Overview

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Overview

• Introduction to projective geometry
• Projective plane
• Projective space of dim 3
• Transformation
• Affine geometry
• Enclidean geometry
• Camera from projective geometry point of view

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Motivation
A camera is modeled as a map from a space pt
(X,Y,Z) to a pixel (u,v) by
homogeneous coordinates have been used to
treat translations multiplicatively in
matrices (translation was additive in vectors),
but this is not the full story
This will be re-used in computer graphics,
computer vision and robotics!
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Basic geometric concepts to understand

• Affine, Euclidean geometries (inhomogeneous
  coordinates)
• projective geometry (homogeneous coordinates)
• plane at infinity affine geometry
• (absolute conic Euclidean geometry)

NB some of the content in this part is only
optional!!!, dont worry ?
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Introduction to projective geometry

• Intuitive ideas from projective geometry
• (Formal definition of projective spaces)

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Intuitive introduction
Naturally everything starts from the known vector
space

• add two vectors
• multiply any vector by any scalar
• zero vector origin
• finite basis

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• Vector space to affine isomorph, one-to-one
• vector to Euclidean as an enrichment scalar
  prod.
• affine to projective as an extension add ideal
  elements

Pts, lines, parallelism
Angle, distances, circles
Pts at infinity
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Points at infinity
Algebraic extension to pts at infinity
introduction of homogeneous coordiantes
Rq the homogeneous coordinates are not unique,
up to a scale.
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On a plane,
Can we see the pts at infinity?
The direction d is a pt at infinity
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Provisional summary
a projective space is an affine space some pts
at infinity
or
a projective space is a space of homogeneous
coordinates
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((Formal) definition of projective geometry)
Given KR or C, can be defined as the
nonzero equivalent classes determined by the
relation on
If there is non-zero real number such that
Any element of the
equivalent class will be called the homogeneous
coordinates of the point.
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• A projective space is nothing but a quotient
  space (space of equivalent classes)
• A space of homogeneous coordinates
• Basic structure linear dependence of points

Definition a pt x is said to be linearly
dependent on a set of pts if
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Relation between Pn (homo) and Rn (in-homo)
Rn --gt Pn, extension, embedded in
Pn --gt Rn, restriction,
P2 and R2
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One example of construction of projective line
by quotient space
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Examples of projective spaces

• Projective plane P2
• Projective line P1
• Projective space P3

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Projective plane
Space of homogeneous coordinates (x,y,t)
Pts are elements of P2
Pts are elements of P2
Pts at infinity (x,y,0), the line at infinity
4 pts determine a projective basis
3 ref. Pts 1 unit pt to fix the scales for
ref. pts
Relation with R2, (x,y,0), line at inf., (0,0,0)
is not a pt
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Lines
Linear combination of two algebraically
independent pts
Operator is span or join
Line equation
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Point/line duality

• Point coordinate, column vector
• A line is a set of linearly dependent points
• Two points define a line

• Line coordinate, row vector
• A point is a set of linearly dependent lines
• Two lines define a point

• What is the line equation of two given points?
• line (a,b,c) has been always homogeneous
  since high school!

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Given 2 points x1 and x2 (in homogeneous
coordinates), the line connecting x1 and x2 is
given by
Given 2 lines l1 and l2, the intersection point x
is given by
NB cross-product is purely a notational device
here.
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Compute the intersection point of two lines, each
defined by two points
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Conics
Conics a curve described by a second-degree
equation

• 33 symmetric matrix
• 5 d.o.f
• 5 pts determine a conic
• affine classification with pts at inf
• the line tangent to a conic at a pt
• dual conic
• pole and polar
• one numerical example

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Tangent to a conic at a pt x on C is given by
lCx
Dual conic (in line coordinates) is given by lT
C-1 l 0
Polar of a pt x is l C x and is also a tangent
on C from x
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Line at infinity
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Projective line
Homogeneous pair (x1,x2)
Finite pts Infinite pts how
many? Topology? A basis by 3 pts Fundamental
inv cross-ratio
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• Euclidean coordinate
• the distance
• Affine coordinate
• the ratio of the distances (x-a/a-o)
• Projective coordinate
• the ratio of the ratio of the distances
• (cross-ratio, double ratio)
• ((x-a)/(a-o)) / ((x-b)(b-o))

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Projective space P3

• Pts, elements of P3
• Relation with R3, plane at inf.
• lines linear comb of 2 pts, but 34 matrix,
  complicated back later
• planes linear comb of 3 pts
• Basis by 4 (ref pts) 1 pts (unit)
• quadrics two classes---ruled and unruled
• (topology of P3)

Line equation?
Plane equation ...
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planes
In practice, take SVD
Homework compute plane normal vector?
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(Plucker coordinates of lines in P3)
How many d.o.f???
6 22 minors,
Two lines intersect in space iff
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(Quadric surfaces)
Ruled hyperboloid of one sheet,
1,1,-1,-1---topo torus
Unruled sphere, ellipsoid, hyperboloid and
paraboloid 1,1,1,-1 ---- topo
sphere
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Key points

• Homo. Coordinates are not unique
• 0 represents no projective pt
• finite points embedded in proj. Space (relation
  between R and P)
• pts at inf. (x,0) missing pts, directions
• hyper-plane (co-dim 1)
• dualily between u and x,

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Introduction to transformation
2D general Euclidean transformation
2D general affine transformation
2D general projective transformation
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Projective transformation
collineation homography
Consider all functions
All linear transformations are represented by
matrices A
Note linear but in homogeneous coordinates!
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Properties

• (n1)(n1) -1 d.o.f.
• all projective properties are left invariant by
  A
• all transformations form a group GL(n,R)

N2 pts to determine a trans. a proj. basis
Check the most important one linear dependency,
i.e. lines into lines as line is just a span
Starting pt for new investigation Kleins
Erlangen program
Inversely, we may also prove that any 1-1 transf.
Preserving lines is a linear trans in homogeneous
coord.
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(Some examples of transformations)
on pts, lines and conics
Transforms contravariantly
Co-variantly to preserve incidence
Co-variantly
NB co-,contra-variance is w.r.t. the basis
trans. Transpose is of no importance, il
accommodates row/column vectors
Some numerical examples of transformation on P2
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How to compute (canonical or standard)
coordinates?--- affine case
Given 4 pts, x1, x2, x3, x4, find the affine
coord of x4 w.r.t. x1, x2 and x3

• by definition,
• vector(x4-x1) a vector(x2-x1) b
  vector(x3-x1)
• by canonical transformation,
• x1-gt(0,0), x2-gt(1,0), x3-gt(0,1), get
  transfromation A, then Ax4

How to solve Axb?
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Canonical projective coordinates?
Given 5 pts, x1, x2, x3, x4, x5find the affine
coord of x5 w.r.t. x1, x2, x3, x4
By canonical transformation
How to solve Ax0?
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A transformation between 2 spaces?
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Exercise
Compute the transformation from (0,0,1),
(1,0,1), (0,1,1) and (1,1,1) into (0,0,1),
(1,1/4,1),(0,1,1) and (1,3/4,1)
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(Geometry as an invariant theory of
transformation groups)

• projective geom. GL(n,R)
  cross-ratio
• affine geom. Subgroup A(n,R)
  ratio
• Euclidean geom. Subgroup E(n,R)
  distance

All proj. Transformations nicely form a group!
Each geometry is associated with a (sub)group!
Hierarchy of geometry
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Example of dim 2
From projective to affine
Affine transformation is a projective one which
leaves the line at inf. invariant x3x30
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From affine to euclidean
Similarity transformation is an affine one which
leaves the circular pts I and J invariant
What are the circular points?
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Circular points
The line at infinity of a usual plane
The pair of circular points
Intuitive introduction of circular pts
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Example of transformation in P3
Affine transformation leaves the plane at inf.
invariant
Similarity (euclidean) leaves the absolute conic
(globally, not point-wise) invariant
What is the absolute conic?
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(Absolute conic)
Euclidean structure in projective space by the
absolute conic

• A space conic on the plane at infinity
• In point coordinates
• In plane coordinates
• rank 3 space quadricabsolute quadric

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Key message from projective geometry for vision

• abstract camera is a projective transformation
  from P3 to P2, so 34 matrix
• the intrinsic parameters of the camera are the
  image of the absolute conic!

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Summary

• transformation and geometry
• group of transformation
• affine group hyper-plane at inf.
• euclidean group absolute pts