palestra matrizes
TRANSCRIPT
![Page 2: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/2.jpg)
Oquevocêencontraránesteguia
ParaquêsevemmatrizesecomouGlizar;
Explicaçãodasfórmulasmaiscomuns;
ExemplospráGcosdejogos;
SituaçõesmaisvoltadasparaRsicaparajogos.
2
(10,-8)
(8,10)
![Page 3: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/3.jpg)
Oquevocênãoencontraráaqui
PráGcasmatemáGcasavançadas;
Nãovamosnosaprofundarmuitonaparte3D;
Computaçãográficaavançada;
Focoemsistemasdeequaçõeslineares.
3
![Page 4: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/4.jpg)
Importante
Assistaaoguiadesobrevivênciadevetoresdaglobalgamejamdoanopassado:
h[ps://www.youtube.com/watch?v=0ScX3TOibKA
h[ps://www.youtube.com/watch?v=V3jwGZhAzMY
4
![Page 5: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/5.jpg)
5
![Page 6: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/6.jpg)
6
![Page 7: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/7.jpg)
7
![Page 8: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/8.jpg)
8 X
Y
Y’
X’
(X’,Y’)
![Page 9: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/9.jpg)
9 X
Y
Y’
X’
(X’,Y’)
![Page 10: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/10.jpg)
10
⇢3x+ 2y = 7�6x+ 6y = 6
![Page 11: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/11.jpg)
11
Álgebra linear
Sistemas de equações lineares
Teoria dos Determinantes
Matrizes
(<< 0)
(XVII - XVIII)
(XIX)
⇢3x+ 2y = 7�6x+ 6y = 6
� =
������
a11 a12 a13a21 a22 a23a31 a32 a33
������
![Page 12: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/12.jpg)
12
⇢3x+ 2y = 7�6x+ 6y = 6
![Page 13: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/13.jpg)
13
O objetivo deste guia é fazê-lo entender como
utilizar a equação matricial abaixo:
![Page 14: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/14.jpg)
14
Seus personagens
Photoshop (2D) Maya (3D)
![Page 15: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/15.jpg)
15
Entendendo as proporções
Photoshop (2D)(200, 0)
(0, 300)v = (200, 0)
u = (0, 300)
![Page 16: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/16.jpg)
16
Entendendo as proporções
(1, 0)
(0, 1)Comproporção1:1,opersonagempossuialturaelargurade200x300.
![Page 17: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/17.jpg)
17
Entendendo as proporções
Comproporção2:2,opersonagempossuialturaelargurade400x600.
(2, 0)
(0, 2)
![Page 18: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/18.jpg)
18
(2, 0)
(0, 2)
Sistema de Coordenadas = Matriz
(1, 0)
(0, 1)
Matriz identidade
![Page 19: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/19.jpg)
19
Transformações lineares
Sãooperaçõesrealizadassobreumsistemadecoordenadas;
PodemoschegaraqualquersistemadecoordenadasexistenteuGlizando3transformaçõesbásicas:Translação,RotaçãoeEscala;
Amatriziden5daderepresentaumsistemadecoordenadassemtransformaçãoalguma:
É,basicamente,opersonagemcomoelefoiexportadodophotoshopoumaya.
![Page 20: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/20.jpg)
20
Multiplicação de matrizes
c11 = a11 • b11 + a12 • b21
![Page 21: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/21.jpg)
21
Escala
![Page 22: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/22.jpg)
22
Como aplicar transformações
(200, 300) ÷ 2 = (100, 150) = (ox, oy)
(0,0)
![Page 23: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/23.jpg)
23
Como aplicar transformações
(ox, oy)
(ox, -oy)(-ox, -oy)
(-ox, oy)
![Page 24: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/24.jpg)
24
Como aplicar transformações
(ox, oy)
(ox, -oy)(-ox, -oy)
(-ox, oy) (-2ox, oy) (2ox, oy)
(-2ox, -oy) (2ox, -oy)
![Page 25: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/25.jpg)
25
Como aplicar transformações
(-2ox, oy) (2ox, oy)
(-2ox, -oy) (2ox, -oy)
ex:
![Page 26: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/26.jpg)
26
Como aplicar transformações
?
![Page 27: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/27.jpg)
27
Como aplicar transformações
A regra é a mesma: Para todo vértice do
personagem, multiplique-o pela
matriz de transformação
![Page 28: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/28.jpg)
28
Rotação
30˚
![Page 29: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/29.jpg)
29
Rotação
(x, y)
(-y, x)(cos, sin)
(-sin, cos)
![Page 30: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/30.jpg)
30
Rotação no 3D Euler Angles
![Page 31: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/31.jpg)
31
Euler Angles
Multiplicando-se o vetor por essas
matrizes, é possível chegar a qualquer
orientação no espaço tridimensional
![Page 32: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/32.jpg)
32
Euler Angles
Proper Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y)
Tait–Bryan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z).
![Page 33: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/33.jpg)
Regradamãodireitaeesquerda
33
![Page 34: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/34.jpg)
Regradamãodireitaeesquerda
34
![Page 35: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/35.jpg)
35
Translação
![Page 36: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/36.jpg)
Sistemadecoordenadashomogêneoeatranslação
36
2 00 2
�2
41 0 00 1 00 0 1
3
5
50100
�
2
42 0 500 2 1000 0 1
3
5
![Page 37: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/37.jpg)
Sistemadecoordenadashomogêneo
37
2 00 2
�2
41 0 00 1 00 0 1
3
5
50100
�
2
42 0 500 2 1000 0 1
3
5
![Page 38: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/38.jpg)
38
Álgebra linear
Estudosdosespaçosvetoriaisedasrelaçõesentreessesespaçosvetoriais:
Matriz–Matriz;
Matriz–Vetor;
Vetor–Vetor.
![Page 39: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/39.jpg)
39
Agrupamento de transformações
MulGplicandoasmatrizesdetransformação,épossívelagrupá-lasemumsistemadecoordenadas,facilitandoaaplicaçãodatransformação;
Aoinvésde:
T•X+R•X+E•X
Temos:
T•R•E•X
![Page 40: Palestra Matrizes](https://reader035.vdocuments.mx/reader035/viewer/2022081420/587361c91a28ab866e8b6d19/html5/thumbnails/40.jpg)
40
Comutatividade das transformações