Projection

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Projection
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Pipeline Review
Focus of this lecture
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Review (Lines in R2)
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Projection (R2)
viewline
viewpoint
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Perspective Projection


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Parallel Projection


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Projection (R3)
See handout for proof!
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Example
Vertices (0,0,0), (2,0,0), (2,3,0), (0,3,0)
(1,1,1), (1,2,1) Parallel projection
onto z 0 plane v (0,0,1,0)T, n (0,0,1,0)T
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Vertices (0,0,0), (2,0,0), (2,3,0), (0,3,0)
(1,1,1), (1,2,1) Perspective
projection onto z 0 plane from viewpoint
(1,5,3) v (1,5,3,1)T, n (0,0,1,0)T
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Viewplane Coordinate Mapping
p
p
O
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Determine Viewplane Transform by Homogeneous
Transformation
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L
L
L left inverse of K
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Example
Viewplane origin (1,2,0) u-axis (3,4,0)
v-axis (-4,3,0)
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Orthographic Projection

• Def direction of projection ? viewplane

is a parallel projection
n
v
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Definitions

• Direction cosine (ref)

• Foreshortening ratio
• (length of projected segment)/(length of
  original segment)

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Theorem

• If the direction cosines of the plane normal (in
  world coordinate system) are n1, n2, and n3, the
  foreshortening ratios in the x-, y-, and z-
  directions are (n22 n32)1/2, (n12 n32)1/2,
  and (n12 n22)1/2, respectively.

• Front, side, top views n (1,0,0,0), (0,1,0,0),
  or (0,0,1,0) as in engineering drawings

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Types of Orthographic Projections

• Axonometric projections attempts to portray
  general 3D shape
• Isometric projection all foreshortening ratio
  are the same
• Dimetric projection exactly two are the same
• Trimetric projection all foreshortening ratio
  are different

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Axonometric Projections
Isometric
Dimetric
Trimetric
f foreshortening ratios
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Example (Dimetric)
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(No Transcript)
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Oblique Projection

• A particular parallel projection where direction
  of projection is not perpendicular to viewplane

n
Oblique projection not available in OpenGL
v
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Cavalier Projection
p/4
n
v
viewplane
Properties
Lines ? viewplane have f 1 Planar faces ?
viewplane appear thicker
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Cabinet Projection
n
f arccot(2)
v
Properties
To overcome thickness problem, choose f ?
viewplane to be 1/2
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Perspective Projection

• A perspective projection maps parallel lines in
  the space to parallel lines in the viewplane IFF
  the lines are parallel to the viewplane.

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Otherwise, they meet
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Vanishing Point

• Suppose (xi, yi, zi) i 1,2,3 are a set of
  mutually perpendicular vectors. The viewplane
  normal (n1, n2, n3) of a perspective projection
  can be perpendicular to (a) none (b) one (c) two
  of the vectors.

(a)
(b)
(c)
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Vanishing Point

• If a perspective projection maps a
  point-at-infinity (x,y,z,0) to a finite point
  (x,y,z,1) on the viewplane, the lines in the
  direction (x,y,z) appear as lines converging to
  point on the (Cartesian) viewplane. The point
  (x,y,z) is called the vanishing point in the
  direction (x,y,z).

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Three-point perspective
Vanishing point
Two-point perspective
One-point perspective
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IMAGE FORMATION Perspective Imaging
The Scholar of Athens, Raphael, 1518
Image courtesy of C. Taylor
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Example

• Determine (and verify it is indeed so) the
  vanishing point of an OpenGL setting.

Eye 15,0,0
Eye 15,0,15
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Numeric Example
Viewpoint (15,0,15,1) Viewplane x z 1 0
How about (1,0,1,0)?
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Summary
Understand how they are differentiated

• Projection
• Parallel projection
• Perspective projection
• Parallel projection
• Orthographic
• Isometric
• Dimetric
• Trimetric
• Oblique
• Cavalier
• Cabinet

• Perspective projection
• Three-point perspective
• Two-point perspective
• One-point perspective

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Fig. 8. Constructing a perspective image of a
house. (a) Drawing the floor plan and defining
the viewing conditions (observer position and
image plane). (b) Constructing a perspective view
of the floor. (c) A reference height (in this
case the height of an external wall) is drawn
from the ground line and the first wall is
constructed in perspective by joining the
reference end points to the horizontal vanishing
point v2. (d) All four external walls are
constructed. (e) The elevations of all other
objects (the door, windows and roofs) are first
defined on the reference segment and then
constructed in the rendered perspective view.
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Exercise

• Hand sketch a perspective drawing of a house
• Use Maxima to compute 2-point perspective
  projection, setting viewplane coordinate system

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Cross Ratio
Cross ratio is preserved in projective
geometry (ratio is NOT preserved)
The cross-ratio of every set of four collinear
points shown in this figure has the same value
z1
z2
z3
z4