Post on 17-Dec-2015
transcript
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Projection
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Model Transform
Viewing Transform
ModelviewMatrix
worldcoordinates
Pipeline Review
Focus of this lecture
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Review (Lines in R2)
2121
2121
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:lines twoofon intersecti asPoint
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:line a determine points Two
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equation line sHomogeneou
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ppllpandlp
lp
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wu yx
4Parallel Projection
Projection (R2)
viewpoint
viewline
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bacabcbac
bacabccba
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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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ExampleVertices (0,0,0), (2,0,0), (2,3,0), (0,3,0) (1,1,1), (1,2,1)
Parallel projection: onto z = 0 planev = (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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321
321
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,,
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p’ p” O
Viewplane Coordinate Mapping
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Determine Viewplane Transform by Homogeneous Transformation
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001
100
010
001333
222
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srq
srq
srq
K4×3
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144313 pVp133414 pKp
pIpKp LL
L: left inverse of K
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ExampleViewplane 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
0,,, :viewpoint
,,, : vectorviewplane
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4321
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nnnnn
v
n
… is a parallel projection
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22
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nnn
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Definitions• Direction cosine (ref)
• Foreshortening ratio= (length of projected segment)/(length of original segment)
1
cos,cos,cos
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Al zyx
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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 (n2
2 + n32)1/2, (n1
2 + n32)1/2, and (n1
2 + 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
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31
31
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,,
0,,,
f
n 3
2232
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37
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,,
0,,,
f
n 75
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153
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,,
0,,,
f
n
Isometric Dimetric Trimetric
f: foreshortening ratios
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Example (Dimetric)
1000
0
0
0
0
98
97
91
97
92
97
91
97
98
31
37
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M
n
TT
TT
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zz
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oo
1' 1100
1' 1010
1' 1001
1000' 1000
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97
91
97
92
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91
97
98
322
32
322
zo
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xo
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Oblique Projection
• A particular parallel projection where direction of projection is not perpendicular to viewplane
v
n
Oblique projection not available in
OpenGL
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Cavalier Projection
Lines viewplane have f = 1Planar faces viewplane appear thicker
v
/4n
Properties:
viewplan
e
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Cabinet Projection
To overcome ‘thickness’ problem, choose f viewplane to be 1/2
Properties:
= arccot(2)
v
n
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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)
n
n
n
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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
Two-point perspective
One-point perspective
Vanishing point
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IMAGE FORMATION – Perspective Imaging
Image courtesy of C. Taylor
“The Scholar of Athens,” Raphael, 1518
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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
TT
TT
TT
T
T
T
M
M
M
Verify
IvnvnM
n
v
003100010
1160150100
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:
30101
1516015
00310
1515016
1101
115015
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How about (1,0,1,0)?
Viewpoint (15,0,15,1)Viewplane: x + z + 1 = 0
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Summary
• 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
Understand how they are
differentiated
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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
The cross-ratio of every set of four collinear points shown in this figure has the same value
Cross ratio is preserved in projective geometry(ratio is NOT preserved)
z1z2 z3 z4