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: p =( p1, p2, p3, p4 )

Planes The plane: A plane in 3-space may be written as p1x1 + p2x2 + p3x3 + p4x4 = 0 pT X = 0 In inhomogeneous coordinates in 3-vector notation Where n =( p1, p2, p3 ), x4 =1 d= p4 , , d /||n|| is the distance of the origin.

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 lA+mB on the line.

(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 p= QX is the polar plane of X w.r.t. Q (iv) The intersection of a plane p 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 planes pT Q* p =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 s(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 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

3D Euclidean motion and the screw decomposition 2

The plane at infinity p2  linf, circular points I,J on linf p3  pinf, absolute conic Winf on pinf The canonical form of pinf = (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 pinf A line is parallel to another line, or to a plane if the point of intersection is on pinf The plane pinf 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 pinf, is fixed under the projective transformation H, if and only if H is an affinity. Consider a Euclidean transformation

The plane at infinity 4 The fixed plane of H are the eigenvectors of HT . The eigenvalues are ( eiq, e –iq, 1, 1) and the corresponding eigenvectors of HT are

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 pinf

The absolute conic The absolute conic, Winf is a point conic on pinf. In a metric frame , pinf = (0,0,0,1)T and points on Winf satisfy x12 + x22 + x32 = 0 x4 = 0 The conic Winf 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 Winf is fixed under the projective transformation H if and only if H is a similarity transformation. In a metric frame, Winf = 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

Absolute conic 3 Winf is only fixed as a set by general similarity; it is not fixed point wise All circles intersect Winf in two points. These two are the circular points of p All spheres intersect pinf inWinf

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:

Metric properties 2 Where d1 and d2 are the points of intersection of the lines with the plane pinf containing the conic Winf 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 Winf = I.

Orthogonality and polarity From (2.23), two directions are orthogonal if Orthogonality is thus encoded by conjugacy w.r.t. Winf.. The main advantage of this is that conjugacy is a projective relation.

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

(b) A plane normal direction d and the intersection line l of the plane with pinf are the pole-polar relation with respect to Winf

The absolute dual quadric Qinf* Winf is defined by two equations – it is a conic on the plane at infinity. The dual of the absolute conic Winf is a degenerate dual quadric in 3-space called the absolute dual quadric, and denoted by Qinf* Geometrically Qinf* consists of planes tangent to Winf .

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

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

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)