CMPUT 498 Projective Geometry

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CMPUT 498Projective Geometry

• Lecturer Sherif Ghali
• Department of Computing Science
• University of Alberta

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Albrecht Dürer, Artist drawing a lute, 1525
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Geometries

• Euclidean Geometry
• A study of object properties that are unchanged
  by rotation, translation, and reflection, e.g.
• segment congruence
• angle congruence
• parallelism
• Projective Geometry
• A study of objects as they are seen

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Distortions inprojective geometry
length, angle, parallelism, and shape distortions
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Distortions inprojective geometry
view of the corner in a room 3 90 360?
view of a checkerboard
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Projectors and the view plane
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Types of perspective

• In computer graphics we are mostly interested in
  linear perspective
• planar projection surface
• linear projectors
• But others exist
• spherical projection
• cylindrical projection

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Relative order

• Computer graphics
• centre of projection on opposite side of image
  plane
• Photography
• image is reversed

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Vanishing points

• One to one mapping between points on object and
  image planes
• A vanishing point is the limit of the images of
  points on both lines

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Locating the vanishing point (1)
find the intersection of the three planes ?1,
?2, and the image plane
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Locating the vanishing point (2)
Horizon plane passes by the eye and is parallel
to the object plane find the intersection of the
horizon plane, the image plane, and (l, l0)
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The vanishing line
object lines do not have to be perpendicular to
the image plane the vanishing points trace the
vanishing line
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Image plane not perpendicular to object plane
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Does seeing a horizon establish that the earth is
spherical?
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Extrinsic vs. Intrinsic geometry
an extrinsic Euclidean plane embedded in 3-space
the intrinsic Euclidean plane
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The projective plane

• What is the interest of planar projections of
  planar regions?
• faceted objects
• One-to-one mapping is a transformation
• how do we handle the vanishing line?

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Inadequacy ofthe Euclidean plane
What do the vanishing line and line m map to?
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Ideal points

• An ideal point is a point at infinity
• Each set of parallel lines meet at a point at
  infinity this ideal point is added to the
  Euclidean plane
• A Euclidean plane its ideal points a
  projective plane

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Attaching ideal points
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Definitions

• A Euclidean line its ideal point
• a projective line
• A Euclidean line is embedded in the Euclidean
  plane
• A projective line is embedded in the projective
  plane
• The set of all ideal points in a projective plane
  form the ideal line (which is a projective line)

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Definitions

• A perspective transformation maps each object
  plane point to a view plane point (using a
  projector through the eye)
• The eye is called the center of perspectivity
• A projective transformation is either a
  perspective transformation or a combination of
  perspective transformations

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Shadows

• Shadows are projective transformations consisting
  of two perspective transformations

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A topological model ofthe projective plane
topologivally equivalent surfaces
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Topology of the Euclidean plane

• A Euclidean plane is equivalent to the interior
  of a disk of radius 1
• T(r, ?) (r/(r1), ?)

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Topology of the Euclidean plane
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to identify

• to identify
• to establish the identity of
• to cause to become identical ?

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Identifying antipodal points
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The projective line
The projective line is topologically equivalent
to a circle
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The projective planeis non-orientable
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The standard embeddedprojective plane
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Vanishing points
vanishing points are the intersections of pairs
of vanishing lines for pairs of object planes
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The projective space
a three dimensional spherical Euclidean ball
a three dimensional spherical projective ball.
Ideal points are added (and antipodal points are
identified)
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Projective propositions

• Two distinct points lie on one and only one line
• One or both points can be ideal
• Two distinct lines meet in one and only one point
• The lines may be parallel
• One of the two lines can be ideal

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Duality ofplanar projective geometry

• Two distinct points lie on one and only one line
• Two distinct lines meet in one and only one point

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Duality ofprojective geometry in space

• Two distinct planes meet in one and only one line
• Two distinct points lie on one and only one line
• Three distinct noncollinear points lie on one and
  only one plane
• Three distinct planes not containing a common
  line meet in one and only one point
• A line and a plane not containing the line meet
  in one and only one point
• A line and a point not on the line are contained
  in one and only one plane

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Perspective transformation(perspectivity)
the eye is the center of perspectivity
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Compositingperspective transformations
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Parallel projection

• The eye or center of projection is an ideal point
• Commonly used by mechanical and civil engineers
• Word usage
• Parallel projections are a special case of
  perspective projections
• Viewing in perspective where perspective and
  parallel (orthographic) are mutually exclusive

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Parallel projection
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Fundamental theorem ofprojective geometry

• For distinct points A,B,C on line l
• and distinct points P,Q,R on line m,
• there is one and only one projective
  transformation taking
• l to m and
• A,B,C to P,Q,R, respectively

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Analytical geometry

• Homogeneous coordinates are to projective
  geometry
• what
• Cartesian coordinates are to Euclidean geometry

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Analytical projective geometry

• We seek new coordinates as Cartesian coordinates
  are insufficient
• Objectives
• Represent ideal points
• Represent Euclidean and ideal points in the same
  manner (homogeneously)
• Ability to determine (easily) whether a point is
  Euclidean or ideal

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Non-zero vectors in R3

• Define RP2 p p1, p2, p3
• RP2 R3 lt0,0,0gt
• p q iff there is k!0,
• p1 k.q1 , p2 k.q2, p3 k.q3
• Fractions also have multiple representations

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Representative vector

• ltq1, q2, q3gt is a representative of
• p1, p2, p3 iff p1, p2, p3 ltq1, q2, q3gt
• An equivalence relationship is defined on RP2
• Decompose RP2 into p1, p2, 1 ? p1, p2, 0
• Can a one-to-one correspondence be defined from
  the projective plane to RP2?

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Mappinghomogeneous coordinates tothe projective
plane

• Consider the projective plane embedded in R3 at
  z1
• p1, p2, 1 maps to the Cartesian point (p1,p2,1)
  
• A line passing through (p1, p2, 1) and (0,0,1)
  is (t.p1, t.p2, 1)
• A representative point for this point is
  (p1,p2,1/t)
• The coordinates of the ideal point in this
  direction are thus defined

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Mappinghomogeneous coordinates tothe projective
plane

• Mapping an ideal point to x,y,0 is consistent
• The limit of any line in the same direction is
  the same ideal point
• The partition p1, p2, 1 ? p1, p2, 0 is the
  partition into Euclidean and ideal points

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Analytical projective line

• Point p1,p2,p3 lies on the line l1,l2,l3
  iff
• l1p1 l2p2 l3p3 0

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Analytical projective line

• The ideal line is 0,0,l3
• Euclidean lines have only one of l10 or l20

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Finding the linepassing by 2 points

• Line passing by qq1,q2,q3 and rr1,r2,r3 has
  equation l1p1 l2p2 l3p3 0, where

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Finding the intersectionof 2 lines

• Two lines with the equations
• intersect at a point pp1,p2,p3, where

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Analytic duality

• There is a one-to-one mapping such that the dual
  of point P x,y,z is the line lx,y,z

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Analytical projective planes

• Point p1,p2,p3,p4 lies on the plane
  ?1,?2,?3,?4 iff
• ?1p1 ?2p2 ?3p3 ?4p4 0

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Finding the plane determined by three points

• A plane with the equation
• ?1p1 ?2p2 ?3p3 ?4p4 0
• passing by q, r, and s
• is determined by

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Finding the intersection of 3 projective planes
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Duality in projective space

• The dual of the origin is the ideal plane
• The dual of an ideal point is a plane passing by
  the origin (orientation?)
• What is the dual of an arbitrary point/plane?

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Properties of duality

• Duality preserves incidence
• If a point p lies on a line l, then the dual of l
  lies on the dual of p

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Properties of duality

• Two points p1 and p2 lie on a line l
• The dual of p1 and p2 meet at the dual of l

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Properties of duality

• The duals of a set of colinear points meet at a
  point

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Detecting colinearity
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Reference

• Penna and Patterson, Projective Geometry and its
  Applications to Computer Graphics, Prentice-Hall,
  1986