An effective integration of subdivision surfaces in a hybri d computer-aided design systemMichele Antonelli1 Carolina Vittoria Beccari2 Giulio Casciola2 Serena Morigi2

1Department of Mathematics - University of Padova, Italy 2Department of Mathematics - University of Bologna, Italy

Subdivision surfaces

Widely supported in nearly all modeling programs

Advantages: flexibility for arbitrary topology + supersetof NURBS “standard”

A lot of theoretical study and many proposedalgorithms potentially useful in CAD, such as surfacefitting, reverse engineering, curve lofting

Their presence in CAD is still negligible due to:• lack of closed-form representation (difficulty of

integration into the system)• quality and regularity issues around extraordinary

points

IBut CAD end-users ask for them!

Project NIIT4CAD

The main objective of the European project NIIT4CADis an effective integration of Catmull-Clark subdivisionsurfaces in a CAD system, which means that:• the desired accuracy is achieved• all the functionalities of the CAD system are inherited

Approach :•Suitable local correction of Catmull-Clark surface,

capable of guaranteeing the required analyticaccuracy•Seamless integration into the modeling workflow,

exploiting an extensible geometric kernel and thepossibility of handling a hybrid boundaryrepresentation

CAD system paradigm

Our integration approach relies on a CAD systembased on an extensible geometric kernel, whichmeans:•Parametrization and evaluation paradigm : any

geometric entity, represented in parametric form, canbe integrated by simply implementing the relatedevaluation algorithm, that acts as an interface with theentire system. In this way, all the functionality of theCAD system are automatically inherited by any newlyintroduced type of representation•Extended boundary representation (B-rep), which

allows non-solid model types, so as to integratesurface modeling into a solid modeling environment(hybrid CAD system)•Hybrid description of the geometry in the B-rep, that

is allowing for the coexistence of different forms ofgeometric representation in the same model

Integration through the Subd B-rep

A Subd B-rep represents a B-rep geometric model inwhich:• the B-rep topology is inferred from the subdivision

control mesh• the geometry of each B-rep face is described as a

subdivision surface patch associated with arectangular parametric domain

Control mesh ←→ Subd B-rep

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

←→

The Subd B-rep maintains an intuitive associa-tion (1-1 almost everywhere) between the controlmesh faces and the B-rep faces

• Geometry: 1 face ←→ 1 parametricpatch Si onQ := [0, 1]2

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

Q ψ Si

(b) Quad face with more than one e.v.

Q0 Q1

Q2Q3

q0 q1

q2q3

φ` Qψ

Si

Si,0 Si,1

Si,2Si,3

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

Q

ψ andscale derivatives Si

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

Q0 Q1

Q2Q3

q0 q1

q2q3

φ` Q

ψ andscale

derivativesSi

where:ψ Catmull-Clark evaluator [4]φ` : Q` −→ Q, φ` := σ2 ◦ ρ

−1`π2◦ τq`−q0

σh scaling by a factor hρ−1

a c.w. rotation around the origin of an angle aτv translation of a vector v

Local correction around extraordinary points

We aim at maintaining the nice shape and B-splinenature of Catmull-Clark surfaces in the widest possiblearea, while improving the analytic properties in thesmallest neighborhood of e.v.

Polynomial blending : the Catmull-Clark surface islocally blended with a suitable polynomial surface [3, 5]

S∗i := w Si + (1− w) P

blending regions

S∗i w Si P

blendedsurface

weightfunction

Catmull-Clarksurface

approximatingpolynomial

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

Our approach generates a parametric surfaceevaluable at arbitrary points

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

•We represent the characteristic map of the subdivisionscheme for valence n as a parametric multipatchsurface ψK[n]

i: Q −→ K[n]

i , i = 0, . . . ,n (n sectors withrotational symmetry)

•Star-shaped transformationκi := ρi 2π

n◦ ψK[n]

0◦ σh ◦ φ

where:

φ(u, v) :=

{

(u, v) patch type (a), (c.1)

φ`(u, v) patch type (b), (c.2)

σh scaling of h = 4(a),2(b),2(c.1), 1(c.2)

ψK[n]

0: Qh

∣

∣

Q −→ K[n]0 , Qh := [0,h]2

ρi 2πn

c.c.w. rotation of angle i 2πn

0 01 112

h

0 1λ[n]

u

v

s

st

t

Q Qh

Qh∣

∣

Q

Si

K[n]0

Dn

Dn

σh ◦ φ

ψSi

ψK[n]0

ρi 2πn

κi

•Star-shaped domain

Kn :=

n−1⋃

i=0

{

(s, t) := κi (u, v)∣

∣

∣(u, v) ∈ Q and σh(φ(u, v)) ∈ Q

}

•Blending regionDn :=

{

(s, t) ∈ Kn

∣

∣

∣‖(s, t)‖2 6 λ[n]

}

where λ[n] = subdominant eigenvalue of the subdivision matrix

•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) elsewhere

The approximating polynomialP is determined by least squarefitting inside the blending regionwith interpolation of the e.v.

Practical modeling

•Workflow for the creation and editing of a Subd B-rep(applies to those faces of the hybrid B-rep model thatare of subdivision type)

•Operations of solid composition generate a B-repwhose faces can have hybrid nature (e.g. NURBS +subdivision) and are editable while maintaining thisfeature

•Hybrid model and quality assessment w.r.t.Catmull-Clark surface

meancurvature

Catmull-Clark local correction

isophotes

(out of scale)

References

[1] ANTONELLI M., BECCARI C. V., CASCIOLA G., CIARLONI R.,MORIGI S.: Subdivision surfaces integrated in a CAD system.Comput. Aided Des. (2013). doi:10.1016/j.cad.2013.06.007.

[2] BECCARI C. V., FARELLA E., LIVERANI A., MORIGI S., RUCCI

M.: A fast interactive reverse-engineering system. Comput. AidedDes. 42, 10 (2010), 860–873.

[3] LEVIN A.: Modified subdivision surfaces with continuouscurvature. In Proc. SIGGRAPH ’06 (2006), pp. 1035–1040.

[4] STAM J.: Exact evaluation of Catmull-Clark subdivision surfacesat arbitrary parameter values. In Proc. SIGGRAPH ’98 (1998),pp. 395–404.

[5] ZORIN D.: Constructing curvature-continuous surfaces byblending. In Proc. Eurographics SGP ’06 (2006), pp. 31–40.

Eurostars project New Interactive and Innovative Technolo gies for CAD (2010–2013)