Projective Geometry- 3D

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Projective Geometry- 3D

• Points, planes,
• lines and quadrics

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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)

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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 )

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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.

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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
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Three planes define a point
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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.

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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 lAmB on
  the line.

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(ii) The the span of the 2D right null space of W
is the pencil of planes with the line as axis
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The dual representation of a line as the
intersection of two planes P and Q
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Examples
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Join and incidence relations from null-space
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Plucker matrices
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Properties of L
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Properties of L 2
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Examples(Plucker matrices)where the point A and
B are the origin and the ideal point in x
direction
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A dual Plucker representation L
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Join and incidence properties
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Examples 2
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Two lines
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The bilinear product (L !L)
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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

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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

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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

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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 QHTDH
  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

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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

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Categorization of point quadrics
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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

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Non-ruled quadrics a sphere and an ellipsoid
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Non ruled quadrics a hyperboloid of two sheets
and a paraboloid
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Ruled quadrics Two examples of hyperboloid of
one sheet are given. A surface is made up of two
sets of disjoint straight lines
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Degenerate quadrics
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The twisted cubic is a 3D analogue of a 2D conic
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Various views of the twisted cubic(t3, t2, t)T
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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.

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2D Euclidean motion and a screw axis
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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

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3D Euclidean motion and the screw decomposition 2
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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

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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

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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

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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

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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

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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.

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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

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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

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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

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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.

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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.

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

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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.

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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

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The absolute dual quadric Qinf (4)

• The above matrix equation holds if and only if
• v 0 and A is a scaled orthogonal matrix

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The absolute dual quadric Qinf (5)