Basic Math

Basic Math - Transcript

CSCE 590E Spring 2007
Basic Math
By Jijun Tang

Applied Trigonometry
? Trigonometric functions
? Defined using right triangle
?
x
yh

Applied Trigonometry
? Angles measured in radians
? Full circle contains 2pi radians

Trigonometry

Trigonometric identities

Inverse trigonometric functions
? Return angle for which sin, cos, or tan
function produces a particular value
? If sin ? = z, then ? = sin-1 z
? If cos ? = z, then ? = cos-1 z
? If tan ? = z, then ? = 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
V
W
V + W
V
W
V
V – W –W

Vectors and Matrices
? Vectors add and subtract
componentwise

Vectors and Matrices
? The magnitude of an n-dimensional
vector V is given by
? In three dimensions, this is

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:
? 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
an n × m matrix
? 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:

Vectors and Matrices
? An n-dimensional vector V can be
thought of as an n × 1 column matrix:
? Or a 1 × n row matrix:

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
? 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

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:

Identity Matrix In
For any n × n matrix M,
the product with the
identity matrix is M itself
? InM = M
? MIn = M

Invertible
? An n × n matrix M is invertible if there
exists another matrix G such that
? The inverse of M is denoted M-1
1 0 0
0 1 0
0 0 1
n
? ?? ?? ?
= = = ? ?? ?? ?? ?? ?
MG GM I
L
L
M M O M
L

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,
det a b a b ad bcc d c d
? ?
= = ?? ?? ?? ?

Determinant
? The determinant of a 3 × 3 matrix is

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
? R is a 3 × 3 rotation matrix, and T is a
translation vector
11 12 13
21 22 23
31 32 33
0 0 0 1
x
y
z
R R R T
R R R T
R R R T
? ?? ?? ?? ?= ? ?? ?? ?? ?
M

Vectors and Matrices
? The inverse of this 4 × 4 matrix is
( )
( )
( )
1 1 1 1
11 12 13
1 1 1 1 1 1
21 22 231
1 1 1 1
31 32 33
1 0 0 0 1
x
y
z
R R R
R R R
R R R
? ? ? ?
? ? ? ? ? ?
?
? ? ? ?
?? ? ? ?? ? ? ?? ? ? ?? ?? ? ? ?
= =? ? ? ?
?? ? ? ?? ? ? ?? ? ? ?
R T
R R T R T
M R T
0

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
? In three dimensions,

The Dot Product
? The dot product can be used to project
one vector onto another
?
V
W

The Dot Product
? The dot product satisfies the formula
? ? 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
? 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
? A helpful tool for remembering this
formula is the pseudodeterminant

The Cross Product
? The cross product can also be
expressed as the matrix-vector product
? The perpendicularity property means

The Cross Product
? The cross product satisfies the
trigonometric relationship
? This is the area of
the parallelogram
formed by
V and W
?
V
W
||V|| sin ?

The Cross Product
? The area A of a triangle with vertices
P1, P2, and P3 is thus given by

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:
? Cross product is anticommutative

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

ILLustration

Transformation matrix
? We transform back to the original
system by inverting the matrix:
? 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:

Transformations
? Homogeneous coordinates
? Four-dimensional space
? Combines 3 × 3 matrix and translation
into one 4 × 4 matrix

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
? 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
? Translates the origin by the vector T
translate
1 0 0
0 1 0
0 0 1
0 0 0 1
x
y
z
T
T
T
? ?? ?? ?
= ? ?? ?? ?? ?? ?
M

Transformations
? Scale matrix
? Scales coordinate axes by a, b, and c
? If a = b = c, the scale is uniform
scale
0 0 0
0 0 0
0 0 0
0 0 0 1
a
b
c
? ?? ?? ?
= ? ?? ?? ?? ?? ?
M

Transformations
? Rotation matrix
? Rotates points about the z-axis
through the angle ?
-rotate
cos sin 0 0
sin cos 0 0
0 0 1 0
0 0 0 1
z
? ?
? ?
?? ?? ?? ?
= ? ?? ?? ?? ?? ?
M

Transformations
? Similar matrices for rotations about x, y
-rotate
1 0 0 0
0 cos sin 0
0 sin cos 0
0 0 0 1
x
? ?
? ?
? ?? ?
?? ?
= ? ?? ?? ?? ?? ?
M
-rotate
cos 0 sin 0
0 1 0 0
sin 0 cos 0
0 0 0 1
y
? ?
? ?
? ?? ?? ?
= ? ?? ??? ?? ?? ?
M

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
? 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
( )t t= +P S V

Geometry
? A plane in 3D space can be defined by
a normal direction N and a point P
? Other points in the plane satisfy
P
Q
N

Geometry
? A plane equation is commonly written
? A, B, and C are the components of the
normal direction N, and D is given by
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 + t V
P
VS
d

Geometry
? Use Pythagorean theorem:
? Taking square root,
? 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
? Careful, S has w-coordinate of 1, and V
has w-coordinate of 0
x x y y z z w
x x y y z z
L S L S L S Lt L V L V L V
+ + +
?
= ? = ?
? + +
L S
L V

Geometry
? If L ? V = 0, the line is parallel to the
plane and no intersection occurs
? Otherwise, the point of intersection is
( )t ?= ?
?
L SP S VL V