Projective Geometry- 3D

Points, planes,

lines and quadrics

Points in Homogeneous coordinates

• X in 3-space is a 4-vector

X= (x1, x2, x3, x4) T with x4 not 0

represents the point ( x, y, z)T

where x = x1/ x4 , y = x1/ x4 z= x1/ x4

For example X = ( x, y, z, 1)

Projective transformation in p3

• A projective transformation H acting on p3 is a linear transformation on homogeneous 4-vectors and is a non-singular 4x4 matrix:

• X’ =HX It has 15 dof

• 2.2.1 Planes with 4 coefficients:• =( 1234•

Planes

• The plane: A plane in 3-space may be written as • x1 + x2 + x3 + x4 = 0

• T X = 0 • In inhomogeneous coordinates in 3-vector

notation

• Where n =( 123x4 =1• d= 4 ,d /||n|| is the distance of the origin.

T),,(X here w0 d X n. zyx

Joins and incidence relation

(1)A plane is uniquely define by three points, or the join of a line and a point in general position.

(2) Two planes meet at a line, three planes meet at a point

Three planes define a point

Lines in 3 space

• A line is defined by the join of two points or the intersection of two planes. A line has 4 dof in 3 space. It is a 5 –vector in homogenous coordinates, and is awkward.

Null space and point representation

• A and B are 2 space points. Then the line joining these points is represented by the span of the row space of the 2x4 matrix W.

• (i)The Span of W is the pencil of points A+B on the line.

T

T

B

AW

(ii) The the span of the 2D right null space of W is the pencil of planes with the line as axis

The dual representation of a line as the intersection of two planes P and Q

Examples

Join and incidence relations from null-space

Plucker matrices

Properties of L

Properties of L 2

Examples(Plucker matrices)where the point A and B are the origin and the

ideal point in x direction

A dual Plucker representation L*

Join and incidence properties

Examples 2

Two lines

The bilinear product (L !L^)

Quadrics and dual quadrics

• A quadric Q is a surface in p3 defined by the equation

• XT Q X = 0• Q is a 4 x 4 matrix

• (i) A quadric has 9 degree of freedom. These corresponds to 10 independent elements of a 4x4 symmetric matrix less one for scale. Nine points in general position define a quadric

Properties of Q• (ii) If the matrix Q is singular, the quadric

degenerates

• (iii) A quadric defines a polarity between a point and plane. The plane

= QX

is the polar plane of X w.r.t. Q

• (iv) The intersection of a plane with a quadric Q is a conic C

Dual quadric

• (v) Under the point transformation X’ =HX, a point quadric transforms as

• Q’ = H-T Q H

• The dual of a quadric is a quadric on planesQ*=0 where Q* = adjoint Q or Q-1 if Q is invertible

A dual quadirc transform as Q*’ = H-T Q* HT

Classification of quadrics

• Decomposition Q = UT D U• Where U is a real orthogonal matrix and D is a

real diagonal matrix.

• By scaling the rows of U, one may write Q=HTDH where D is a diagonal with entries 0,1, or –1.

• H is equivalent to a projective transform. Then up to a projective equivalence, the quadric is represented by D

Classification of quadrics 2

• Signature of D denoted by (D) = Number of 1 entries minus number of –1 entries

• A quadric with diag(d1,, d2,, d3,, d4 ,) corresponds to a set of point given by

d1x2 + d2y2 + d3z2 + d4T2 =0

Categorization of point quadrics

Some examples of quadrics

• The sphere, ellipsoid, hyperboloid of two sheets and paraboloid are allprojectively equivalent.

• The two examples of ruled quadrics are also projectively equivalent. Their equations are

• x2 + y2 = z2 + 1• xy = z

Non-ruled quadrics: a sphere and an ellipsoid

Non ruled quadrics: a hyperboloid of two sheets and a paraboloid

Ruled quadrics: Two examples of hyperboloid of one sheet are given. A surface is made up

of two sets of disjoint straight lines

Degenerate quadrics

The twisted cubic is a 3D analogue of a 2D conic

Various views of the twisted cubic(t3, t2, t)T

The screw decomposition

• Any particular translation and rotation is equivalent to a rotation about a screw axis together with a translation along the screw axis. The screw axis is parallel to the original rotation axis.

• In the case of a translation and an orthogonal rotation axis ( termed planar motion), the motion is equivalent to a rotation about the screw axis.

2D Euclidean motion and a screw axis

3D Euclidean motion and the screw decomposition.

• Since t can be decomposed into tll and

(components parallel to the rotation axis and perpendicular to the rotation axis).

• Then a rotation about the screw axis is equivalent to a rotation about the original and a translation

t

t

3D Euclidean motion and the screw decomposition 2

The plane at infinity

• p2 linf, circular points I,J on linf

• p3 inf, absolute conic inf on inf

The canonical form of inf = (0,0,0,1)T

in affine space.

It contains the directions D = (x1, x2, x3, 0)T

The plane at infinity 2

• Two planes are parallel if and only if , their line of intersection is on inf

• A line is parallel to another line, or to a plane if the point of intersection is on inf

• The plane inf has 3 dof and is a fixed plane under affine transformation but is moved by a general projective transform

The plane at infinity 3

• Result 2.7 The plane at infinity inf, is fixed under the projective transformation H, if and only if H is an affinity.

• Consider a Euclidean transformation

•

1000

0100

00cossin

00sin-cos

10

0R H TE

The plane at infinity 4

• The fixed plane of H are the eigenvectors of HT .

• The eigenvalues are ( eie –iand the corresponding eigenvectors of HT are

1

0

0

0

E

0

1

0

0

E

0

0

1

E

0

0

1

E 1111

ii

The plane at infinity 5

• E1 and E2 are not real planes.

• E3 and E4 are degenerate. Thus there is a pencil of fixed planes which is spanned by these eigenvectors. The axis of this pencil is the line of intersection of the planes with inf

The absolute conic

• The absolute conic, inf is a point conic on inf. In a metric frame , inf = (0,0,0,1)T and points on inf satisfy

• x12 + x2

2 + x32 = 0

• x4 = 0

• The conic inf is a geometric representation of the 5 additional dof required to specify metric properties in an affine coordinate frame.

The absolute conic 2

The absolute conic inf is fixed under the projective transformation H if and only if H is a similarity transformation.

In a metric frame, inf = I3 x 3 and is fixed by HA. One has A-T I A-1 = I (up to scale)

Taking inverse gives AAT =I implying A is orthogonal•

10

tA H TA

Absolute conic 3

inf is only fixed as a set by general similarity; it is not fixed point wise

All circles intersect inf in two points. These two are the circular points of

ll spheres intersect inf ininf

Metric properties

• Two lines with directions d1 and d2 ( 3-vectors). The angle between these two directions in a Euclidean world frame is given by

• This may be written as:

(2.22)

dddd

dd cos

2T22

T1

2T1

(2.23)

dddd

dd cos

2T22

T1

2T1

Metric properties 2

• Where d1 and d2 are the points of intersection of the lines with the plane inf containing the conic inf

• The expression (2.23) is valid in any projective coordinate frame

• The expression (2.23) reduces to (2.22) in a Euclidean world frame where inf = I.

Orthogonality and polarity

• From (2.23), two directions are orthogonal if

• Orthogonality is thus encoded by conjugacy w.r.t. inf..

• The main advantage of this is that conjugacy is a projective relation.

0 d d 2T1

(a) On inf orthogonal directions d1, d2 are conjugate w.r.t. inf

(b) plane normal direction d and the intersection line l of the plane with inf are the pole-polar relation

with respect to inf

The absolute dual quadric Qinf*

inf is defined by two equations – it is a conic on the plane at infinity.

The dual of the absolute conic inf is a degenerate dual quadric in 3-space called the absolute dual quadric, and denoted by Qinf*

Geometrically Qinf* consists of planes tangent to inf .

The absolute dual quadric Qinf* (2)

Qinf* is a 4 x 4 homogeneous matrix of rank 3, which in metric space has the canonical form

The dual quadric Qinf* is a degenerate quadric and has 8 dof. Qinf* has a significant advantage over inf in algebra

manipulations because both inf ( 5 dof) and inf (3 dof )are contained in a single geometric object.

00

0I Q T

*

The absolute dual quadric Qinf* (3)

• The absolute dual quadric Qinf* is fixed under a projective transformation H if and only if H is a similarity. That is T** H Q H Q

The absolute dual quadric Qinf* (4)

• The above matrix equation holds if and only if

v = 0 and A is a scaled orthogonal matrix

The absolute dual quadric Qinf* (5)