Transcript Document

CSCE 590E Spring 2007
Basic Math
By Jijun Tang
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
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
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
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
W
V–W
V
V
–W
Vectors and Matrices

Vectors add and subtract
componentwise
V  W  V1  W1 , V2  W2 ,
, Vn  Wn
V  W  V1  W1 ,V2  W2 ,
,Vn  Wn
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
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
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
1 2 3
an n  m matrix
M

 4 5 6 
At the right, M is a
2  3 matrix
If n = m, the matrix is a square matrix
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 
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
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
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
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
1
R23
R321
R331
0
0
  R 1T  x 

1
  R T y 

  R 1T  z 


1

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
The Dot Product

The dot product can be used to project
one vector onto another
V
a
V cos a 
VW
W
W
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
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
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)
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
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
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
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
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
W  V  V  W
Reversing order of vectors negates the
cross product:
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
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 
ILLustration
Transformation matrix

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
Transformations


A 3  3 matrix can reorient the
coordinate axes in any way, but it
leaves the origin fixed
We must add a translation component
D to move the origin:
 Rx

W   Ry

 Rz
Sx
Sy
Sz
Tx  Vx   Dx 
   
Ty  Vy    Dy 
   
Tz  Vz   Dz 
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 
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
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
Transformations

Transformation matrices are often the
result of combining several simple
transformations




Translations
Scales
Rotations
Transformations are combined by
multiplying their matrices together
Transformation Steps
Orderings
Orderings


Orderings of different type is important
A rotation followed by a translation is
different from a translation followed by
a rotation
Orderings of the same type does not
matter
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
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
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







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














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
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
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
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
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
Geometry

Distance d from a point P to a line
S+tV
P
P S
d
S
P  S  V
V
V
Geometry

Use Pythagorean theorem:
 P  S  V 
d 2   P  S 2  

V



Taking square root,
d   P  S 2 

2
 P  S   V  2
V2
If V is unit length, then V 2 = 1
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
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