Application of coordinate system and vectors in the real life
1 Math Review Coordinate systems 2-D, 3-D Vectors Matrices Matrix operations.
-
Upload
willis-cook -
Category
Documents
-
view
230 -
download
3
Transcript of 1 Math Review Coordinate systems 2-D, 3-D Vectors Matrices Matrix operations.
1
Math Review
Coordinate systems 2-D, 3-D
VectorsMatrices Matrix operations
CS-321Dr. Mark L. Hornick
2
Math Review
Cornerstone of graphics Basis for most algorithms Systematic notation Simplifying communication Organizing ideas Compact representation
CS-321Dr. Mark L. Hornick
3
2-D Coordinate Systems
0 x
yP
yP
0 x
Rectangular or Cartesian
CS-321Dr. Mark L. Hornick
4
Points and Vectors
A point at (x,y) can be represented by vector P from the
origin (0,0).
0 x
y
PIn general, a vector represents
the difference (directed distance) between two points.
2 1
2 1 2 1,
,x y
x x y y
V V
V P P
0 x
y
P1
P2
V
CS-321Dr. Mark L. Hornick
5
2-D Vector Representations
Cartesian components
Magnitude and direction angle
,x yV VVVx
Vy
V
|V|
2 2
1tan
x y
y
x
V V
V
V
V
CS-321Dr. Mark L. Hornick
6
2-D Vector Operations
Addition
1 2 1 2 1 2,x x y y
V V V V V V
V1V2
V1+V2
Subtraction
1 2 1 2 1 2,x x y y
V V V V V V V1
-V2
V1-V2 V2
CS-321Dr. Mark L. Hornick
7
2-D Vector Operations
Scalar multiply
,x ya aV aVV
Vx
Vy
V
aVx
aVy
aV
CS-321Dr. Mark L. Hornick
8
2-D Unit Vector
For any vector V, V can also be written as av
2 2
2 2
,
1
x y
x y
x y
av av
a V V
v v
V
VVx
Vy
V
CS-321Dr. Mark L. Hornick
9
Direction cosines
2 2
cos ,cos
cos cos 1
yxVV
V V
α
β
Vx
Vy
V
CS-321Dr. Mark L. Hornick
10
3-D Coordinate Systems
z x
y
P
In a Right-handed coordinate system, the z axisdefined by the vector cross product of the x and y axes.
CS-321Dr. Mark L. Hornick
11
3-D Vector Operations
Addition
Scalar multiply
1 2 1 2 1 2 1 2, ,x x y y z z
V V V V V V V V
, ,x y za aV aV aVV
CS-321Dr. Mark L. Hornick
12
3-D Vector Representations
Cartesian components
Magnitude and direction cosines
, ,x y zV V VV
2 2 2
2 2 2
cos ,cos ,cos
cos cos cos 1
x y z
yx z
V V V
VV V
V
V V Vx
Vz
y
α
β
γ
CS-321Dr. Mark L. Hornick
13
3-D Vector Operations
1 2 1 2 1 2 1 2
1 2 cos
x x y y z zV V V V V V
V V
V V
Inner (dot) product
θV1 V2
1
θcosV
VVV
212
2V 1VPortion of in direction
Projections
CS-321Dr. Mark L. Hornick
14
3-D Vector Cross Product
1 2 1 2
1 2 1 2 1 2
1 2 1 2
1 1 1
2 2 2
,
,y z z y
z x x z
x y y x
x y z
x y z
x y z
V V V V
V V V V
V V V V
V V V
V V V
V V
u u u
“Right-hand rule”
θV1
V2
V1 x V2
CS-321Dr. Mark L. Hornick
15
Matrices
11 12 1
21 22 2
1 2
n
n
m m mn
a a a
a a a
a a a
A
Rectangular matrix(m x n)(rows x
cols)
1 2 3a a aRRow vector
1
2
3
a
a
a
CColumn vector
CS-321Dr. Mark L. Hornick
16
Scalar Matrix Multiplication
u v w
x y z
M
au av awa
ax ay az
M
CS-321Dr. Mark L. Hornick
17
Matrix Addition
u v w
x y z
N
a u b v c w
d x e y f z
M N
a b c
d e f
M
Matrices must have same dimensions
CS-321Dr. Mark L. Hornick
18
Matrix Transpose
u v w
x y z
M
T
u x
v y
w z
M
CS-321Dr. Mark L. Hornick
19
Matrix Multiplication
11 12 1
21 22 2
1 2
n
n
m m mn
a a a
a a a
a a a
A
11 12 1
21 22 2
1 2
q
q
p p pq
b b b
b b b
b b b
B
ijc C AB1
n
ij ik kjk
c a b
Matrices must be conformable (n=p)
CS-321Dr. Mark L. Hornick
20
Matrix Multiplication Example
1 2 3
4 5 6
2 1
3 4
5 7
=
A B C
C row = A row, C column = B column
23
1 2 2 3 3 5 2 6 15 23
66
53
30
CS-321Dr. Mark L. Hornick
21
Identity Matrix and Inverse
1 1 AA A A IInverse computed by Gaussian elimination,
determinants, or other methods; used directly or indirectly to solve sets of linear equations
1 0 0
0 1 0
0 0 1
I
1
Ax b
x A b
CS-321Dr. Mark L. Hornick
22
Determinants
1
A-1 = adj(A)
det(A)
CS-321Dr. Mark L. Hornick
23
Determinants
A
A
Gaussian elimination is the best method
Swapping two rows changes the sign of Multiplying a row by s, multiplies by s Adding row multiples has no effect
n
1kkkaA
Only on square matrices For an upper triangular matrix