Transcript Document

CSCE 552 Spring 2011
Math
By Jijun Tang
Layered
Initialization/Shutdown

Resource Acquisition Is Initialization



A useful rule to minimalize mismatch errors in the
initialization and shutdown steps
Means that creating an object acquires and
initializes all the necessary resources, and
destroying it destroys and shuts down all those
resources
Optimizations


Fast shutdown
Warm reboot
Main Game Loop

Structure




Hard-coded loops
Multiple game loops: for each major game state
Consider steps as tasks to be iterated through
Coupling



Can decouple the rendering step from simulation
and update steps
Results in higher frame rate, smoother animation,
and greater responsiveness
Implementation is tricky and can be error-prone
Math
Applied Trigonometry

Trigonometric functions

Defined using right triangle
y
sin a 
h
h
y
x
cos a 
h
tan a 
a
x
y sin a

x cos a
Applied Trigonometry

Angles measured in radians
radians 
degrees 

p
180
180
p
 degrees 
 radians 
Full circle contains 2p radians
Applied Trigonometry

Sine and cosine used to decompose a
point into horizontal and vertical
components
y
r
r sin a
a
r cos a
x
Trigonometry
Trigonometric identities
sin  a    sin a
cos a  sin a  p 2 
cos  a   cos a
tan  a    tan a
sin a  cos a  p 2 
cos a   sin a  p 2 
sin 2 a  cos 2 a  1
sin a   cos a  p 2 
sin a   sin a  p    sin a  p 
cos a   cos a  p    cos a  p 
Inverse trigonometric functions

Return angle for which sin, cos, or tan
function produces a particular value

If sin a = z, then a = sin-1 z

If cos a = z, then a = cos-1 z

If tan a = z, then a = tan-1 z
arcs
Applied Trigonometry

Law of sines
a
b
c


sin a sin  sin g
a
c

Law of cosines
c2  a 2  b2  2ab cos g
b

g
a
Reduces to Pythagorean theorem when
g = 90 degrees
Vectors and Scalars

Scalars represent quantities that can
be described fully using one value




Mass
Time
Distance
Vectors describe a magnitude and
direction together using multiple values
Examples of vectors

Difference between two points



Velocity of a projectile



Magnitude is the distance between the points
Direction points from one point to the other
Magnitude is the speed of the projectile
Direction is the direction in which it’s traveling
A force is applied along a direction
Visualize Vectors



The length represents the magnitude
The arrowhead indicates the direction
Multiplying a vector by a scalar
changes the arrow’s length
2V
V
–V
Vectors Add and Subtraction


Two vectors V and W are added by
placing the beginning of W at the end
of V
Subtraction reverses the second
vector
V
W
V+W
W
V–W
V
V
–W
High-Dimension Vectors



An n-dimensional vector V is
represented by n components
In three dimensions, the components
are named x, y, and z
Individual components are expressed
using the name as a subscript:
V  1, 2,3
Vx  1
Vy  2
Vz  3
Add/Subtract Componentwise
V  W  V1  W1 , V2  W2 ,
, Vn  Wn
V  W  V1  W1 ,V2  W2 ,
,Vn  Wn
Vector Magnitude

The magnitude of an n-dimensional
vector V is given by
V 
n
2
V
i
i 1

In three dimensions, this is
V  Vx2  Vy2  Vz2
Vector Normalization


A vector having a magnitude of 1 is
called a unit vector
Any vector V can be resized to unit
length by dividing it by its magnitude:
V
ˆ
V
V

This process is called normalization
Matrices


A matrix is a rectangular array of
numbers arranged as rows and
columns
A matrix having n rows and m columns
is an n  m matrix


1 2 3
M

 4 5 6 
At the right, M is a
2  3 matrix
If n = m, the matrix is a square matrix
Matrix Representation


The entry of a matrix M in the i-th row
and j-th column is denoted Mij
For example,
1 2 3
M

 4 5 6 
M 11  1
M 21  4
M 12  2 M 22  5
M 13  3 M 23  6
Matrix Transpose
The transpose of a matrix M is denoted
MT and has its rows and columns
exchanged:
1 2 3
M

 4 5 6 
1 4


T
M  2 5


 3 6 
Vectors and Matrices

An n-dimensional vector V can be
thought of as an n  1 column matrix:
V  V1 , V2 ,

V1 
 
V2 
, Vn   
 
 
Vn 
Or a 1  n row matrix:
V T  V1 V2
Vn 
Vectors and Matrices

Product of two matrices A and B


Number of columns of A must equal
number of rows of B
Entries of the product are given by
m
 AB ij   Aik Bkj
k 1

If A is a n  m matrix, and B is an m  p
matrix, then AB is an n  p matrix
Vectors and Matrices

Example matrix product
 2 3   2 1   8 13
M



1 1  4 5  6 6 
M 11  2   2   3  4

M 12  2 1  3   5 
 13
8
M 21  1   2    1  4  6
M 22  1 1   1   5  
6
Vectors and Matrices


Matrices are used to transform vectors
from one coordinate system to another
In three dimensions, the product of a
matrix and a column vector looks like:
 M11

 M 21

 M 31
M12
M 22
M 32
M13  Vx   M11Vx  M 12Vy  M 13Vz 
  

M 23  Vy    M 21Vx  M 22Vy  M 23Vz 
  

M 33  Vz   M 31Vx  M 32Vy  M 33Vz 
Identity Matrix In
For any n  n matrix M,
the product with the
identity matrix is M itself


I nM = M
MIn = M
Invertible

An n  n matrix M is invertible if there
exists another matrix G such that
1 0

0 1
MG  GM  I n  


0 0

0

0



1 
The inverse of M is denoted M-1
Properties of Inverse



Not every matrix has an inverse
A noninvertible matrix is called singular
Whether a matrix is invertible can be
determined by calculating a scalar
quantity called the determinant
Determinant



The determinant of a square matrix M
is denoted det M or |M|
A matrix is invertible if its determinant
is not zero
For a 2  2 matrix,
a b  a b
det 
 ad  bc

 c d  c d
Determinant

The determinant of a 3  3 matrix is
m11
m12
m13
m21
m22
m23  m11
m31
m32
m33
m22
m23
m32
m33
 m12
m21
m23
m31
m33
 m13
m21
m22
m31
m32
 m11  m22 m33  m23 m32   m12  m21m33  m23 m31 
 m13  m21m32  m22 m31 
Inverse
Explicit formulas exist for matrix inverses


These are good for small matrices, but
other methods are generally used for
larger matrices
In computer graphics, we are usually
dealing with 2  2, 3  3, and a special
form of 4  4 matrices
Inverse of 2x2 and 3x3

The inverse of a 2  2 matrix M is
1  M 22  M12 
M 


det M   M 21 M11 
1

The inverse of a 3  3 matrix M is
 M 22 M 33  M 23 M 32
1 
1
M 
 M 23 M 31  M 21M 33
det M 
 M 21M 32  M 22 M 31
M 13 M 32  M 12 M 33
M 11M 33  M 13 M 31
M 12 M 31  M 11M 32
M 12 M 23  M 13 M 22 

M 13 M 21  M 11M 23 

M 11M 22  M 12 M 21 
4x4

A special type of 4  4 matrix used in
computer graphics looks like
 R11

 R21
M
 R31

 0


R12
R13
R22
R23
R32
R33
0
0
Tx 

Ty 

Tz 

1

R is a 3  3 rotation matrix, and T is a
translation vector
Inverse of 4x4



M 1  




R 1
0
  R111
 
1
R T   R211

  R311
 
1   0
 
R121
R131
R221
1
R23
R321
R331
0
0
  R 1T  x 

1
  R T y 

  R 1T  z 


1

General Matrix Inverse


(AB) -1 = B-1A-1
A general nxn matrix can be inverted
using methods such as



Gauss-Jordan elimination
Gaussian elimination
LU decomposition
Gauss-Jordan Elimination
Gaussian Elimination
The Dot Product



The dot product is a product between
two vectors that produces a scalar
The dot product between two
n-dimensional vectors
V and W is
n
given by V  W   VW
i i
i 1
In three dimensions,
V  W  VxWx  VyWy  VzWz
Dot Product Usage

The dot product can be used to project
one vector onto another
V
a
V cos a 
VW
W
W
Dot Product Properties

The dot product satisfies the formula
V  W  V W cos a



a is the angle between the two vectors
Dot product is always 0 between
perpendicular vectors
If V and W are unit vectors, the dot
product is 1 for parallel vectors pointing in
the same direction, -1 for opposite
Self Dot Product

The dot product of a vector with itself
produces the squared magnitude
VV  V V  V

2
Often, the notation V 2 is used as
shorthand for V  V
The Cross Product


The cross product is a product
between two vectors the produces a
vector
The cross product only applies in three
dimensions


The cross product is perpendicular to
both vectors being multiplied together
The cross product between two parallel
vectors is the zero vector (0, 0, 0)
Illustration

The cross product satisfies the
trigonometric relationship
V  W  V W sin a

This is the area of
the parallelogram
formed by
V
V and W
||V|| sin a
a
W
Area and Cross Product

The area A of a triangle with vertices
P1, P2, and P3 is thus given by
1
A   P2  P1    P3  P1 
2
Rules

Cross products obey the right hand
rule



If first vector points along right thumb,
and second vector points along right
fingers,
Then cross product points out of right
palm
W  V  V  W
Reversing order of vectors negates the
cross product:
Rules in Figure
Formular

The cross product between V and W is
V  W  VyWz  VzWy , VzWx  VxWz , VxWy  VyWx

A helpful tool for remembering this
formula is the pseudodeterminant (x, y,
z) are unit vectors
Alternative

The cross product can also be
expressed as the matrix-vector product
 0

V  W   Vz

 Vy

Vz
0
Vx
Vy  Wx 
 
Vx  Wy 
 
0  Wz 
The perpendicularity property means
 V  W  V  0
 V  W  W  0