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CS 551 / 645:
Introductory Computer Graphics
David Luebke
[email protected]
http://www.cs.virginia.edu/~cs551
David Luebke
7/8/2017
Administrivia
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Office hours:
– Luebke:
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2:30 - 3:30 Monday, Wednesday
Olsson 219
– Dale / Jinze / Derek:
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3 - 4 Tuesday
2 - 3 Thursday
2 - 3 Friday
Small Hall Unixlab for now
Might move to Olsson 227 later
Assignment 1 postponed till Wednesday
David Luebke
7/8/2017
UNIX Section: 4 PM, Olsson 228
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Optional UNIX section led by Dale today
– Getting around
– Using make and makefiles
– Using gdb (if time)
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We will use 2 libraries: OpenGL and Xforms
– OpenGL native on SGIs; on other platforms Mesa
– Xforms: available on all platforms of interest
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XForms Intro
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Xforms: a toolkit for easily building Graphical
User Interfaces, or GUIs
– See http://bragg.phys.uwm.edu/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
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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
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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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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
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Trigonometry
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Sine: “opposite over hypotenuse”
Cosine: “adjacent over hypotenuse”
Tangent: “opposite over adjacent”
Unit circle definitions:
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David Luebke
sin () = x
cos () = y
tan () = x/y
Etc…
(x, y)
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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
7/8/2017
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
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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
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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
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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
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Vectors And Point
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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
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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
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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 points and integers in C
– Be careful to avoid nonsensical operations
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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
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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
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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
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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
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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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 
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 Mmn 
By (OpenGL) convention,
vectors are columns:
David Luebke
 v1 
v   v 2 
 v 3 
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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
7/8/2017
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
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David Luebke
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The End
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Next: drawing lines on a raster display
Reading: FvD 3.2
David Luebke
7/8/2017