Post on 30-Dec-2015
transcript
Lecture Summary
Matrices Some fundamental operations
Vectors Some fundamental operations
Geometric Primitives: Points, Lines, Curves, Polygons
2D Modeling Transformations
ScaleRotate
Translate
ScaleTranslate
x
y
World Coordinates
ModelingCoordinates
2D Modeling Transformations
x
y
ModelingCoordinates
Initial locationat (0, 0) withx- and y-axesaligned
2D Modeling Transformations
x
y
ModelingCoordinates
Scale .3, .3Rotate -90
Translate 5, 3
World Coordinates
Matrices A matrix is a rectangular array of elements (numbers,
expression, or function) A matrix with m rows and n columns is said to be an m-by-n
matirx ( matrix), e.g
In general, we can write an m-by-n matrix as
z
y
x
cbaxe
xex
x
,,,63.100.046.5
00.201.06.322
mnmm
n
n
aaa
aaa
aaa
A
21
22221
11211
nm
Matrices A matrix with a single row or a single column represent a vector Single row : 1-by-n matrix is a row vector
Single column : n-by-1 matrix is a column vector
A square matrix is a matrix has the same number of rows as columns
In graphics, we frequently work with two-by-two, three-by-three, and four-by-four matrices
The zero matrix The identity matrix A diagonal matrix
321V
6
5
4
V
42
31A
00
00A
10
01I
Scalar Multiplication To multiply a martix A by a scalar value s, we multiply each
element amn by the scalar
Ex. , find 3A = ?
mnmm
n
n
sasasa
sasasa
sasasa
sA
21
22221
11211
654
321A
Matrix Addition Two matrices A and B may be added together when these two
matrices have the same number of rows and column the same shape
The sum is obtained by adding corresponding elements.
Ex. , find A+B = ?
Ex. , find A+B = ?
654
321A
121110
987B
0.100.6
5.10.0
654
221BA
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 bababa
b
b
b
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababa
babababa
bb
bb
aa
aa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababa
bababa
bababa
b
b
b
aaa
aaa
aaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 bababa
b
b
b
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababa
babababa
bb
bb
aa
aa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababa
bababa
bababa
b
b
b
aaa
aaa
aaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 bababa
b
b
b
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababa
babababa
bb
bb
aa
aa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababa
bababa
bababa
b
b
b
aaa
aaa
aaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 bababa
b
b
b
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababa
babababa
bb
bb
aa
aa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababa
bababa
bababa
b
b
b
aaa
aaa
aaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 bababa
b
b
b
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababa
babababa
bb
bb
aa
aa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababa
bababa
bababa
b
b
b
aaa
aaa
aaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 bababa
b
b
b
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababa
babababa
bb
bb
aa
aa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababa
bababa
bababa
b
b
b
aaa
aaa
aaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 bababa
b
b
b
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababa
babababa
bb
bb
aa
aa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababa
bababa
bababa
b
b
b
aaa
aaa
aaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 bababa
b
b
b
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababa
babababa
bb
bb
aa
aa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababa
bababa
bababa
b
b
b
aaa
aaa
aaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 bababa
b
b
b
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababa
babababa
bb
bb
aa
aa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababa
bababa
bababa
b
b
b
aaa
aaa
aaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 bababa
b
b
b
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababa
babababa
bb
bb
aa
aa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababa
bababa
bababa
b
b
b
aaa
aaa
aaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 bababa
b
b
b
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababa
babababa
bb
bb
aa
aa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababa
bababa
bababa
b
b
b
aaa
aaa
aaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
15514123
5
4
2
113
32
41
13020312
14010411
10
01
32
41
1
8
6
115040
135140
125041
1
5
4
100
310
201
e.g.:
Matrix Multiplication
15514123
5
4
2
113
32
41
13020312
14010411
10
01
32
41
1
8
6
115040
135140
125041
1
5
4
100
310
201
e.g.:
Matrix Multiplication
15514123
5
4
2
113
32
41
13020312
14010411
10
01
32
41
1
8
6
115040
135140
125041
1
5
4
100
310
201
e.g.:
Matrix Multiplication
15514123
5
4
2
113
32
41
13020312
14010411
10
01
32
41
1
8
6
115040
135140
125041
1
5
4
100
310
201
e.g.:
Matrix Multiplication
15514123
5
4
2
113
32
41
13020312
14010411
10
01
32
41
1
8
6
115040
135140
125041
1
5
4
100
310
201
e.g.:
Matrix Multiplication
15514123
5
4
2
113
32
41
13020312
14010411
10
01
32
41
1
8
6
115040
135140
125041
1
5
4
100
310
201
e.g.:
Matrix Multiplication
15514123
5
4
2
113
32
41
13020312
14010411
10
01
32
41
1
8
6
115040
135140
125041
1
5
4
100
310
201
e.g.:
Matrix Multiplication
15514123
5
4
2
113
32
41
13020312
14010411
10
01
32
41
1
8
6
115040
135140
125041
1
5
4
100
310
201
e.g.:
Matrix Multiplication
15514123
5
4
2
113
32
41
13020312
14010411
10
01
32
41
1
8
6
115040
135140
125041
1
5
4
100
310
201
e.g.:
Matrix Multiplication
15514123
5
4
2
113
32
41
13020312
14010411
10
01
32
41
1
8
6
115040
135140
125041
1
5
4
100
310
201
e.g.:
Matrix Multiplication
15514123
5
4
2
113
32
41
13020312
14010411
10
01
32
41
1
8
6
115040
135140
125041
1
5
4
100
310
201
e.g.: