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Projective 3D geometry
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Projective 3D geometry

Dec 31, 2015

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ava-gillespie

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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Page 1: Projective 3D geometry

Projective 3D geometry

Page 2: 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

Page 3: Projective 3D geometry

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

Page 4: Projective 3D geometry

Projective 3D Geometry

• Points, lines, planes and quadrics

• Transformations

• П∞, ω∞ and Ω ∞

Page 5: Projective 3D geometry

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

Page 6: Projective 3D geometry

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

Page 7: Projective 3D geometry

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

Page 8: Projective 3D geometry

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

Page 9: Projective 3D geometry

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

Page 10: Projective 3D geometry

Points, lines and planes

TX

WM 0π M

W*

M 0X M

W

X

*W

π

Page 11: Projective 3D geometry

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

Page 12: Projective 3D geometry

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)

Page 13: Projective 3D geometry

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)

Page 14: Projective 3D geometry

Quadrics and dual quadrics

(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

5. (plane ∩ quadric)=conic

6. transformation

Q

QXπ QMMC T MxX:π

-1-TQHHQ'

0πQπ * T

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

2. transformation THHQQ' **

Page 15: Projective 3D geometry

Quadric classification

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

Page 16: Projective 3D geometry

Quadric classificationProjectively equivalent to sphere:

Ruled quadrics:

hyperboloids of one sheet

hyperboloid of two sheets

paraboloidsphere ellipsoid

Degenerate ruled quadrics:

cone two planes

Page 17: Projective 3D geometry

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

Page 18: Projective 3D geometry

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

Page 19: Projective 3D geometry

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

Page 20: Projective 3D geometry

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∞

Page 21: Projective 3D geometry

The absolute conic

2211

21

dddd

ddcos

TT

T

2211

21

dddd

ddcos

TT

T

0dd 21 T

Euclidean:

Projective:

(orthogonality=conjugacy)

plane

normal

Page 22: Projective 3D geometry

The absolute dual quadric

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