Kobbelt, Slides on Surface Representations
)NPUT $ATA 2ANGE 3CAN #!$ 4OMOGRAPHY 2EMOVAL OF TOPOLOGICAL AND GEOMETRICAL ERRORS !NALYSIS OF SURFACE QUALITY 3URFACE SMOOTHING FOR NOISE REMOVAL 0ARAMETERIZATION Surface Representations 3IMPLIFICATION FOR COMPLEXITY REDUCTION Leif Kobbelt RWTH Aachen University 2EMESHING FOR IMPROVING MESH QUALITY &REEFORM AND MULTIRESOLUTION MODELING 1 Outline • (mathematical) geometry representations – parametric vs. implicit • approximation properties • types of operations – distance queries – evaluation – modification / deformation • data structures Leif Kobbelt RWTH Aachen University 2 2 Outline • (mathematical) geometry representations – parametric vs. implicit • approximation properties • types of operations – distance queries – evaluation – modification / deformation • data structures Leif Kobbelt RWTH Aachen University 3 3 Mathematical Representations • parametric – range of a function – surface patch 2 3 f : R → R , SΩ = f(Ω) • implicit – kernel of a function – level set 3 F : R → R, Sc = {p : F (p)=c} Leif Kobbelt RWTH Aachen University 4 4 2D-Example: Circle • parametric r cos(t) f : t !→ , S = f([0, 2π]) ! r sin(t) " • implicit 2 2 2 F (x, y)=x + y − r S = {(x, y):F (x, y)=0} Leif Kobbelt RWTH Aachen University 5 5 2D-Example: Island • parametric r cos(???t) f : t !→ , S = f([0, 2π]) ! r sin(???t) " • implicit 2 2 2 F (x, y)=x +???y − r S = {(x, y):F (x, y)=0} Leif Kobbelt RWTH Aachen University 6 6 Approximation Quality • piecewise parametric r cos(???t) f : t !→ , S = f([0, 2π]) ! r sin(???t) " • piecewise implicit 2 2 2 F (x, y)=x +???y − r S = {(x, y):F (x, y)=0} Leif Kobbelt RWTH Aachen University 7 7 Approximation Quality • piecewise parametric r cos(???t) f : t !→ , S = f([0, 2π]) ! r sin(???t) " • piecewise implicit 2 2 2 F (x, y)=x +???y − r S = {(x, y):F (x, y)=0} Leif Kobbelt RWTH Aachen University 8 8 Requirements / Properties • continuity – interpolation / approximation f(ui,vi) ≈ pi • topological consistency – manifold-ness • smoothness – C0, C1, C2, ..
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