Homogeneous Coordinates and Transformation

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Homogeneous Coordinates and Transformation
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Line in R2
General line equation
Normalize
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Line in R2
Parametric equation of a line
Corresponding implicit form
Implicitize
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Linear (affine) Transformation

• Properties
• Collinearity (maps a line to a line)
• Preserve ratio of distances (midpoint stays in
  the middle after transformation)

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Common 2D Linear Transformations

• Translation
• Scaling
• Reflection (QI2uuT)
• Rotation about origin
• Shear

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Homogeneous Coordinate

• Motivation to unify representations of affine
  map (esp. translation)

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Definitions
Equivalence relation on the set S R3 \
(0,0,0)
Ex Show that this relation is reflexive,
symmetric, and transitive
Equivalence classes of the relation
Homogeneous coordinates
Projective plane P2 the set of all equivalence
classes
An equivalence class is referred to as a point in
the projective plane.
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Definitions
Points on P2
I. (u,v,w) with w ?0
Choose a representative (u/w, v/w, 1) 1-1
correspondence with Cartesian plane
II. (u,v,w) with w 0
Corresponds to points-at-infinity, each with a
specific direction
Points on P2 the plane R2 plus all the points at
infinity
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Points at Infinity
(x,y,0)
Points at infinity (x,y,0)
Reach the same point (at ?), from any starting
point
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Parallel Lines Intersect at Infinity
(-2,1,0)
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Visualization

• Line model and spherical model

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Line in Cartesian Space
(or any multiple of it)
(or any multiple of it)
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Examples (cases in R2)

• The line passes through (3,1) and (-4,5)

• Intersection of

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• Two parallel lines

• Defining a line with a point at infinity

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Plane in Cartesian Space
Extend to P3 and R3
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Intersection of Three Planes
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Line in R3(Plücker Coordinate)
Line in parametric form
Define
Plucker coordinate of the line (q, q0)
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Space Transformation

• Translation
• Scaling
• Rotation about coordinate axes
• Rotation about arbitrary line
• Reflection about arbitrary plane (QI2uuT)

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Transformed Equations

• If transformation T is applied to geometry
  (line/plane), whats the transformed equation?
• Apply T to homogenous line/plane equation?! NOT
  !!
• Answers
• See handout p.3 (convert to parametric form
  transform the points then to implicit equation)
• More detailed version see homogeneous-transforma
  tion.ppt from R. Paul
• Also related to the normal matrix in OpenGL.

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From Richard Paul Ch.1
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Summary
Point u on a plane
After transformation H
Point u becomes v Hu Plane P becomes PH-1
Reason
Note if P is written as a column vector, the
formula becomes P H-TP
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(No Transcript)
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Transformation
v Hu
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From Opengl-1.ppt
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Vectors and Points are Different!
glNormal
glVertex

• Point
• Homogenenous coordinate
• p x y z 1
• M affine transform (translate, rotate, scaling,
  reflect, )
• p M p

• Vector
• Homogeneous coordinate
• v x y z 0
• Affine transform (applicable when M is invertible
  (not full rank projection to 2D is not)
• v (M-1)T v

(ref)
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vMv wont work
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On (M-1)T

• The w (homogeneous coord) of vectors are 0
  hence, the translation part (3?1 vector) plays no
  role
• For rotation, M-1MT, hence (MT)T M rotate the
  vector as before
• For scaling

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Hence
This is known as the normal matrix (ref)