subdivision surfaces for cad: integration through ...antonelm/talks/milano2012.pdf · difﬁculty...

Workshop:New trends in subdivision and related applications

September 4–7, 2012Department of Mathematics and Applications, University of Milano-Bicocca, Italy

Subdivision surfaces for CAD:

integration through

parameterization and local

correction

Michele Antonelli1, Carolina Beccari2, Giulio Casciola2, Serena Morigi2

1Department of Mathematics, University of Padova, Italy2Department of Mathematics, University of Bologna, Italy

Milano, September 6, 2012

Subdivision surfaces and CADSS 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 inCAD:

• surface fitting (Ma, Zhao, 2002)• reverse engineering (Ma, Zhao, 2000; Beccari, Farella, Liverani, Morigi, Rucci, 2010)• curve lofting (Nasri, 2001; Schaefer, Warren, Zorin, 2004)

Their presence in CAD is still negligible:• no closed-form representation• quality and regularity issues

Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi





. . . But CAD end-users ask for them!






. . . But CAD end-users ask for them!

2010–2013: Eurostars Project NIIT4CAD(New Interactive and Innovative Technologies for CAD)


Objective• Seamless integration of subdivision surfaces in a CAD system


Objective• Seamless integration of subdivision surfaces in a CAD systemg

Catmull-Clark



Catmull-Clark

• Main roadblocks:lack of quality and precisiondifficulty of integration into the modeling workflow



Catmull-Clark

• Main roadblocks:lack of quality and precisiondifficulty of integration into the modeling workflow

Seamless integration

• The desired accuracy is achieved

• All the functionalities of the CAD system are inherited


SS seamlessly integrated in our system



Meancurvature

<−0.2035 > 0.07465<−0.2053

> 0.07465

Catmull-Clark Local correction

Isophotes



Meancurvature

<−0.2035 > 0.07465<−0.2053

> 0.07465


Isophotes

Meancurvature

<−0.2053

> 0.07465


Isophotes


Outline1 Features that a CAD system must possess to allow the integration of SS

2 Integration of subdivision solids through parameterization and boundaryrepresentation

3 Local correction of regularity issues

4 Examples


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 surfacesSolids: B-rep

• Heterogeneous geometric description

• Extensible geometric kernel


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 surfacesSolids: B-rep

• Heterogeneous geometric description

• Extensible geometric kernel

Boundary representation:a geometric model is described through itsgeometric limits, storing information ontopology 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 ofloops of connected edges

• Vertex: limit of an edge

The B-rep induces a structure of graphbetween the different components.

Cube with spherical hollow


IntegrationSubd-B-rep

A Subd-B-rep represents a B-rep geometric model in which:

• each B-rep face is a subdivision surface patch associated with a rectangularparametric 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) be-tween the control mesh faces and the B-rep faces






Idea: the Subd-B-rep should maintain an intuitive association (possibly 1-1) be-tween the control mesh faces and the B-rep faces

Control mesh Subd-B-rep

• Topology: 1 quad face ←→ 1 face1 n-sided face ←→ n (quad) faces

←→

• Geometry: 1 face ←→1 parametric quad patchon base domain Q := [0,1]2


B-rep geometry description (I)• 1 face←→ 1 parametric quad patch Si 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 ψSi: Q → Si

Limit surface structure around an extraordinary vertex

Stam, 1998; Yamaguchi, 2001;Lai, Cheng, 2006


B-rep geometry description (I)• 1 face←→ 1 parametric quad patch Si 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 ψSi: Q → Si

Limit surface structure around an extraordinary vertex

Stam, 1998; Yamaguchi, 2001;Lai, Cheng, 2006

(a) Quad face with a single e.v.

0 10

1

u

vQ ψ Si


B-rep geometry description (II)(b) Quad face with more than one e.v.

0 12

10

12

1

u0

v0 u1

v1

u2

v2u3

v3

Q0 Q1

Q2Q3

q0 q1

q2q3

φ`,`= 0,. . . ,3 Q ψ

Si

Si,0 Si,1

Si,2Si,3

φ` := σ2 ◦ρ−1`

π2◦ τq`−q0

σh scaling by a factor hρ−1

a c.w. rotation around theorigin of an angle a

τv translation of a vector v


B-rep geometry description (II)(b) Quad face with more than one e.v.

0 12

10

12

1

u0

v0 u1

v1

u2

v2u3

v3

Q0 Q1

Q2Q3

q0 q1

q2q3

φ`,`= 0,. . . ,3 Q ψ

Si

Si,0 Si,1

Si,2Si,3

(c.1) Non-quad face without e.v.

0 10

1

u

vQ ψ and scale derivatives

Si

(c.2) Non-quad face with at least one e.v.

0 12

10

12

1

u0

v0 u1

v1

u2

v2u3

v3

Q0 Q1

Q2Q3

q0 q1

q2q3

φ`,`= 0,. . . ,3 Q

ψand scalederivatives

Si

Si,0 Si,1

Si,2Si,3

φ` := σ2 ◦ρ−1`

π2◦ τq`−q0

σh scaling by a factor hρ−1

a c.w. rotation around theorigin of an angle a

τv translation of a vector v


Surface tuningObjective: local correction of quality issues



• The shape of CC surfaces is satisfactory =⇒ we are interested in maintainingtheir appearance and B-spline nature in the widest possible area, whiletuning their analytical properties in the smallest neighborhood of e.v.

Catmull-Clark




Idea: local correction through polynomial blending (A. Levin, 2006; Zorin, 2006)

S∗i := w Si +(1−w) P


blendingregion

S∗i w Si P

blended surface

weight function

Catmull-Clark surface

approximating polynomial





S∗i := w Si +(1−w) P


blendingregion

S∗i w Si P

blended surface

weight function



• w s.t. C2-transition between S∗i and Si −1 0 1−1 0 1

0

1

−1 0 1−1 0 10

1

A. Levin, 2006 Zero in a circular neigh-borhood of the origin





S∗i := w Si +(1−w) P


blendingregion

S∗i w Si P

blended surface

weight function



• w s.t. C2-transition between S∗i and Si −1 0 1−1 0 1

0

1

−1 0 1−1 0 10

1

A. Levin, 2006 Zero in a circular neigh-borhood of the origin

Approach: definition of a parametric surface evaluable at arbitrary pointsA. Levin, Zorin: discrete vs. Our: parametric surface with exact evaluation


A common parameterization domain

• Si,P,w must be parameterized over a common domain



• Si,P,w must be parameterized over a common domain=⇒ Characteristic map of valence n, regarded as a parametric multipatch

surface: ψK[n]

0: Q −→ K[n]

0one sector




surface: ψK[n]

0: Q −→ K[n]

0one sector

• Star-shaped transformationκi := ρi 2π

n◦ψ

K[n]0◦σh ◦φ

• φ(u,v) :=

{

(u,v) if patch of type (a) or (c.1)φ`(u,v) if patch of type (b) or (c.2)

• σh scaling of hpatch type (a) (b) (c.1) (c.2)

h 4 2 2 1

• ψK[n]0

: Qh∣

∣

Q −→ K[n]0 , Qh := [0,h]2

• ρi 2π

nc.c.w. rotation of angle i

2πn




surface: ψK[n]

0: Q −→ K[n]

0one sector


n◦ψ

K[n]0◦σh ◦φ

• φ(u,v) :=

{



h 4 2 2 1

• ψK[n]0

: Qh∣

∣

Q −→ K[n]0 , Qh := [0,h]2

• ρi 2π


2πn

0 1u

v

Q

Si

ψSi




surface: ψK[n]

0: Q −→ K[n]

0one sector


n◦ψ

K[n]0◦σh ◦φ

• φ(u,v) :=

{



h 4 2 2 1

• ψK[n]0

: Qh∣

∣

Q −→ K[n]0 , Qh := [0,h]2

• ρi 2π


2πn

0 01 hu

v

Q Qh

Si

σh ◦φ

ψSi




surface: ψK[n]

0: Q −→ K[n]

0one sector


n◦ψ

K[n]0◦σh ◦φ

• φ(u,v) :=

{



h 4 2 2 1

• ψK[n]0

: Qh∣

∣

Q −→ K[n]0 , Qh := [0,h]2

• ρi 2π


2πn

0 01 1 h

0 1

u

v

s

t

Q Qh

Qh∣∣

Q

Si

K[n]0

σh ◦φ

ψSi

ψK[n]0




surface: ψK[n]

0: Q −→ K[n]

0one sector


n◦ψ

K[n]0◦σh ◦φ

• φ(u,v) :=

{



h 4 2 2 1

• ψK[n]0

: Qh∣

∣

Q −→ K[n]0 , Qh := [0,h]2

• ρi 2π


2πn

0 01 1 h

0 1

u

v

s

st

t

Q Qh

Qh∣∣

Q

Si

K[n]0

σh ◦φ

ψSi

ψK[n]0

ρi 2π

n




surface: ψK[n]

0: Q −→ K[n]

0one sector


n◦ψ

K[n]0◦σh ◦φ

• φ(u,v) :=

{



h 4 2 2 1

• ψK[n]0

: Qh∣

∣

Q −→ K[n]0 , Qh := [0,h]2

• ρi 2π


2πn

0 01 1 h

0 1

u

v

s

st

t

Q Qh

Qh∣∣

Q

Si

K[n]0

σh ◦φ

ψSi

ψK[n]0

ρi 2π

n

κi




surface: ψK[n]

0: Q −→ K[n]

0one sector


n◦ψ

K[n]0◦σh ◦φ

• φ(u,v) :=

{



h 4 2 2 1

• ψK[n]0

: Qh∣

∣

Q −→ K[n]0 , Qh := [0,h]2

• ρi 2π


2πn

• Star-shaped domain

Kn :=n−1⋃

i=0

{

κi (u,v)∣

∣

∣(u,v) ∈ Q and σh(φ(u,v)) ∈ Q

}

0 01 1 h

0 1

u

v

s

st

t

Q Qh

Qh∣∣

Q

Si

K[n]0

σh ◦φ

ψSi

ψK[n]0

ρi 2π

n

κi




surface: ψK[n]

0: Q −→ K[n]

0one sector


n◦ψ

K[n]0◦σh ◦φ

• φ(u,v) :=

{



h 4 2 2 1

• ψK[n]0

: Qh∣

∣

Q −→ K[n]0 , Qh := [0,h]2

• ρi 2π


2πn

• Star-shaped domain

Kn :=n−1⋃

i=0

{

κi (u,v)∣

∣


}

• Blending region Dn :={

(s,t) ∈ Kn

∣

∣

∣‖(s,t)‖2 6 λ [n]

}

(s,t) := κi(u,v), λ [n] subdominant eigenvalue of the subdivision matrix

0 01 112

h

0 1λ [n]

u

v

s

st

t

Q Qh

Qh∣∣

Q

Si

K[n]0

Dn

Dn

D̃nσh ◦φ

ψSi

ψK[n]0

ρi 2π

n

κi


Blended surface on Kn

• Blended surface S∗i (u,v) :=

{

w(s,t)Si(u,v)+(1−w(s,t))P(s,t) (s,t) ∈ Dn

Si(u,v) elsewherewhere (s,t) := κi(u,v)


Blended surface on Kn

• Blended surface S∗i (u,v) :=

{

w(s,t)Si(u,v)+(1−w(s,t))P(s,t) (s,t) ∈ Dn

Si(u,v) elsewherewhere (s,t) := κi(u,v)

• Preimage of Dn under κi

D̃n,i :={

(u,v) ∈ Q∣

∣

∣σh(φ(u,v)) ∈ Q and ‖κi(u,v)‖2 6 λ [n]

}

D̃n,i ⊂

{

Q18 if patch of type (a) or (b)

Q14 if patch of type (c.1) or (c.2)

=⇒ blending regions surrounding e.v. ofthe same face are well separated

=⇒ most of the surface is spline!

(a)

(a)(a)

(b)(b)

(c.1)(c.1)(c.1)(c.1)

(c.2)


LS approximating polynomial• Interpolate the e.v. P(s,t) = pev+Cm(s,t), m(s,t) =

(

s,t,s2,st,t2, . . .)

• Coefficients C are computed by least squares fitting

12 uniformly distributedapproximation pointsper sector in D̃n,i ⊂ Q



(

s,t,s2,st,t2, . . .)


• V TV c =V T (p−pev) =⇒ precompute and store(

V TV)−1

V T

for each valence n




(

s,t,s2,st,t2, . . .)


• V TV c =V T (p−pev) =⇒ precompute and store(

V TV)−1

V T

for each valence n

• E.v. on boundary: fan-shaped domain

K̂n :=n−2⋃

i=0

{

ψ̂K̂[n]

i(σh (φ(u,v)))

∣

∣


}



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

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

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• Operations of solid composition =⇒ B-rep whose faces can have heteroge-neous nature (NURBS + subdivision)and are editable while maintaining thisfeature

−→ the workflow applies to those faces of the heterogeneous B-rep modelthat are of subdivision type


Validation fromIntegration between conceptual design and engineering phase

Subd-B-rep solid

Model with thickness Division of the object in two parts


Thankyou!


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