Download R n

Week 3 - Friday
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What did we talk about last time?
Vertex shaders
Geometry shaders
Pixel shaders
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The Utah teapot was modeled in 1975 by
graphics pioneer Martin Newell at the
University of Utah
It's actually taller than it looks
 They distorted the model so that it would look
right on their non-square pixel displays
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Original
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Modern
Yeah… just what do you know about vectors?
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We refer to n-dimensional real Euclidean
space as Rn
A vector v in this space is an n-tuple, an
ordered list of n real numbers
To match the book (and because we are
computer scientists), we'll index these from 0
to n – 1
We will generally write our vectors as column
vectors rather than row vectors
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We will be interested in a number of
operations on vectors, including:
 Addition
 Scalar multiplication
 Dot product
 Norm
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Addition of two vectors is just element-byelement addition
 u0   v0   v0  u0 

 
 

 u1   v1   u1  v1 
n
uv 



R









 
 

u  v  u  v 
 n 1   n 1   n 1 n 1 
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Vector addition is associative:
(u  v)  w  u  (v  w)
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Vector addition is commutative:
u v  v u
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There is a unique vector for Rn which is 0 =
(0,0,…,0) with a total of n zeroes
o is additive identity:
0 v  v
For vector v, there is a unique inverse –v =
(-v0, -v1, … -vn-1)
-v is the additive inverse:
v  (  v)  0
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Multiplication by a scalar is just element-byelement multiplication of that scalar
 au0 


 au1  n
au  

R
 


 au 
 n1 
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Rules for scalar multiplication can easily be
inferred from the normal properties of reals
under addition and multiplication:
(ab)u  a(bu)
(a  b)u  au  bu
a(u  v)  au  av
1u  u
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The dot product is a form of multiplication
between two vectors that produces a scalar
n 1
u  v   u i v i R
i 0
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Dot product rules are slightly less obvious
than scalar product
 𝐮 ∙ 𝐮 ≥ 0 and is 0 only when 𝐮 = 𝟎
 (𝐮 + 𝐯) ∙ 𝐰 = 𝐮 ∙ 𝐰 + 𝐯 ∙ 𝐰
 (𝑎𝐮) ∙ 𝐯 = 𝑎 𝐮 ∙ 𝐯
 𝐮∙𝐯=0↔𝐮⊥𝐯
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A norm is a way of measuring the magnitude of
a vector
We are actually only interested in the L2 norm,
but there are many (infinite) norms for any given
vector space
We'll denote the norm of u as ||u||
𝑛−1
𝑢𝑖2
𝐮 = 𝐮∙𝐮=
𝑖=0
𝐮 =0↔𝐮=𝟎
 𝑎𝐮 = 𝑎 𝐮
 𝐮+𝐯 ≤ 𝐮 + 𝐯
 𝐮∙𝐯 ≤ 𝐮 𝐯
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Mathematical rules are one thing
Understanding how they are interpreted in
geometry is something else
Unfortunately, this means getting more math
to link up the existing math with geometry
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A set of vectors u0, u1, … un-1 is linearly
independent if the only scalars that satisfy
the following identity are v0 = v1 = … = vn-1 = 0
v0uo  v1u1  ...  vn1un1  0
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In other words, you can't make any one
vector out of any of the others
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A set of vectors u0, u1, … un-1 spans Rn if any
vector v Rn can be written:
n 1
v   v i ui
i 0
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In addition, if v0, v1, … , vn-1 are uniquely
determined for all v Rn, then u0, u1, … un-1
form a basis of Rn
To properly describe Rn, the
vectors we give are actually
scalar multiplied by each of
the basis vectors ui
 By convention, we leave off
the basis vectors ui,
because it would be
cumbersome to show them
 Also, they are often boring:
(1,0,0), (0,1,0), and (0,0,1)
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A vector can either be a point in space or an
arrow (direction and distance)
The norm of a vector is its distance from the
origin (or the length of the arrow)
In R2 and R3, the dot product is:
u  v  u v cosφ
where  is the smallest angle between u and
v
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A basis is orthogonal if every vector in the
basis is orthogonal to every other (has dot
product 0)
An orthogonal basis is orthonormal if every
vector in it has length 1
The standard basis is orthonormal and made
up of vectors ei which are all 0's except a 1 at
location i
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We can find the orthogonal projection w of
vector u onto vector v
Essentially, this means the part of u that's in v
 u v   u v 
w   2 v  
v  tv

 v   vv 


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The cross product of two vectors finds a
vector that is orthogonal to both
For 3D vectors u and v in an orthonormal
basis, the cross product w is:
 u y v z  u zv y 
wx 


 
w   w y   u  v   uzv x  u x v z 
u v u v 
w 
y x
 z
 x y
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𝐰 = 𝐮 × 𝐯 = 𝐮 𝐯 sin θ
𝐮 × 𝐯 = −𝐯 × 𝐮
𝑎𝐮 + 𝑏𝐯 × 𝐰 = 𝑎 𝐮 × 𝐰 + 𝑏(𝐯 × 𝐰)
In addition
 wu and wv
 u, v, and w form a right-handed system
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More linear algebra
 Matrices
 Homogeneous notation
 Geometric techniques
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Keep reading Appendix A
Assignment 1 due tonight by midnight!
Keep working on Project 1, due next Friday,
February 10 by 11:59