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Chapter 4.1
Mathematical Concepts
Applied Trigonometry

Trigonometric functions

Defined using right triangle
y
sin a 
h
h
y
a
x
x
cos a 
h
y sin a
tan a  
x cos a
2
Applied Trigonometry

Angles measured in radians
radians 
degrees 

p
180
180
p
 degrees 
 radians 
Full circle contains 2p radians
3
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
4
Applied Trigonometry

Trigonometric identities
sin  a    sin a
cos a  sin a  p 2 
cos  a   cos a
sin a  cos a  p 2 
tan  a    tan a
sin 2 a  cos 2 a  1
cos a   sin a  p 2 
sin a   cos a  p 2 
sin a   sin a  p    sin a  p 
cos a   cos a  p    cos a  p 
5
Applied Trigonometry

Inverse trigonometric functions

Return angle for which sin, cos, or tan
function produces a particular value
a = z, then a = sin-1 z
-1 z
 If cos a = z, then a = cos
-1 z
 If tan a = z, then a = tan

If sin
6
Applied Trigonometry

Law of sines
a
b
c


sin a sin  sin g

a
c
Law of cosines
c 2  a 2  b2  2ab cos g
b

g
a

Reduces to Pythagorean theorem when
g = 90 degrees
7
Vectors and Matrices

Scalars represent quantities that can be
described fully using one value




Mass
Time
Distance
Vectors describe a magnitude and
direction together using multiple values
8
Vectors and Matrices

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
9
Vectors and Matrices

Vectors can be visualized by an arrow



The length represents the magnitude
The arrowhead indicates the direction
Multiplying a vector by a scalar changes
the arrow’s length
2V
V
–V
10
Vectors and Matrices


Two vectors V and W are added by
placing the beginning of W at the end
of V
Subtraction reverses the second vector
W
V
V+W
V
W
V–W
–W
V
11
Vectors and Matrices



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
12
Vectors and Matrices

Vectors add and subtract
componentwise
V  W  V1  W1 ,V2  W2 ,
,Vn  Wn
V  W  V1  W1 ,V2  W2 ,
,Vn  Wn
13
Vectors and Matrices

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
14
Vectors and Matrices


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
15
Vectors and 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
At the right, M is a
M

2  3 matrix
 4 5 6 
If n = m, the matrix is a square matrix
16
Vectors and Matrices


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
17
Vectors and Matrices

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 
18
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 
19
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
20
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
21
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:
 M 11

 M 21

 M 31
M 12
M 22
M 32
M 13  Vx   M 11Vx  M 12Vy  M 13Vz 
  

M 23  Vy    M 21Vx  M 22Vy  M 23Vz 
  

M 33  Vz   M 31Vx  M 32Vy  M 33Vz 
22
Vectors and Matrices

The n  n identity matrix is denoted In

For any n  n matrix M, the product with
the identity matrix is M itself




I nM = M
MIn = M
The identity matrix is the matrix analog of
the number one
In has entries of 1 along the main diagonal
and 0 everywhere else
23
Vectors and Matrices

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
24
Vectors and Matrices



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
25
Vectors and Matrices



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
26
Vectors and Matrices

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  m23m31 
 m13  m21m32  m22 m31 
27
Vectors and Matrices

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
28
Vectors and Matrices

The inverse of a 2  2 matrix M is
1  M 22  M 12 
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
M13 M 32  M12 M 33
M11M 33  M13 M 31
M 12 M 31  M11M 32
M12 M 23  M13 M 22 

M13 M 21  M11M 23 

M11M 22  M12 M 21 
29
Vectors and Matrices

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
30
Vectors and Matrices

The inverse of this 4  4 matrix is



M 1  




R 1
0
  R111
 
1
R T   R211

  R311
 
1   0
 
R121
R131
R221
R231
R321
R331
0
0
  R 1T  x 

1
 R T y 

  R 1T  z 


1

31
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 given by
n
V  W   VW
i i
i 1

In three dimensions,
V  W  VxWx  VyWy  VzWz
32
The Dot Product

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
33
The 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
34
The Dot Product

The dot product can be used to project
one vector onto another
V
a
V cos a 
VW
W
W
35
The Cross Product

The cross product is a product between
two vectors that 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)
36
The Cross Product

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
ˆi
ˆj
kˆ
V  W  Vx
Vy
Vz
Wx Wy Wz
37
The Cross Product

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
38
The Cross Product

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
39
The Cross Product

The area A of a triangle with vertices
P1, P2, and P3 is thus given by
A
1
 P2  P1    P3  P1 
2
40
The Cross Product

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
Reversing order of vectors negates the
cross product:
W  V  V  W

Cross product is anticommutative
41
Transformations



Calculations are often carried out in
many different coordinate systems
We must be able to transform
information from one coordinate system
to another easily
Matrix multiplication allows us to do this
42
Transformations


Suppose that the coordinate axes in
one coordinate system correspond to
the directions R, S, and T in another
Then we transform a vector V to the
RST system as follows
 Rx

W   R S T  V   Ry

 Rz
Sx
Sy
Sz
Tx  Vx 
 
Ty  Vy 
 
Tz  Vz 
43
Transformations

We transform back to the original
system by inverting the matrix:
 Rx

V   Ry

 Rz

Sx
Sy
Sz
1
Tx 

Ty  W

Tz 
Often, the matrix’s inverse is equal to
its transpose—such a matrix is called
orthogonal
44
Transformations


A 3  3 matrix can reorient the
coordinate axes in any way, but it
leaves the origin fixed
We must at a translation component D
to move the origin:
 Rx

W   Ry

 Rz
Sx
Sy
Sz
Tx  Vx   Dx 
   
Ty  Vy    Dy 
   
Tz  Vz   Dz 
45
Transformations

Homogeneous coordinates


Four-dimensional space
Combines 3  3 matrix and translation into
one 4  4 matrix
 Rx

 Ry
W
 Rz

 0
Sx
Tx
Sy
Ty
Sz
Tz
0
0
Dx  Vx 
 
Dy  Vy 
 
Dz  Vz 
 
1  Vw 
46
Transformations

V is now a four-dimensional vector




The w-coordinate of V determines whether
V is a point or a direction vector
If w = 0, then V is a direction vector and
the fourth column of the transformation
matrix has no effect
If w  0, then V is a point and the fourth
column of the matrix translates the origin
Normally, w = 1 for points
47
Transformations

The three-dimensional counterpart of a
four-dimensional homogeneous vector
V is given by
Vx Vy Vz
V3D 
, ,
Vw Vw Vw

Scaling a homogeneous vector thus has
no effect on its actual 3D value
48
Transformations

Transformation matrices are often the
result of combining several simple
transformations




Translations
Scales
Rotations
Transformations are combined by
multiplying their matrices together
49
Transformations

Translation matrix
M translate

1

0

0

0
0 0 Tx 

1 0 Ty 

0 1 Tz 

0 0 1 
Translates the origin by the vector T
50
Transformations

Scale matrix
M scale


a

0

0

 0
0 0 0

b 0 0

0 c 0

0 0 1 
Scales coordinate axes by a, b, and c
If a = b = c, the scale is uniform
51
Transformations

Rotation matrix
M z -rotate

 cos q

 sin q

 0

 0
 sin q
0
0
cos q
0
0
0
1
0
0
0
1







Rotates points about the z-axis through
the angle q
52
Transformations

Similar matrices for rotations about x, y
M x -rotate
M y -rotate







1
0
0
0
0
cos q
 sin q
0
0
sin q
cos q
0
0
0
0
1
0
sin q
0
1
0
0
0
cos q
0
0
0
1
 cos q

 0

  sin q

 0














53
Transformations

Normal vectors transform differently
than do ordinary points and directions



A normal vector represents the direction
pointing out of a surface
A normal vector is perpendicular to the
tangent plane
If a matrix M transforms points from one
coordinate system to another, then normal
vectors must be transformed by (M-1)T
54
Geometry

A line in 3D space is represented by
P  t   S  tV



S is a point on the line, and V is the
direction along which the line runs
Any point P on the line corresponds to a
value of the parameter t
Two lines are parallel if their direction
vectors are parallel
55
Geometry


A plane in 3D space can be defined by a
normal direction N and a point P
Other points in the plane satisfy
N  Q  P   0
N
Q
P
56
Geometry

A plane equation is commonly written
Ax  By  Cz  D  0

A, B, and C are the components of the
normal direction N, and D is given by
D  N  P
for any point P in the plane
57
Geometry



A plane is often represented by the 4D
vector (A, B, C, D)
If a 4D homogeneous point P lies in the
plane, then (A, B, C, D)  P = 0
If a point does not lie in the plane, then
the dot product tells us which side of
the plane the point lies on
58
Geometry

Distance d from a point P to a line
S+tV
P
P S
d
S
P  S  V
V
V
59
Geometry

Use Pythagorean theorem:
 P  S  V 
d   P  S  

V


2

2
Taking square root,
d   P  S 2 

2
 P  S   V  2
V2
If V is unit length, then V 2 = 1
60
Geometry

Intersection of a line and a plane



Let P(t) = S + t V be the line
Let L = (N, D) be the plane
We want to find t such that L  P(t) = 0
Lx S x  Ly S y  Lz S z  Lw
L S
t

LV
LxVx  LyVy  LzVz

Careful, S has w-coordinate of 1, and V
has w-coordinate of 0
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Geometry


If L  V = 0, the line is parallel to the
plane and no intersection occurs
Otherwise, the point of intersection is
L S
P t   S 
V
LV
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