ Z2=1 Sphere

0 (1,1,-1,-1) X2+ Y2= Z2+1 Hyperboloid (1S)

3 3 (1,1,1,0) X2+ Y2+ Z2=0 Single point

1 (1,1,-1,0) X2+ Y2= Z2 Cone

2 2 (1,1,0,0) X2+ Y2= 0 Single line

0 (1,-1,0,0) X2= Y2 Two planes

1 1 (1,0,0,0) X2=0 Single plane

hyperboloids of one sheet

hyperboloid of two sheets

paraboloidsphere ellipsoid

cone two planes

Hierarchy of transformations

vTv

tAProjective15dof

Affine12dof

Similarity7dof

Euclidean6dof

Intersection and tangency

Parallellism of planes,Volume ratios, centroids,The plane at infinity π∞

The absolute conic Ω∞

Volume

10

tAT

10

tRT

s

10

tRT

Screw decompositionAny particular translation and rotation is

equivalent to a rotation about a screw axis and a translation along the screw axis.

ttt //

screw axis // rotation axis

The plane at infinity

π

1

0

0

0

1t

0ππ

A

AH

TT

A

The plane at infinity π is a fixed plane under a projective transformation H iff H is an affinity

1. canical position2. contains directions 3. two planes are parallel line of intersection in π∞

4. line // line (or plane) point of intersection in π∞

T1,0,0,0π

T0,,,D 321 XXX

The absolute conic

The absolute conic Ω∞ is a fixed conic under the projective transformation H iff H is a similarity

04

23

22

21

X

XXX

The absolute conic Ω∞ is a (point) conic on π.

In a metric frame:

T321321 ,,I,, XXXXXXor conic for directions:(with no real points)

1. Ω∞ is only fixed as a set2. Circle intersect Ω∞ in two points3. Spheres intersect π∞ in Ω∞

The absolute conic

2211

21

dddd

ddcos

TT

T

2211

21

dddd

ddcos

TT

T

0dd 21 T

Euclidean:

Projective:

(orthogonality=conjugacy)

plane

normal

00

0I*T

The absolute conic Ω*∞ is a fixed conic under the

projective transformation H iff H is a similarity

1. 8 dof2. plane at infinity π∞ is the nullvector of Ω∞

3. Angles:

2*

21*

1

2*

1

ππππ

ππcos

TT

T