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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 f : R 2 → R 3 , S Ω = f ( Ω ) • implicit – kernel of a function – level set F : R 3 → R, S c = { p : F ( p ) = c } Leif Kobbelt RWTH Aachen University 4 4
2D-Example: Circle • parametric � r cos( t ) � f : t �→ S = f ([0 , 2 π ]) , r sin( t ) • implicit F ( x, y ) = x 2 + y 2 − r 2 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 , 2 π ]) , r sin( t ) ??? • implicit F ( x, y ) = x 2 + y 2 − r 2 ??? 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 , 2 π ]) , r sin( t ) ??? • piecewise implicit F ( x, y ) = x 2 + y 2 − r 2 ??? 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 , 2 π ]) , r sin( t ) ??? • piecewise implicit F ( x, y ) = x 2 + y 2 − r 2 ??? S = { ( x, y ) : F ( x, y ) = 0 } Leif Kobbelt RWTH Aachen University 8 8
Requirements / Properties • continuity f ( u i , v i ) ≈ p i – interpolation / approximation • topological consistency – manifold-ness • smoothness – C 0 , C 1 , C 2 , ... C k • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 9 9
Requirements / Properties • continuity f ( u i , v i ) ≈ p i – interpolation / approximation • topological consistency – manifold-ness • smoothness – C 0 , C 1 , C 2 , ... C k • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 10 10
Requirements / Properties • continuity f ( u i , v i ) ≈ p i – interpolation / approximation • topological consistency – manifold-ness • smoothness – C 0 , C 1 , C 2 , ... C k • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 11 11
Requirements / Properties • continuity f ( u i , v i ) ≈ p i – interpolation / approximation • topological consistency – manifold-ness • smoothness – C 0 , C 1 , C 2 , ... C k • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 12 12
Requirements / Properties • continuity f ( u i , v i ) ≈ p i – interpolation / approximation • topological consistency – manifold-ness • smoothness – C 0 , C 1 , C 2 , ... C k • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 13 13
Topological Consistency Leif Kobbelt RWTH Aachen University 14 14
Topological Consistency Leif Kobbelt RWTH Aachen University 14 14
Topological Consistency Mesh Repair ... Leif Kobbelt RWTH Aachen University 14 14
Closed 2-Manifolds • parametric – disk-shaped neighborhoods – + injectivity f ( D ε [ u, v ]) = D δ [ f ( u, v )] • implicit – surface of a “physical” solid F ( x, y, z ) = c, �∇ F ( x, y, z ) � � = 0 – Leif Kobbelt RWTH Aachen University 15 15
Closed 2-Manifolds • parametric – disk-shaped neighborhoods – + injectivity f ( D ε [ u, v ]) = D δ [ f ( u, v )] • implicit – surface of a “physical” solid F ( x, y, z ) = c, �∇ F ( x, y, z ) � � = 0 – Leif Kobbelt RWTH Aachen University 16 16
Closed 2-Manifolds • parametric – disk-shaped neighborhoods – + injectivity f ( D ε [ u, v ]) = D δ [ f ( u, v )] • implicit – surface of a “physical” solid F ( x, y, z ) = c, �∇ F ( x, y, z ) � � = 0 – Leif Kobbelt RWTH Aachen University 17 17
Closed 2-Manifolds • parametric – disk-shaped neighborhoods f ( D ε [ u, v ]) = D δ [ f ( u, v )] – • implicit – surface of a “physical” solid F ( x, y, z ) = c, �∇ F ( x, y, z ) � � = 0 – Leif Kobbelt RWTH Aachen University 18 18
Closed 2-Manifolds • parametric – disk-shaped neighborhoods f ( D ε [ u, v ]) = D δ [ f ( u, v )] – • implicit – surface of a “physical” solid F ( x, y, z ) = c, �∇ F ( x, y, z ) � � = 0 – Leif Kobbelt RWTH Aachen University 19 19
Smoothness • position continuity : C 0 • tangent continuity : C 1 • curvature continuity : C 2 Leif Kobbelt RWTH Aachen University 20 20
Smoothness • position continuity : C 0 • tangent continuity : C 1 • curvature continuity : C 2 Leif Kobbelt RWTH Aachen University 21 21
Smoothness • position continuity : C 0 • tangent continuity : C 1 • curvature continuity : C 2 Leif Kobbelt RWTH Aachen University 22 22
Fairness • minimum surface area • minimum curvature • minimum curvature variation Leif Kobbelt RWTH Aachen University 23 23
Outline • (mathematical) geometry representations – parametric vs. implicit • approximation properties • types of operations – distance queries – evaluation – modification / deformation • data structures Leif Kobbelt RWTH Aachen University 24 24
Polynomials • computable functions p p c i t i = � � c � p ( t ) = i Φ i ( t ) i =0 i =0 • Taylor expansion p 1 i ! f ( i ) (0) h i + O ( h p +1 ) � f ( h ) = i =0 • interpolation error (mean value theorem) p ( t i ) = f ( t i ) , t i ∈ [0 , h ] p 1 ( p + 1)! f ( p +1) ( t ∗ ) � ( t − t i ) = O ( h ( p +1) ) � f ( t ) − p ( t ) � = i =0 Leif Kobbelt RWTH Aachen University 25 25
Polynomials • computable functions p p c i t i = � � c � p ( t ) = i Φ i ( t ) i =0 i =0 • Taylor expansion p 1 i ! f ( i ) (0) h i + O ( h p +1 ) � f ( h ) = i =0 • interpolation error (mean value theorem) p ( t i ) = f ( t i ) , t i ∈ [0 , h ] p 1 ( p + 1)! f ( p +1) ( t ∗ ) � ( t − t i ) = O ( h ( p +1) ) � f ( t ) − p ( t ) � = i =0 Leif Kobbelt RWTH Aachen University 26 26
Polynomials • computable functions p p c i t i = � � c � p ( t ) = i Φ i ( t ) i =0 i =0 • Taylor expansion p 1 i ! f ( i ) (0) h i + O ( h p +1 ) � f ( h ) = i =0 • interpolation error (mean value theorem) p ( t i ) = f ( t i ) , t i ∈ [0 , h ] p 1 ( p + 1)! f ( p +1) ( t ∗ ) � ( t − t i ) = O ( h ( p +1) ) � f ( t ) − p ( t ) � = i =0 Leif Kobbelt RWTH Aachen University 27 27
Implicit Polynomials • interpolation error of the function values � F ( x, y, z ) − P ( x, y, z ) � = O ( h ( p +1) ) • approximation error of the contour F ( p + � p ) − F ( p ) � p = λ ∇ F ( p ) ≈ �∇ F ( p ) � �� p � Leif Kobbelt RWTH Aachen University 28 28
Implicit Polynomials • interpolation error of the function values � F ( x, y, z ) − P ( x, y, z ) � = O ( h ( p +1) ) p p+ Δ p • approximation error of the contour F ( p + � p ) − F ( p ) � p = λ ∇ F ( p ) ≈ �∇ F ( p ) � �� p � Leif Kobbelt RWTH Aachen University 29 29
Implicit Polynomials • interpolation error of the function values � F ( x, y, z ) − P ( x, y, z ) � = O ( h ( p +1) ) p p+ Δ p • approximation error of the contour �� p � ≈ F ( p + � p ) − F ( p ) � p = λ ∇ F ( p ) �∇ F ( p ) � (gradient bounded from below) Leif Kobbelt RWTH Aachen University 30 30
Implicit Polynomials F(x,y,z) F(x,y,z) x,y,z x,y,z large small gradient gradient Leif Kobbelt RWTH Aachen University 31 31
Polynomial Approximation • approximation error is O(h p+1 ) • improve approximation quality by – increasing p ... higher order polynomials – decreasing h ... smaller / more segments • issues – smoothness of the target data ( max t f (p+1) (t) ) – handling higher order patches (e.g. boundary conditions) Leif Kobbelt RWTH Aachen University 32 32
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