Lecture 12 - Spline surfaces (slides)

# Lecture 12 spline surfaces slides

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Unformatted text preview: Interactive Computer Graphics:   Lecture 12 " Introduction to Surface Construction ! Teapot Subdivision: Russ Fish Teapot Subdivision: Russ Fish " Graphics Lecture 12: Slide 2! 2 / 35 Non Parametric Surface" •  Surfaces can be constructed from Cartesian equations directly, and this is acceptable for speciﬁc applications, usually involving interpolation. ! •  As before, using a simple polynomial surface is a quick and easy approach. ! Graphics Lecture 12: Slide 3! Non Parametric Polynomial Surface Non on Parametric Polynomial Surface " N Parametric Polynomial Surface 00 11 011 0 a a b b c c d d xx BB e f g CC BCC b b e f C By C g C By C 0 B x x y y z z 1 1BB @@ f h j AA @AA = 0 @z = cc f h j z ddgg j j 11 11 whichwhich multiplies out to: ! •  multiplies out to: which multiplies out to: 2 ey 2 2 hz 2 2 2 ax2 ++ ey++ hz++bxy ++cxz ++f yz ++dx ++gy ++jz ++ 1 = 0 ax 2bxy 2 2cxz 2 2f yz 2 2dx 2 2gy 2 2jz 1 = 0 Because ofof theof the symmetryare 9 scalar 9 scalar inin the •  Because symmetry there 9 scalar unknowns the Because the symmetry there arethere are unknownsunknowns in equation equation ! the equation •  So we need to specify nine points through which the So we need toto specify nine points through which the surface will So we need will pass ! points through which the surface will surface specify nine pass pass Graphics Lecture 12: Slide 4! As Before " •  This formulation suffers the same problems as the nonparametric spline curve. It is a ﬁxed surface for a given set of nine points. ! •  We need more ﬂexibility for the design of surfaces. ! Graphics Lecture 12: Slide 5! Simple Parametric surfaces Simple Parametric surfaces Simple Parametric surfaces " We can extend the formulation to simple parametric surfaces using •  can extend the equation: WeWe can extendformulation to simplesimple parametric using the vector the formulation to parametric surfaces urfaces using ! thesvector equation:the vector equation: 0 10 1 abc µ 0 10 1 a 1) c µ P(µ, ⌫ ) = (µ, ⌫, b @b d eA @⌫ A P(µ, ⌫ ) = (µ, ⌫, 1) @b d eA @⌫ A cef 1 cef 1 2 2 P(µ, ⌫ ) = aµ + d⌫ + 2bµ⌫ + 2cµ + 2...
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• Spring '14
• Surface, Parametric equation, Parametric surface

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