aula17aconci/CG-Aula17-2017.pdf · 2017. 11. 12. · Microsoft PowerPoint -...
Transcript of aula17aconci/CG-Aula17-2017.pdf · 2017. 11. 12. · Microsoft PowerPoint -...
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Curvas de Bezier
2017/2 – IC / UFF
aula 17
https://pt.wikipedia.org/wiki/Curva_de_Bezier
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Curva de Bezier
pontos de controle = Pi
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Onde se usa: Qualquer representação de
curvas
O circulo ao lado é formado por 8
segmentos. Os quadrados são os pontos de
controle de extremidade dos segmentos e
os anéis os de controle do interior destes
segmentos
Por exemplo:
Os contornos dos caracteres em fontes TrueType são
feitas de segmentos de retas e curvas Bézier quadrática.
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Mesmo segmentos lineares podem ser definido por Bezier:
Um segmento de curva quadrática de Bézier é
definido por 2 pontos extremos e 1 de controle.
pontos de controle = P0 , P1
pontos de controle, i =0,1,2 , Pi
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Forma geral pode ter n+1 pontos de controle,
vamos chamar esses agora de Vi e P(t) os pontos
da curva:
Lembre que o Fatorial de um número = n ! = n (n-1) .... 1
pontos de controle = Vi
i =0,1,2 ,
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Como vimos na aula anterior
• As curvas cúbicas são
especialmente úteis
(porque mesmo ????)
(esqueceu? )
( Vai lá, na aula passada, ver....)
Assim vamos nos concentrar na...pontos de controle = Pi
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Bezier cúbica:
pontos de controle = Vi
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Os interpoladores usados são chamados de
Polinômios cúbicos de
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A soma dos
resulta:
Sempre 1 !!!!!
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Fecho convexo?
• Falamos isso na aula passada ! ! ! !
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pontos de controle = Pi
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Quando se tem muitas curvas para
formar uma única como fica o Fecho
convexo?
Convex hullpontos de controle = Vi
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Cont. pontos de controle = Pi
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Demonstrando essas propriedades para uma
Bezier cúbica:
pontos de controle = Vi
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Se fosse pedido para
reparametrizar de forma especial
• por exemplo com mais pontos onde a derivada da curva fosse maior, ou ela tivesse maior curvatura ?
• Com as expressões do slide anterior isso poderia ser feito!
• (simples não??)
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A ordem e posição dos pontos controla a curva!
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Representação
matricial :
pontos de controle = Vi
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pontos de controle = Gi
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Outras formas de Bezier
pontos de controle = Pi
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Outras formas de Bezier
pontos de controle = Pi
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Outras formas de Bezier
pontos de controle = Pi
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Algoritmo geométrico
pontos de controle = pi
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pontos de controle = pi
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pontos de controle = pi
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pontos de controle = pi
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Outras formas de Bezier
pontos de controle = Pi
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Cont.
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pontos de controle = pi
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pontos de controle = pi
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Vamos falar em coordenadas:
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O usuário
• Definirá os pontos iniciais finais e os intermediários nas coordenadas dele.
• A curva de Bezier pode ser desenhada no seu trabalho, agora!!
Socorro!!Socorro!!
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e iniciar com seus pontos de vértices limitando o convex hull,
mas pode ir incluindo novos pontos de controle a vontade .Até seu objeto 2D ficar como você achar adequado!
Quando isso tiver legal transforme ele em 3D usando a mesma técnica que transformou seu objeto anterior de 2D
para 3D.
De muitas maneiras!!!
Mas vamos ficar com as de graus 3 :
??????Como faço isso
Finalmente, renderize ele usando um shading suave....