��������� ��� ��������� ���� ������� �������������� ��� ���������� ��������� ������� ��������� ��� ����� ������� �������� �� ������� ������� �������� ��� ��������������� �������� ������� �� ����������� ��� ����������� ������ ����� ���� ��� ���������� ���������� ���������������� Differential Geometry Mark Pauly
Outline • Differential Geometry – curvature – fundamental forms – Laplace-Beltrami operator • Discretization • Visual Inspection of Mesh Quality 2 Mark Pauly
Differential Geometry • Continuous surface x ( u, v ) R 2 x ( u, v ) = y ( u, v ) , ( u, v ) ∈ I z ( u, v ) • Normal vector n = ( x u × x v ) / � x u × x v � – assuming regular parameterization, i.e. x u × x v � = 0 3 Mark Pauly
Differential Geometry • Normal Curvature x u × x v n = n � x u × x v � x u x v p t t = cos φ x u � x u � + sin φ x v � x v � 4 Mark Pauly
Differential Geometry • Normal Curvature x u × x v n = n � x u × x v � c t p t = cos φ x u � x u � + sin φ x v � x v � 5 Mark Pauly
Differential Geometry • Principal Curvatures κ 1 = max κ n ( φ ) – maximum curvature φ – minimum curvature κ 2 = min φ κ n ( φ ) t ) = κ n ( φ ) = κ 1 cos 2 φ + κ 2 sin 2 φ κ n (¯ • Euler Theorem: � 2 π H = κ 1 + κ 2 = 1 • Mean Curvature κ n ( φ ) dφ 2 2 π 0 • Gaussian Curvature K = κ 1 · κ 2 6 Mark Pauly
Differential Geometry • Normal curvature is defined as curvature of the normal curve at a point c ∈ x ( u, v ) p ∈ c • Can be expressed in terms of fundamental forms as ea 2 + 2 fab + gb 2 t T II ¯ ¯ t κ n (¯ t ) = = Ea 2 + 2 Fab + Gb 2 t T I ¯ ¯ t n t = a x u + b x v t c p 7 Mark Pauly
Differential Geometry • First fundamental form � � � x T x T � E F u x u u x v I = := x T x T F G u x v v x v • Second fundamental form � � � x T x T � e f uu n uv n II = := x T x T f g uv n vv n 8 Mark Pauly
Differential Geometry • I and II allow to measure – length, angles, area, curvature – arc element ds 2 = Edu 2 + 2 Fdudv + Gdv 2 – area element � EG − F 2 dudv dA = 9 Mark Pauly
Differential Geometry • Intrinsic geometry: Properties of the surface that only depend on the first fundamental form – length – angles – Gaussian curvature (Theorema Egregium) 6 πr − 3 C ( r ) K = lim πr 3 r → 0 10 Mark Pauly
Differential Geometry • A point x on the surface is called – elliptic , if K > 0 – parabolic , if K = 0 – hyperbolic , if K < 0 – umbilical , if κ 1 = κ 2 • Developable surface ⇔ K = 0 11 Mark Pauly
Laplace Operator gradient Laplace 2nd partial operator operator derivatives ∂ 2 f � ∆ f = div ∇ f = ∂x 2 i i function in Cartesian divergence Euclidean space coordinates operator 12 Mark Pauly
Laplace-Beltrami Operator • Extension of Laplace to functions on manifolds gradient Laplace- operator Beltrami ∆ S f = div S ∇ S f function on divergence S manifold operator 13 Mark Pauly
Laplace-Beltrami Operator • Extension of Laplace to functions on manifolds gradient Laplace- mean operator Beltrami curvature ∆ S x = div S ∇ S x = − 2 H n surface coordinate divergence normal function operator 14 Mark Pauly
Outline • Differential Geometry – curvature – fundamental forms – Laplace-Beltrami operator • Discretization • Visual Inspection of Mesh Quality 15 Mark Pauly
Discrete Differential Operators • Assumption: Meshes are piecewise linear approximations of smooth surfaces • Approach: Approximate differential properties at point x as spatial average over local mesh neighborhood N(x) , where typically – x = mesh vertex – N(x) = n -ring neighborhood or local geodesic ball 16 Mark Pauly
Discrete Laplace-Beltrami • Uniform discretization 1 � ( f ( v i ) − f ( v )) ∆ uni f ( v ) := |N 1 ( v ) | v i ∈N 1 ( v ) – depends only on connectivity → simple and efficient – bad approximation for irregular triangulations 17 Mark Pauly
Discrete Laplace-Beltrami • Cotangent formula 2 � ∆ S f ( v ) := (cot α i + cot β i ) ( f ( v i ) − f ( v )) A ( v ) v i ∈N 1 ( v ) v A ( v ) v α i v β i v i v i v i 18 Mark Pauly
Discrete Laplace-Beltrami • Cotangent formula 2 � ∆ S f ( v ) := (cot α i + cot β i ) ( f ( v i ) − f ( v )) A ( v ) v i ∈N 1 ( v ) • Problems – negative weights – depends on triangulation 19 Mark Pauly
Discrete Curvatures • Mean curvature H = � ∆ S x � • Gaussian curvature A � G = (2 π − θ j ) /A j θ j • Principal curvatures � � H 2 − G H 2 − G κ 2 = H − κ 1 = H + 20 Mark Pauly
Links & Literature • P. Alliez: Estimating Curvature Tensors on Triangle Meshes (source code) – http://www-sop.inria.fr/geometrica/team/ Pierre.Alliez/demos/curvature/ • Wardetzky, Mathur, Kaelberer, Grinspun: Discrete Laplace Operators: No free lunch , SGP 2007 principal directions 21 Mark Pauly
Outline • Differential Geometry – curvature – fundamental forms – Laplace-Beltrami operator • Discretization • Visual Inspection of Mesh Quality 22 Mark Pauly
Mesh Quality • Smoothness – continuous differentiability of a surface ( C k ) • Fairness – aesthetic measure of “well-shapedness” – principle of simplest shape – fairness measures from physical models � 2 � 2 � � ∂κ 1 � ∂κ 2 � κ 2 1 + κ 2 2 dA dA + ∂ t 1 ∂ t 2 S S strain energy variation of curvature 23 Mark Pauly
Mesh Quality � 2 � 2 � � ∂κ 1 � ∂κ 2 � κ 2 1 + κ 2 2 dA dA + ∂ t 1 ∂ t 2 S S strain energy variation of curvature 24 Mark Pauly
Mesh Quality • Visual inspection of “sensitive” attributes – Specular shading 25 Mark Pauly
Mesh Quality • Visual inspection of “sensitive” attributes – Specular shading – Reflection lines 26 Mark Pauly
Mesh Quality • Visual inspection of “sensitive” attributes – Specular shading – Reflection lines • differentiability one order lower than surface • can be efficiently computed using graphics hardware C 0 C 1 C 2 27 Mark Pauly
Mesh Quality • Visual inspection of “sensitive” attributes – Specular shading – Reflection lines – Curvature • Mean curvature 28 Mark Pauly
Mesh Quality • Visual inspection of “sensitive” attributes – Specular shading – Reflection lines – Curvature • Mean curvature • Gauss curvature 29 Mark Pauly
Mesh Quality Criteria • Smoothness – Low geometric noise 30 Mark Pauly
Mesh Quality Criteria • Smoothness – Low geometric noise • Adaptive tessellation – Low complexity 31 Mark Pauly
Mesh Quality Criteria • Smoothness – Low geometric noise • Adaptive tessellation – Low complexity • Triangle shape – Numerical robustness 32 Mark Pauly
Triangle Shape Analysis • Circum radius / shortest edge r 1 r 1 < r 2 e 2 e 1 e 2 r 2 e 1 • Needles and caps Needle Cap 33 Mark Pauly
Mesh Quality Criteria • Smoothness – Low geometric noise • Adaptive tessellation – Low complexity • Triangle shape – Numerical robustness • Feature preservation – Low normal noise 34 Mark Pauly
Normal Noise Analysis 35 Mark Pauly
Mesh Optimization • Smoothness ➡ Mesh smoothing • Adaptive tessellation ➡ Mesh decimation • Triangle shape ➡ Repair, remeshing 36 Mark Pauly
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