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parametric surface patches
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implicit representation
implicit surface representation e.g. sphere:
f(P) = 0 f(P) = 0 → + + − = 0 x2 y2 z2 r2
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parametric surface patches 1 implicit representation implicit - - PowerPoint PPT Presentation
parametric surface patches 1 implicit representation implicit - - PowerPoint PPT Presentation
parametric surface patches 1 implicit representation implicit surface representation f ( P ) = 0 e.g. sphere: x 2 y 2 z 2 r 2 f ( P ) = 0 + + = 0 2 parametric representation parametric surface representation P ( u ) = ( f x ( u ),
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parametric representation
parametric surface representation e.g. sphere:
P(u) = ( (u), (u), (u)) fx fy fz P(u) = (rcos ϕsinθ,rsinϕsinθ,rcos θ)
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parametric representation
goals when defining smoothness, efficiency, local control same as curves (1) combine curves to obtain surfaces easy for simple solids, but not general (2) extend parametric curves to surface patches general formulation used heavily in CAD
f
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surfaces from curves
example: extrude curve in XY along Z example: revolve curve in YZ along Z
C(u) P(u,v) = ( (u), (u),v) Cx Cy C(u) P(u,v) = ( (u) cos(v), (u) sin(v), (u)) Cy Cy Cz
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SLIDE 6
patches
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SLIDE 7
patches
spline curves: 1D blending functions surface patches: 2D blending functions cross product of 1D blendering functions
P(t) = (t) ∑
i
bi Pi P(u,v) = (u,v) ∑
ij
bij Pij (u,v) = (u) (v) bij bi bj
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patches
bicubic Bezier patches
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SLIDE 9
patches
joining is hard
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SLIDE 10
- ther patch functions
just like curves uniform, non-uniform non-uniform rational B-splines (NURBS) ratios of B-splines invariance under perspective can represent conic sections exactly
- ften used in 3D
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SLIDE 11
rendering parametric surfaces
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SLIDE 12
rendering parametric surfaces
tesselation: approximate surfaces with triangles/quads meshes are efficient to draw in hardware and software more faces to provide better approximation uniform tesselation: split intervals uniformly fast to compute and simple to implement generates many segments adaptive tesselation: split recursively until good enough Bezier patches can use De Castaljeu
(u,v)
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uniform tesselation
split in points uniformly at tesselate into quads by creating a quad grid set vertices to join vertices to avoid holes if function wraps (e.g. cylinder) normals evaluate analytically if possible (e.g. sphere)
- r evaluate by computing partial derivatives
- r smooth the mesh by averaging face normals over
vertices
K × K ( , ) = (1/ ,1/ ) uk1 vk2 k1 k2 K × K P( , ) uk1 vk2 n = ∂P/∂u × ∂P/∂v
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