Subdivision surfaces for CAD: integration through parameterization and local correction

Transcription

1 Workshop: New trends in subdivision and related applications September 4 7, 212 Department of Mathematics and Applications, University of Milano-Bicocca, Italy Subdivision surfaces for CAD: integration through parameterization and local correction Michele Antonelli 1, Carolina Beccari 2, Giulio Casciola 2, Serena Morigi 2 1 Department of Mathematics, University of Padova, Italy 2 Department of Mathematics, University of Bologna, Italy Milano, September 6, 212

2 Subdivision surfaces and CAD SS widely supported in nearly all modeling programs Advantages: flexibility for arbitrary topology + superset of NURBS standard A lot of theoretical study and many proposed algorithms potentially useful in CAD: surface fitting (Ma, Zhao, 22) reverse engineering (Ma, Zhao, 2; Beccari, Farella, Liverani, Morigi, Rucci, 21) curve lofting (Nasri, 21; Schaefer, Warren, Zorin, 24) Their presence in CAD is still negligible: no closed-form representation quality and regularity issues

3 Subdivision surfaces and CAD SS widely supported in nearly all modeling programs Advantages: flexibility for arbitrary topology + superset of NURBS standard A lot of theoretical study and many proposed algorithms potentially useful in CAD: surface fitting (Ma, Zhao, 22) reverse engineering (Ma, Zhao, 2; Beccari, Farella, Liverani, Morigi, Rucci, 21) curve lofting (Nasri, 21; Schaefer, Warren, Zorin, 24) Their presence in CAD is still negligible: no closed-form representation quality and regularity issues... But CAD end-users ask for them!

4 Subdivision surfaces and CAD SS widely supported in nearly all modeling programs Advantages: flexibility for arbitrary topology + superset of NURBS standard A lot of theoretical study and many proposed algorithms potentially useful in CAD: surface fitting (Ma, Zhao, 22) reverse engineering (Ma, Zhao, 2; Beccari, Farella, Liverani, Morigi, Rucci, 21) curve lofting (Nasri, 21; Schaefer, Warren, Zorin, 24) Their presence in CAD is still negligible: no closed-form representation quality and regularity issues... But CAD end-users ask for them! : Eurostars Project NIIT4CAD (New Interactive and Innovative Technologies for CAD)

5 Objective Seamless integration of subdivision surfaces in a CAD system

6 Objective Catmull-Clark Seamless integration of subdivision surfaces in a CAD system

7 Objective Catmull-Clark Seamless integration of subdivision surfaces in a CAD system Main roadblocks: lack of quality and precision difficulty of integration into the modeling workflow

8 Objective Catmull-Clark Seamless integration of subdivision surfaces in a CAD system Main roadblocks: lack of quality and precision difficulty of integration into the modeling workflow Seamless integration The desired accuracy is achieved All the functionalities of the CAD system are inherited

9 SS seamlessly integrated in our system

10 SS seamlessly integrated in our system >.7465 Mean curvature <.235 >.7465 Catmull-Clark Local correction <.253 Isophotes

11 SS seamlessly integrated in our system >.7465 Mean curvature Catmull-Clark Local correction <.253 >.7465 Mean curvature Isophotes <.235 >.7465 Catmull-Clark Local correction <.253 Isophotes

12 Outline 1 Features that a CAD system must possess to allow the integration of SS 2 Integration of subdivision solids through parameterization and boundary representation 3 Local correction of regularity issues 4 Examples

13 The CAD system paradigm Geometric kernel: set of geometric representations (NURBS, planes, cylinders, quadrics) set of tools which operate on them (intersections, projections, Boolean operations, offsets, fillets, etc.) Parametric curves and surfaces Solids: B-rep Heterogeneous geometric description Extensible geometric kernel

14 The CAD system paradigm Geometric kernel: set of geometric representations (NURBS, planes, cylinders, quadrics) set of tools which operate on them (intersections, projections, Boolean operations, offsets, fillets, etc.) Parametric curves and surfaces Solids: B-rep Heterogeneous geometric description Boundary representation: a geometric model is described through its geometric Extensible limits, storingeometric information kernel on topology and geometry. Solid: volume limited by shells (boundary solid/non-solid) Shell: collection of surface patches, called B-rep faces The boundary of a face is composed of loops of connected edges Vertex: limit of an edge The B-rep induces a structure of graph between the different components. Cube with spherical hollow

15 Subd-B-rep A Subd-B-rep represents a B-rep geometric model in which: Integration each B-rep face is a subdivision surface patch associated with a rectangular parametric domain; the B-rep topology is inferred from the subdivision control mesh.

16 Subd-B-rep A Subd-B-rep represents a B-rep geometric model in which: Integration each B-rep face is a subdivision surface patch associated with a rectangular parametric domain; the B-rep topology is inferred from the subdivision control mesh. Idea: the Subd-B-rep should maintain an intuitive association (possibly 1-1) between the control mesh faces and the B-rep faces

17 Subd-B-rep A Subd-B-rep represents a B-rep geometric model in which: Integration each B-rep face is a subdivision surface patch associated with a rectangular parametric domain; the B-rep topology is inferred from the subdivision control mesh. Idea: the Subd-B-rep should maintain an intuitive association (possibly 1-1) between the control mesh faces and the B-rep faces Control mesh Subd-B-rep Topology: 1 quad face 1 face 1 n-sided face n (quad) faces Geometry: 1 face 1 parametric quad patch on base domain Q :=[,1] 2

18 B-rep geometry description (I) 1 face 1 parametric quad patch S i on Q For Catmull-Clark surfaces, we are able to evaluate two types of patches: regular (bi-cubic tensor-product B-splines) quadrilateral and containing a single extraordinary vertex parameterization function ψ Si : Q S i Stam, 1998; Yamaguchi, 21; Lai, Cheng, 26 Limit surface structure around an extraordinary vertex

19 B-rep geometry description (I) 1 face 1 parametric quad patch S i on Q For Catmull-Clark surfaces, we are able to evaluate two types of patches: regular (bi-cubic tensor-product B-splines) quadrilateral and containing a single extraordinary vertex parameterization function ψ Si : Q S i Stam, 1998; Yamaguchi, 21; Lai, Cheng, 26 Limit surface structure around an extraordinary vertex (a) Quad face with a single e.v. 1 v Q ψ S i u 1

20 B-rep geometry description (II) (b) Quad face with more than one e.v. v 3 u 2 1 q 3 q 2 Q 3 Q 2 u 3 v v u 1 Q Q 1 q q 1 1 u v φ l,l=,...,3 φ l := σ 2 ρ 1 l 2 π τ ql q σ h scaling by a factor h ρa 1 c.w. rotation around the origin of an angle a τ v translation of a vector v Q ψ S i,3 S i,2 S i, S i,1 S i

21 B-rep geometry description (II) (b) Quad face with more than one e.v. (c.1) Non-quad face without e.v. v 3 u 2 1 q 3 q 2 Q 3 Q 2 u 3 v v u 1 Q Q 1 q q 1 1 u v Q v φ l,l=,...,3 φ l := σ 2 ρ 1 l 2 π τ ql q σ h scaling by a factor h ρa 1 c.w. rotation around the origin of an angle a τ v translation of a vector v Q ψ and scale derivatives ψ S i,3 S i,2 S i, S i,1 S i S i u 1 (c.2) Non-quad face with at least one e.v. v 3 u 2 1 q 3 q 2 Q 3 Q 2 u 3 v v u 1 Q Q 1 q q 1 1 u v φ l,l=,...,3 Q ψ and scale derivatives S i,3si,2 S i, S i,1 S i

22 Objective: local correction of quality issues Surface tuning

23 Objective: local correction of quality issues Surface tuning The shape of CC surfaces is satisfactory= we are interested in maintaining their appearance and B-spline nature in the widest possible area, while tuning their analytical properties in the smallest neighborhood of e.v. Catmull-Clark

24 Objective: local correction of quality issues Surface tuning The shape of CC surfaces is satisfactory= we are interested in maintaining their appearance and B-spline nature in the widest possible area, while tuning their analytical properties in the smallest neighborhood of e.v. Idea: local correction through polynomial blending (A. Levin, 26; Zorin, 26) blended surface Catmull-Clark surface blending region S i := w S i +(1 w) P weight function approximating polynomial Catmull-Clark Local correction

25 Objective: local correction of quality issues Surface tuning The shape of CC surfaces is satisfactory= we are interested in maintaining their appearance and B-spline nature in the widest possible area, while tuning their analytical properties in the smallest neighborhood of e.v. Idea: local correction through polynomial blending (A. Levin, 26; Zorin, 26) blending blended surface region Catmull-Clark surface S i := w S i +(1 w) P weight function approximating polynomial Catmull-Clark Local correction 1 1 w s.t. C 2 -transition between S i and S i A. Levin, 26 Zero in a circular neighborhood of the origin

26 Objective: local correction of quality issues Surface tuning The shape of CC surfaces is satisfactory= we are interested in maintaining their appearance and B-spline nature in the widest possible area, while tuning their analytical properties in the smallest neighborhood of e.v. Idea: local correction through polynomial blending (A. Levin, 26; Zorin, 26) blending blended surface region Catmull-Clark surface S i := w S i +(1 w) P weight function approximating polynomial Catmull-Clark Local correction 1 1 w s.t. C 2 -transition between S i and S i A. Levin, 26 Zero in a circular neighborhood of the origin Approach: definition of a parametric surface evaluable at arbitrary points A. Levin, Zorin: discrete vs. Our: parametric surface with exact evaluation

27 A common parameterization domain S i,p,w must be parameterized over a common domain

28 A common parameterization domain S i,p,w must be parameterized over a common domain = Characteristic map of valence n, regarded as a parametric multipatch surface: ψ K [n] : Q K [n] one sector

29 A common parameterization domain S i,p,w must be parameterized over a common domain = Characteristic map of valence n, regarded as a parametric multipatch surface: ψ K [n] : Q K [n] one sector Star-shaped transformation κ i := ρ i 2π ψ [n] n K σ h φ { (u,v) if patch of type (a) or (c.1) φ(u,v) := φ l (u,v) if patch of type (b) or (c.2) patch type (a) (b) (c.1) (c.2) σ h scaling of h h ψ [n] : Q h K Q K [n], Qh :=[,h] 2 ρ i 2π n c.c.w. rotation of angle i 2π n

30 A common parameterization domain S i,p,w must be parameterized over a common domain = Characteristic map of valence n, regarded as a parametric multipatch surface: ψ K [n] : Q K [n] one sector Star-shaped transformation κ i := ρ i 2π ψ [n] n K σ h φ { (u,v) if patch of type (a) or (c.1) φ(u,v) := φ l (u,v) if patch of type (b) or (c.2) patch type (a) (b) (c.1) (c.2) σ h scaling of h h ψ [n] : Q h K Q K [n], Qh :=[,h] 2 v Q 1 u ψ Si ρ i 2π n c.c.w. rotation of angle i 2π n S i

31 A common parameterization domain S i,p,w must be parameterized over a common domain = Characteristic map of valence n, regarded as a parametric multipatch surface: ψ K [n] : Q K [n] one sector Star-shaped transformation κ i := ρ i 2π ψ [n] n K σ h φ { (u,v) if patch of type (a) or (c.1) φ(u,v) := φ l (u,v) if patch of type (b) or (c.2) patch type (a) (b) (c.1) (c.2) σ h scaling of h h ψ [n] : Q h K Q K [n], Qh :=[,h] 2 v Q 1 u h ψ Si σ h φ Q h ρ i 2π n c.c.w. rotation of angle i 2π n S i

32 A common parameterization domain S i,p,w must be parameterized over a common domain = Characteristic map of valence n, regarded as a parametric multipatch surface: ψ K [n] : Q K [n] one sector Star-shaped transformation κ i := ρ i 2π ψ [n] n K σ h φ { (u,v) if patch of type (a) or (c.1) φ(u,v) := φ l (u,v) if patch of type (b) or (c.2) patch type (a) (b) (c.1) (c.2) σ h scaling of h h ψ [n] : Q h K Q K [n], Qh :=[,h] 2 ρ i 2π n c.c.w. rotation of angle i 2π n v Q Q h Q Q h 1 u 1 h ψ t [n] K ψ Si S i σ h φ K [n] 1 s

33 A common parameterization domain S i,p,w must be parameterized over a common domain = Characteristic map of valence n, regarded as a parametric multipatch surface: ψ K [n] : Q K [n] one sector Star-shaped transformation κ i := ρ i 2π ψ [n] n K σ h φ { (u,v) if patch of type (a) or (c.1) φ(u,v) := φ l (u,v) if patch of type (b) or (c.2) patch type (a) (b) (c.1) (c.2) σ h scaling of h h ψ [n] : Q h K Q K [n], Qh :=[,h] 2 ρ i 2π n c.c.w. rotation of angle i 2π n v Q Q h Q Q h 1 u 1 h ψ t [n] K ψ Si S i σ h φ K [n] 1 s t ρ i 2π n s

34 A common parameterization domain S i,p,w must be parameterized over a common domain = Characteristic map of valence n, regarded as a parametric multipatch surface: ψ K [n] : Q K [n] one sector Star-shaped transformation κ i := ρ i 2π ψ [n] n K σ h φ { (u,v) if patch of type (a) or (c.1) φ(u,v) := φ l (u,v) if patch of type (b) or (c.2) patch type (a) (b) (c.1) (c.2) σ h scaling of h h ψ [n] : Q h K Q K [n], Qh :=[,h] 2 ρ i 2π n c.c.w. rotation of angle i 2π n v Q Q h Q Q h 1 u 1 h ψ t [n] K ψ Si S i σ h φ K [n] 1 s t ρ i 2π n s κ i

35 A common parameterization domain S i,p,w must be parameterized over a common domain = Characteristic map of valence n, regarded as a parametric multipatch surface: ψ K [n] : Q K [n] one sector Star-shaped transformation κ i := ρ i 2π ψ [n] n K σ h φ { (u,v) if patch of type (a) or (c.1) φ(u,v) := φ l (u,v) if patch of type (b) or (c.2) patch type (a) (b) (c.1) (c.2) σ h scaling of h h ψ [n] : Q h K Q K [n], Qh :=[,h] 2 ρ i 2π n c.c.w. rotation of angle i 2π n Star-shaped domain n 1 { } K n := κ i (u,v) (u,v) Q and σ h (φ(u,v)) Q i= v Q Q h Q Q h 1 u 1 h ψ t [n] K ψ Si S i σ h φ K [n] 1 s t ρ i 2π n s κ i

36 A common parameterization domain S i,p,w must be parameterized over a common domain = Characteristic map of valence n, regarded as a parametric multipatch surface: ψ K [n] : Q K [n] one sector Star-shaped transformation κ i := ρ i 2π ψ [n] n K σ h φ { (u,v) if patch of type (a) or (c.1) φ(u,v) := φ l (u,v) if patch of type (b) or (c.2) patch type (a) (b) (c.1) (c.2) σ h scaling of h h ψ [n] : Q h K Q K [n], Qh :=[,h] 2 ρ i 2π n c.c.w. rotation of angle i 2π n Star-shaped domain n 1 { } K n := κ i (u,v) (u,v) Q and σ h (φ(u,v)) Q i= (s,t) 2 Blending region D n := {(s,t) K n λ [n]} (s,t) := κ i (u,v), 1 u 1 1 h 2 ψ t [n] K ψ Si λ [n] subdominant eigenvalue of the subdivision matrix v D n Q S i σ h φ Q h Q D n Q h K [n] λ [n] 1 s t ρ i 2π n s D n κ i

37 Blended surface on K n { Blended surface S i (u,v) := w(s,t)s i (u,v)+(1 w(s,t))p(s,t) (s,t) D n where (s,t) := κ i (u,v) S i (u,v) elsewhere

38 Blended surface on K n { Blended surface S i (u,v) := w(s,t)s i (u,v)+(1 w(s,t))p(s,t) (s,t) D n where (s,t) := κ i (u,v) S i (u,v) Preimage { of D n under κ i D n,i := (u,v) Q σ h (φ(u,v)) Q and κ i (u,v) 2 λ [n]} { Q 1 8 if patch of type (a) or (b) D n,i if patch of type (c.1) or (c.2) Q 1 4 = blending regions surrounding e.v. of the same face are well separated = most of the surface is spline! (c.1)(c.1) (c.1) (c.1) (c.2) (a) (a) elsewhere (b) (b) (a)

39 LS approximating polynomial Interpolate the e.v. P(s,t)=p ev +Cm(s,t), m(s,t)= Coefficients C are computed by least squares fitting ( ) s,t,s 2,st,t 2, uniformly distributed approximation points per sector in D n,i Q

40 LS approximating polynomial Interpolate the e.v. P(s,t)=p ev +Cm(s,t), m(s,t)= Coefficients C are computed by least squares fitting ( ) s,t,s 2,st,t 2, uniformly distributed approximation points per sector in D n,i Q V T V c= V T (p p ev ) = precompute and store for each valence n ( ) 1V V T T V

41 LS approximating polynomial Interpolate the e.v. P(s,t)=p ev +Cm(s,t), m(s,t)= Coefficients C are computed by least squares fitting ( ) s,t,s 2,st,t 2, uniformly distributed approximation points per sector in D n,i Q V T V c= V T (p p ev ) = precompute and store for each valence n ( ) 1V V T T V E.v. on boundary: fan-shaped domain n 2 { } ˆK n := ˆψ ˆK [n] (σ h (φ(u,v))) (u,v) Q and σ h (φ(u,v)) Q i i=

42 Handling of heterogeneous rep. Workflow for the creation and editing of a Subd-B-rep

43 Handling of heterogeneous rep. Workflow for the creation and editing of a Subd-B-rep Operations of solid composition= B-rep whose faces can have heterogeneous nature (NURBS + subdivision) and are editable while maintaining this feature the workflow applies to those faces of the heterogeneous B-rep model that are of subdivision type

44 Validation from Integration between conceptual design and engineering phase Subd-B-rep solid Model with thickness Division of the object in two parts

45 Thank you!