CS 551 / 645:
Introductory Computer Graphics
Mathematical Foundations
The Rendering Pipeline
David Luebke
7/27/2016
Administrivia
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Sorry about the canceled class
OpenGL text: v1.1 or v1.2?
Teddy link fixed
Go over Assignment 1
David Luebke
7/27/2016
XForms Intro
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Xforms: a toolkit for easily building Graphical
User Interfaces, or GUIs
– See http://www.westworld.com/~dau/xforms
– Lots of widgets: buttons, sliders, menus, etc.
– Plus, an OpenGL canvas widget that gives us a
viewport or context to draw into with GL or Mesa.
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Quick tour now
You’ll learn the details yourself in
Assignment 1
David Luebke
7/27/2016
Mathematical Foundations
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FvD appendix gives good review
I’ll give a brief, informal review of some of the
mathematical tools we’ll employ
– Geometry (2D, 3D)
– Trigonometry
– Vector and affine spaces
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Points, vectors, and coordinates
– Dot and cross products
– Linear transforms and matrices
David Luebke
7/27/2016
2D Geometry
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Know your high-school geometry:
– Total angle around a circle is 360° or 2π radians
– When two lines cross:
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Opposite angles are equivalent
Angles along line sum to 180°
– Similar triangles:
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David Luebke
All corresponding angles are equivalent
Corresponding pairs of sides have the same length ratio
and are separated by equivalent angles
Any corresponding pairs of sides have same length ratio
7/27/2016
Trigonometry
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Sine: “opposite over hypotenuse”
Cosine: “adjacent over hypotenuse”
Tangent: “opposite over adjacent”
Unit circle definitions:
–
–
–
–
David Luebke
sin () = x
cos () = y
tan () = x/y
Etc…
(x, y)
7/27/2016
3D Geometry
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To model, animate, and render 3D scenes,
we must specify:
– Location
– Displacement from arbitrary locations
– Orientation
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We’ll look at two types of spaces:
– Vector spaces
– Affine spaces
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We will often be sloppy about the distinction
David Luebke
7/27/2016
Vector Spaces
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Two types of elements:
– Scalars (real numbers): a, b, g, d, …
– Vectors (n-tuples): u, v, w, …
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Supports two operations:
– Addition operation u + v, with:
 Identity 0
v+0=v
 Inverse v + (-v) = 0
– Scalar multiplication:
 Distributive rule:
a(u + v) = a(u) + a(v)
(a + b)u = au + bu
David Luebke
7/27/2016
Vector Spaces
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A linear combination of vectors results in a
new vector:
v = a1v1 + a2v2 + … + anvn
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If the only set of scalars such that
a1v1 + a2v2 + … + anvn = 0
is
a1 = a2 = … = a3 = 0
then we say the vectors are linearly independent
The dimension of a space is the greatest number of
linearly independent vectors possible in a vector set
For a vector space of dimension n, any set of n
linearly independent vectors form a basis
David Luebke
7/27/2016
Vector Spaces:
A Familiar Example
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Our common notion of vectors in a 2D plane
is (you guessed it) a vector space:
– Vectors are “arrows” rooted at the origin
– Scalar multiplication “streches” the arrow, changing
its length (magnitude) but not its direction
– Addition uses the “trapezoid rule”:
u+v
y
v
u
x
David Luebke
7/27/2016
Vector Spaces: Basis Vectors
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Given a basis for a vector space:
– Each vector in the space is a unique linear
combination of the basis vectors
– The coordinates of a vector are the scalars from
this linear combination
– Best-known example: Cartesian coordinates
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Draw example on the board
– Note that a given vector v will have different
coordinates for different bases
David Luebke
7/27/2016
Vectors And Points
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We commonly use vectors to represent:
– Points in space (i.e., location)
– Displacements from point to point
– Direction (i.e., orientation)
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But we want points and directions to behave
differently
– Ex: To translate something means to move it
without changing its orientation
– Translation of a point = different point
– Translation of a direction = same direction
David Luebke
7/27/2016
Affine Spaces
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To be more rigorous, we need an explicit
notion of position
Affine spaces add a third element to vector
spaces: points (P, Q, R, …)
Points support these operations
– Point-point subtraction:
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v
Result is a vector pointing from P to Q
– Vector-point addition:
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Q-P=v
Q
Result is a new point
P+v=Q
P
– Note that the addition of two points is not defined
David Luebke
7/27/2016
Affine Spaces
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Points, like vectors, can be expressed in
coordinates
– The definition uses an affine combination
– Net effect is same: expressing a point in terms of a
basis
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Thus the common practice of representing
points as vectors with coordinates (see FvD)
Analogous to equating pointers & integers in C
– Be careful to avoid nonsensical operations
David Luebke
7/27/2016
Affine Lines: An Aside
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Parametric representation of a line with a
direction vector d and a point P1 on the line:
P(a) = Porigin + ad
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Restricting 0  a produces a ray
Setting d to P - Q and restricting 0  a  1
produces a line segment between P and Q
David Luebke
7/27/2016
Dot Product
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The dot product or, more generally, inner
product of two vectors is a scalar:
v1 • v2 = x1x2 + y1y2 + z1z2 (in 3D)
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Useful for many purposes
– Computing the length of a vector: length(v) = sqrt(v • v)
– Normalizing a vector, making it unit-length
– Computing the angle between two vectors:
u • v = |u| |v| cos(θ)
– Checking two vectors for orthogonality
– Projecting one vector onto another
v
θ
u
David Luebke
7/27/2016
Cross Product
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The cross product or vector product of two
vectors is a vector:
 y1 z 2  y 2 z1 
v1  v 2   ( x1 z 2  x 2 z1)
 x1 y 2  x 2 y1 
The cross product of two vectors is
orthogonal to both
Right-hand rule dictates direction of cross
product
David Luebke
7/27/2016
Linear Transformations
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A linear transformation:
– Maps one vector to another
– Preserves linear combinations
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Thus behavior of linear transformation is
completely determined by what it does to a
basis
Turns out any linear transform can be
represented by a matrix
David Luebke
7/27/2016
Matrices
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By convention, matrix element Mrc is located
at row r and column c:
 M11 M12
 M21 M22
M
 
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
Mm1 Mm2
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 M1n 
 M2n 
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 

 Mmn 
By (OpenGL) convention,
vectors are columns:
David Luebke
 v1 


v   v 2
 v 3 
7/27/2016
Matrices
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Matrix-vector multiplication applies a linear
transformation to a vector:
 M11 M12 M13  vx 
M  v  M 21 M 22 M 23  vy 
M31 M32 M33  vz 
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Recall how to do matrix multiplication
David Luebke
7/27/2016
Matrix Transformations
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A sequence or composition of linear
transformations corresponds to the product
of the corresponding matrices
– Note: the matrices to the right affect vector first
– Note: order of matrices matters!
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The identity matrix I has no effect in
multiplication
Some (not all) matrices have an inverse:
M 1 Mv   v
David Luebke
7/27/2016
The Rendering Pipeline:
A Whirlwind Tour
Transform
Illuminate
Transform
Clip
Project
Rasterize
Model & Camera
Parameters
David Luebke
Rendering Pipeline
Framebuffer
Display
7/27/2016
The Display You Know
Transform
Illuminate
Transform
Clip
Project
Rasterize
Model & Camera
Parameters
David Luebke
Rendering Pipeline
Framebuffer
Display
7/27/2016
The Framebuffer You Know
Transform
Illuminate
Transform
Clip
Project
Rasterize
Model & Camera
Parameters
David Luebke
Rendering Pipeline
Framebuffer
Display
7/27/2016
The Rendering Pipeline
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Why do we call it a pipeline?
Transform
Illuminate
Transform
Clip
Project
Rasterize
Model & Camera
Parameters
David Luebke
Rendering Pipeline
Framebuffer
Display
7/27/2016
2-D Rendering: Rasterization
(Coming Soon)
Transform
Illuminate
Transform
Clip
Project
Rasterize
Model & Camera
Parameters
David Luebke
Rendering Pipeline
Framebuffer
Display
7/27/2016
The Rendering Pipeline: 3-D
Transform
Illuminate
Transform
Clip
Project
Rasterize
Model & Camera
Parameters
David Luebke
Rendering Pipeline
Framebuffer
Display
7/27/2016
The Rendering Pipeline: 3-D
Scene graph
Object geometry
Result:
Modeling
Transforms
• All vertices of scene in shared 3-D “world” coordinate system
Lighting
Calculations
• Vertices shaded according to lighting model
Viewing
Transform
• Scene vertices in 3-D “view” or “camera” coordinate system
Clipping
Projection
Transform
David Luebke
• Exactly those vertices & portions of polygons in view frustum
• 2-D screen coordinates of clipped vertices
7/27/2016
The Rendering Pipeline: 3-D
Scene graph
Object geometry
Result:
Modeling
Transforms
• All vertices of scene in shared 3-D “world” coordinate system
Lighting
Calculations
• Vertices shaded according to lighting model
Viewing
Transform
• Scene vertices in 3-D “view” or “camera” coordinate system
Clipping
Projection
Transform
David Luebke
• Exactly those vertices & portions of polygons in view frustum
• 2-D screen coordinates of clipped vertices
7/27/2016
Rendering: Transformations
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So far, discussion has been in screen space
But model is stored in model space
(a.k.a. object space or world space)
Three sets of geometric transformations:
– Modeling transforms
– Viewing transforms
– Projection transforms
David Luebke
7/27/2016
Rendering: Transformations
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Modeling transforms
– Size, place, scale, and rotate objects parts of the
model w.r.t. each other
– Object coordinates  world coordinates
Y
Y
X
Z
X
Z
David Luebke
7/27/2016
Rendering: Transformations
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Viewing transform
– Rotate & translate the world to lie directly in front of
the camera
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Typically place camera at origin
Typically looking down -Z axis
– World coordinates  view coordinates
David Luebke
7/27/2016
Rendering: Transformations
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Projection transform
– Apply perspective foreshortening
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Distant = small: the pinhole camera model
– View coordinates  screen coordinates
David Luebke
7/27/2016
Rendering: Transformations
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All these transformations involve shifting
coordinate systems (i.e., basis sets)
That’s what matrices do…
Represent coordinates as vectors, transforms
as matrices
X  cos q
    q
Y  sin
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David Luebke
sin q  X 
 
q
cos  Y 
Multiply matrices = concatenate transforms!
7/27/2016
Rendering: Transformations
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Homogeneous coordinates: represent
coordinates in 3 dimensions with a 4-vector
– Denoted [x, y, z, w]T
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Note that w = 1 in model coordinates
– To get 3-D coordinates, divide by w:
[x’, y’, z’]T = [x/w, y/w, z/w]T
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David Luebke
Transformations are 4x4 matrices
Why? To handle translation and projection
7/27/2016
Rendering: Transformations
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OpenGL caveats:
– All modeling transforms and the viewing transform
are concatenated into the modelview matrix
– A stack of modelview matrices is kept
– The projection transform is stored separately in
the projection matrix
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See Chapter 3 of the OpenGL book
David Luebke
7/27/2016
The Rendering Pipeline: 3-D
Scene graph
Object geometry
Result:
Modeling
Transforms
• All vertices of scene in shared 3-D “world” coordinate system
Lighting
Calculations
• Vertices shaded according to lighting model
Viewing
Transform
• Scene vertices in 3-D “view” or “camera” coordinate system
Clipping
Projection
Transform
David Luebke
• Exactly those vertices & portions of polygons in view frustum
• 2-D screen coordinates of clipped vertices
7/27/2016
Rendering: Lighting
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Illuminating a scene: coloring pixels
according to some approximation of lighting
– Global illumination: solves for lighting of the whole
scene at once
– Local illumination: local approximation, typically
lighting each polygon separately
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Interactive graphics (e.g., hardware) does
only local illumination at run time
David Luebke
7/27/2016
The Rendering Pipeline: 3-D
Scene graph
Object geometry
Result:
Modeling
Transforms
• All vertices of scene in shared 3-D “world” coordinate system
Lighting
Calculations
• Vertices shaded according to lighting model
Viewing
Transform
• Scene vertices in 3-D “view” or “camera” coordinate system
Clipping
Projection
Transform
David Luebke
• Exactly those vertices & portions of polygons in view frustum
• 2-D screen coordinates of clipped vertices
7/27/2016
Rendering: Clipping
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Clipping a 3-D primitive returns its
intersection with the view frustum:
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See Foley & van Dam section 19.1
David Luebke
7/27/2016
Rendering: Clipping
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Clipping is tricky!
David Luebke
Clip
In: 3 vertices
Out: 6 vertices
Clip
In: 1 polygon
Out: 2 polygons
7/27/2016
The Rendering Pipeline: 3-D
Transform
Illuminate
Transform
Clip
Project
Rasterize
Model & Camera
Parameters
David Luebke
Rendering Pipeline
Framebuffer
Display
7/27/2016
Modeling: The Basics
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Common interactive 3-D primitives: points,
lines, polygons (i.e., triangles)
Organized into objects
– Collection of primitives, other objects
– Associated matrix for transformations
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Instancing: using same geometry for multiple
objects
– 4 wheels on a car, 2 arms on a robot
David Luebke
7/27/2016
Modeling: The Scene Graph
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The scene graph captures transformations
and object-object relationships in a DAG
Objects in black; blue arrows indicate
instancing and each have a matrix
Robot
Head
Mouth
David Luebke
Body
Eye
Leg
Trunk
Arm
7/27/2016
Modeling: The Scene Graph
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Traverse the scene graph in depth-first order,
concatenating transformations
Maintain a matrix stack of transformations
Robot
Visited
Head
Body
Unvisited
Active
Matrix
Stack
Mouth
Eye
Leg
Trunk
Arm
Foot
David Luebke
7/27/2016
Modeling: The Camera
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Finally: need a model of the virtual camera
– Can be very sophisticated
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Field of view, depth of field, distortion, chromatic
aberration…
– Interactive graphics (OpenGL):
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Pinhole camera model
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David Luebke
Field of view
Aspect ratio
Near & far clipping planes
Camera pose: position & orientation
7/27/2016
Modeling: The Camera
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These parameters are encapsulated in a
projection matrix
– Homogeneous coordinates  4x4 matrix!
– See OpenGL Appendix F for the matrix
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The projection matrix premultiplies the
viewing matrix, which premultiplies the
modeling matrices
– Actually, OpenGL lumps viewing and modeling
transforms into modelview matrix
David Luebke
7/27/2016
Rigid-Body Transforms
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Goal: object coordinatesworld coordinates
Idea: use only transformations that preserve
the shape of the object
– Rigid-body or Euclidean transforms
– Includes rotation, translation, and scale
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To reiterate: we will represent points as
column vectors:
 x


( x, y , z )   y 
 z 
David Luebke
7/27/2016
Vectors and Matrices
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Vector algebra operations can be expressed
in this matrix form
– Dot product:
– Cross product:
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Note: use
right-hand
rule!
a  b  ax ay
bx 
az by   a
bz 
 0  az ay  bx  cx 
a  b   az
0  ax  by   cy   c
 ay ax
0  bz  cz 
ac  0
bc  0
David Luebke
7/27/2016
Translations
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For convenience we usually describe objects
in relation to their own coordinate system
– Solar system example
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We can translate or move points to a new
position by adding offsets to their coordinates:
 x'  x  tx 
 y '   y   ty 
     
 z '   z  tz 
– Note that this translates all points uniformly
David Luebke
7/27/2016
3-D Rotations
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Rotations in 2-D are easy:
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3-D is more complicated
 x' cosq  sin q  x 
 y '   sin q cosq   y 
  
 
– Need to specify an axis of rotation
– Common pedagogy: express rotation about this
axis as the composition of canonical rotations
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David Luebke
Canonical rotations: rotation about X-axis, Y-axis, Z-axis
7/27/2016
3-D Rotations
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Basic idea:
– Using rotations about X, Y, Z axes, rotate model
until desired axis of rotation coincides with Z-axis
– Perform rotation in the X-Y plane (i.e., about Z-axis)
– Reverse the initial rotations to get back into the
initial frame of reference
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Objections:
– Difficult & error prone
– Ambiguous: several combinations about the
canonical axis give the same result
– For a different approach, see McMillan’s lecture 12
David Luebke
7/27/2016
The End
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Next: Basic OpenGL
David Luebke
7/27/2016