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January 30, 2003Frank PfenningCarnegie Mellon University
http://www.cs.cmu.edu/~fp/courses/graphics/
Shear TransformationCamera PositioningSimple Parallel ProjectionsSimple Perspective Projections
[Angel, Ch. 5.2-5.4][Red’s Dream, Pixar, 1987]
Shear TransformationCamera PositioningSimple Parallel ProjectionsSimple Perspective Projections
[Angel, Ch. 5.2-5.4][Red’s Dream, Pixar, 1987]
Viewing and ProjectionViewing and Projection
15-462 Computer Graphics ILecture 5
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01/30/2003 15-462 Graphics I 2
Transformation Matrices in OpenGLTransformation Matrices in OpenGL
• Transformation matrices in OpenGl are vectors of 16 values (column-major matrices)
• In glLoadMatrixf(GLfloat *m);
• Some books transpose all matrices!
m = {m1, m2, ..., m16} represents
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01/30/2003 15-462 Graphics I 3
Pondering TransformationsPondering Transformations
• Derive transformation given some parameters– Choose parameters carefully– Consider geometric intuition, basic trigonometry
• Compose transformation from others– Use translations to and from origin
• Test if matrix describes some transformation– Determine action on basis vectors
• Meaning of dot product and cross product
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01/30/2003 15-462 Graphics I 4
Shear TransformationsShear Transformations
• x-shear scales x proportional to y• Leaves y and z values fixed
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01/30/2003 15-462 Graphics I 5
Specification via AngleSpecification via Angle
• cot(θ) = (x’-x)/y• x’ = x + y cot(θ)• y’ = y• z’ = z
(x,y) (x’,y’)
θ x
y
y
x’-x
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01/30/2003 15-462 Graphics I 6
Specification via RatiosSpecification via Ratios
• Shear in both x and z direction• Leave y fixed• Slope α for x-shear, γ for z-shear• Solve
• Yields
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01/30/2003 15-462 Graphics I 7
Composing TransformationsComposing Transformations
• Every affine transformation is a composition of rotations, scalings, and translations
• How do we compose these to form an x-shear?• Exercise!
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01/30/2003 15-462 Graphics I 8
Thinking in FramesThinking in Frames
• Action on frame determines affine transfn.• Frame given by basis vectors and origin• xz-shear: preserve basis vectors ux and uz
• Move uy = [0 1 0 0]T
to uv’ = [α 1 γ 0]T
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01/30/2003 15-462 Graphics I 9
Preservation of OriginPreservation of Origin
• Preserve origin P0
• Results comprise columns of the transfn. matrix
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01/30/2003 15-462 Graphics I 10
OutlineOutline
• Shear Transformation• Camera Positioning• Simple Parallel Projections• Simple Perspective Projections
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01/30/2003 15-462 Graphics I 11
Camera in Modeling CoordinatesCamera in Modeling Coordinates
• Camera position is identified with a frame• Either move and rotate the objects• Or move and rotate the camera• Initially, pointing in negative z-direction• Initially, camera at origin
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01/30/2003 15-462 Graphics I 12
Moving Camera and World FrameMoving Camera and World Frame
• Move world frame relative to camera frame• glTranslatef(0.0, 0.0, -d); moves world frame
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01/30/2003 15-462 Graphics I 13
Order of Viewing TransformationsOrder of Viewing Transformations
• Think of moving the world frame• Viewing transfn. is inverse of object transfn.• Order opposite to object transformations
glMatrixMode(GL_MODELVIEW);glLoadIdentity();glTranslatef(0.0, 0.0, -d); /*T*/glRotatef(-90.0, 0.0, 1.0, 0.0); /*R*/
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01/30/2003 15-462 Graphics I 14
The Look-At FunctionThe Look-At Function
• Convenient way to position camera• gluLookAt(ex, ey, ez, ax, ay, az, px, py, pz);• e = eye point• a = at point• p = up vector
ae
p
view plane
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01/30/2003 15-462 Graphics I 15
Implementing the Look-At FunctionImplementing the Look-At Function
• (1) Transform world frame to camera frame• Compose a rotation R with translation T• W = T R• (2) Invert W to obtain viewing transformation V• V = W-1 = (T R)-1 = R-1 T-1
• Derive R, then T, then R-1 T-1
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01/30/2003 15-462 Graphics I 16
World Frame to Camera Frame IWorld Frame to Camera Frame I
• Camera points in negative z direction• n = (a – e) / |a – e| is unit normal to view plane• R maps [0 0 -1 0]T to [nx ny nz 0]T
ae
p
view plane
n
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01/30/2003 15-462 Graphics I 17
World Frame to Camera Frame IIWorld Frame to Camera Frame II
• R maps y to projection of p onto view plane• α = (p ¢ n) / |n| = p ¢ n• v0 = p – α n• v = v0 / |v0|
ae
p
view plane
nV0
α
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01/30/2003 15-462 Graphics I 18
World Frame to Camera Frame IIIWorld Frame to Camera Frame III
• x is orthogonal to n and v in view plane• u = n £ v• (u, v, -n) is right-handed
ae
p
view plane
nv
u
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01/30/2003 15-462 Graphics I 19
Summary of RotationSummary of Rotation
• gluLookAt(ex, ey, ez, ax, ay, az, px, py, pz);• n = (a – e) / |a – e|• v = (p – (p ¢ n) n) / |p – (p ¢ n) n|• u = n £ v
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01/30/2003 15-462 Graphics I 20
World Frame to Camera Frame IVWorld Frame to Camera Frame IV
• Translation of origin to e = [ex ey ez 1]T
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01/30/2003 15-462 Graphics I 21
Camera Frame to World FrameCamera Frame to World Frame
• V = W-1 = (T R)-1 = R-1 T-1
• R is rotation, so R-1 = RT
• T is translation, so T-1 negates displacement
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01/30/2003 15-462 Graphics I 22
Putting it TogetherPutting it Together
• Calculate V = R-1 T-1
• This is different from book [Angel, Ch. 5.2.2]• There, u, v, n are right-handed (here: u, v, -n)
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01/30/2003 15-462 Graphics I 23
Other Viewing FunctionsOther Viewing Functions
• Roll (about z), pitch (about x), yaw (about y)
• Assignment 2 poses related problem
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01/30/2003 15-462 Graphics I 24
OutlineOutline
• Shear Transformation• Camera Positioning• Simple Parallel Projections• Simple Perspective Projections
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01/30/2003 15-462 Graphics I 25
Projection MatricesProjection Matrices
• Recall geometric pipeline
• Projection takes 3D to 2D• Projections are not invertible• Projections also described by matrix• Homogenous coordinates crucial• Parallel and perspective projections
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01/30/2003 15-462 Graphics I 26
Orthographic ProjectionsOrthographic Projections
• Parallel projection• Projectors perpendicular to projection plane• Simple, but not realistic• Used in blueprints (multiview projections)
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01/30/2003 15-462 Graphics I 27
Orthographic Projection MatrixOrthographic Projection Matrix
• Project onto z = 0• xp = x, yp = y, zp = 0• In homogenous coordinates
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01/30/2003 15-462 Graphics I 28
PerspectivePerspective
• Perspective characterized by foreshortening• More distant objects appear smaller• Parallel lines appear to converge• Rudimentary perspective in cave drawings
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01/30/2003 15-462 Graphics I 29
Discovery of PerspectiveDiscovery of Perspective
• Foundation in geometry (Euclid)
Mural fromPompeii
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01/30/2003 15-462 Graphics I 30
Middle AgesMiddle Ages
• Art in the service of religion• Perspective abandoned or forgotten
Ottonian manuscript,ca. 1000
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01/30/2003 15-462 Graphics I 31
RenaissanceRenaissance
• Rediscovery, systematic study of perspectiveFilippo BrunelleschiFlorence, 1415
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01/30/2003 15-462 Graphics I 32
Perspective Viewing MathematicallyPerspective Viewing Mathematically
• More on history of perspective (icscis) http://www.cyberus.ca/~icscis/icscis.htm
• y/z = yp/d so yp = y/(z/d)• Note this is non-linear!
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01/30/2003 15-462 Graphics I 33
Exploiting the 4th DimensionExploiting the 4th Dimension
• Perspective projection is not affine:
• Idea: represent point [x y z 1]T by line in 4D
has no solution for M
for arbitrary w ≠ 0
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01/30/2003 15-462 Graphics I 34
Perspective Projection MatrixPerspective Projection Matrix
• Represent multiple of point
• Solve
with
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01/30/2003 15-462 Graphics I 35
Perspective DivisionPerspective Division
• Normalize [x y z w]T to [(x/w) (y/w) (z/w) 1]T
• Perform perspective division after projection
• Projection in OpenGL is more complex
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01/30/2003 15-462 Graphics I 36
Parallel Viewing in OpenGLParallel Viewing in OpenGL
• glOrtho(xmin, xmax, ymin, ymax, near, far)
zmin = near, zmax = far
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01/30/2003 15-462 Graphics I 37
Perspective Viewing in OpenGLPerspective Viewing in OpenGL
• Two interfaces: glFrustum and gluPerspective• glFrustum(xmin, xmax, ymin, ymax, near, far);
zmin = near, zmax = far
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01/30/2003 15-462 Graphics I 38
Field of View InterfaceField of View Interface
• gluPerspective(fovy, aspect, near, far);• near and far as before• Fovy specifies field of view as height (y) angle
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01/30/2003 15-462 Graphics I 39
Matrices for Projections in OpenGLMatrices for Projections in OpenGL
• Next lecture:– Use shear for predistortion– Use projections for “fake” shadows– Other kinds of projections
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01/30/2003 15-462 Graphics I 40
AnnouncementsAnnouncements
• Assignment 1 due Thursday midnight (100 pts)• Late policy
– Up to 3 days any time, no penalty– No other late hand-in permitted
• Assignment 2 out Thursday (1 week, 50 pts)• Extra credit policy
– Up to 20% of assignment value– Recorded separately– Weighed for “borderline” cases
• Remember: no collaboration on assignments!
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01/30/2003 15-462 Graphics I 41
Looking AheadLooking Ahead
• Lighting and shading• Video: Red’s Dream, John Lasseter, Pixar,1987
http://www.pixar.com/shorts/rd/index.html