Wonwoo Lee

1
Projective geometry and Transformations of 3D

• Wonwoo Lee
• 2006.01.03
• GIST U-VR Lab.
• Gwangju 500-712, S. Korea
• http//uvr.gist.ac.kr

2
Outline

• Point and projective Transformation
• Planes in P3
• Points and a plane
• Plane at infinity
• Absolute conic

3
Point and projective Transform.

• A point in P3 in homogeneous coord.
• If X40 ? point at infinity
• Projective transformation on P3
• Represented as 4x4 matrix H
• 15 degrees of freedom

4
Planes in P3

• Planes are defined as

Homogeneous representation of a plane
The point X is on the plane
Normal vector of the plane
5
Points and a plane

• Suppose three points X1, X2, X3 are incident with
  the plane, then

• For mxn matrix
• Null space and row space are sub-spaces of Rn
• Left null space and column space are sub-spaces
  of Rm

• X1, X2, X3 are linearly independent in general.
• Dimension of null space 4 3 1
• The solution is unique. (up to scale)

6
Points and a plane

• In P2

• In P3

A plane which passes three points, X1, X2, X3
7
Planes and a point

• Three planes define a point
• Dual to the case three points defining a plane.
• Intersection point of three planes

The point X is null space of 3x4 matrix

• Parameterized points on a plane

8
Lines

• Null-space and span representation
• A line is a pencil of collinear points
• A line is the axis of a pencil of planes
• A line as the span of two vectors

A, B Two points in space

• The span of WT is the pencil of points
• The span of the 2-dimensional right null-space of
  W is the pencil of planes with the line as axis

9
Quadrics and dual quadrics

• Quadrics
• A surface in P3 defined by the equation

(Q 4x4 symmetric matrix)

• 9 DOF, 9 points define a quadric
• A surface in P3 defined by the equation
• Det(Q) 0 ?? degenerate quadric
• The plane is the polar plane of X
  wrt. Q
• Intersection btw a quadric and a plane is a conic
  
• Projective transformation
• Dual of a quadric is also a quadric

10
Classification of quadrics
Projectively equivalent to sphere
sphere
ellipsoid
paraboloid
hyperboloid of two sheets
11
Hierarchy of transformations
12
Hierarchy of transformations
13
Plane at infinity

• Corresponding to the line at infinity in P2
• Two planes are parallel if, and only if their
  line of intersection is on a plane at infinity
• A line is parallel to another line or to a plane,
  if the point of intersection is on a plane at
  infinity
• The plane at infinity is a fixed plane under the
  affine transformation

Line at infinity
Plane at infinity
14
Absolute conic

• A conic is a curve described by a 2nd degree
  equation in a plane
• Absolute conic ??8 on the plane at infinity
• For the points on the plane at infinity
• A conic of purely imaginary points on ??8

15
Absolute conic

• Properties
• ??8 is fixed as a set by a general similarity
  transformation, not pointwise
• All circles intersect ??8 in two points
• All spheres intersect ??8 in ??8
• Metric properties

d1, d2 intersection with ??8
d1, d2 line directions