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Mathematics for Computer Graphics
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
World Coordinates
ModelingCoordinates
Let’s lookat this indetail…
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
2D Modeling Transformations
x
y
ModelingCoordinates
Scale .3, .3Rotate -90
Translate 5, 3
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
zyx
cbaxexe
x
x
,,,63.100.046.5
00.201.06.322
mnmm
n
n
aaa
aaaaaa
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
654
V
4231
A
0000
A
1001
I
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
sasasasasasa
sA
21
22221
11211
654321
A
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 = ?
654321
A
121110987
B
0.100.65.10.0
654221BA
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 babababbb
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababababababa
bbbb
aaaa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababababababababa
bbb
aaaaaaaaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 babababbb
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababababababa
bbbb
aaaa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababababababababa
bbb
aaaaaaaaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 babababbb
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababababababa
bbbb
aaaa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababababababababa
bbb
aaaaaaaaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 babababbb
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababababababa
bbbb
aaaa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababababababababa
bbb
aaaaaaaaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 babababbb
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababababababa
bbbb
aaaa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababababababababa
bbb
aaaaaaaaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 babababbb
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababababababa
bbbb
aaaa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababababababababa
bbb
aaaaaaaaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 babababbb
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababababababa
bbbb
aaaa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababababababababa
bbb
aaaaaaaaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 babababbb
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababababababa
bbbb
aaaa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababababababababa
bbb
aaaaaaaaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 babababbb
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababababababa
bbbb
aaaa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababababababababa
bbb
aaaaaaaaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 babababbb
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababababababa
bbbb
aaaa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababababababababa
bbb
aaaaaaaaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
1
0
N
kkjikij bac
332211
3
2
1
321 babababbb
aaa
2222122121221121
2212121121121111
2221
1211
2221
1211
babababababababa
bbbb
aaaa
333232131
311222121
311212111
3
2
1
333231
232221
131211
bababababababababa
bbb
aaaaaaaaa
1x1 1x3 3x1
2x2 2x2 2x2
3x3 3x1 3x1
Matrix Multiplication
?542
113
?1001
3241
?154
100310201
e.g.:
Matrix Multiplication
15514123542
113
3241
1302031214010411
1001
3241
186
115040135140125041
154
100310201
e.g.:
Matrix Multiplication
15514123542
113
3241
1302031214010411
1001
3241
186
115040135140125041
154
100310201
e.g.:
Matrix Multiplication
15514123542
113
3241
1302031214010411
1001
3241
186
115040135140125041
154
100310201
e.g.:
Matrix Multiplication
15514123542
113
3241
1302031214010411
1001
3241
186
115040135140125041
154
100310201
e.g.:
Matrix Multiplication
15514123542
113
3241
1302031214010411
1001
3241
186
115040135140125041
154
100310201
e.g.:
Matrix Multiplication
15514123542
113
3241
1302031214010411
1001
3241
186
115040135140125041
154
100310201
e.g.:
Matrix Multiplication
15514123542
113
3241
1302031214010411
1001
3241
186
115040135140125041
154
100310201
e.g.:
Matrix Multiplication
15514123542
113
3241
1302031214010411
1001
3241
186
115040135140125041
154
100310201
e.g.:
Matrix Multiplication
15514123542
113
3241
1302031214010411
1001
3241
186
115040135140125041
154
100310201
e.g.:
Matrix Multiplication
15514123542
113
3241
1302031214010411
1001
3241
186
115040135140125041
154
100310201
e.g.:
Matrix Multiplication
15514123542
113
3241
1302031214010411
1001
3241
186
115040135140125041
154
100310201
e.g.:
Warning!!! but (AB)C = A(BC)
A(B+C) = AB + AC
(A+B)C = AC + BC
(AB)T = BTAT
A(sB) = sAB
BAAB
Determinant of a Matrix
Matrix Inverse IAA 1