Subdivision surfaces for CAD: integration through ...antonelm/talks/Milano2012.pdf · difficulty...
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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
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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
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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
. . . But CAD end-users ask for them!
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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
. . . But CAD end-users ask for them!
2010–2013: Eurostars Project NIIT4CAD(New Interactive and Innovative Technologies for CAD)
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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Objective• Seamless integration of subdivision surfaces in a CAD system
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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Objective• Seamless integration of subdivision surfaces in a CAD systemg
Catmull-Clark
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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Objective• Seamless integration of subdivision surfaces in a CAD systemg
Catmull-Clark
• Main roadblocks:lack of quality and precisiondifficulty of integration into the modeling workflow
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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Objective• Seamless integration of subdivision surfaces in a CAD systemg
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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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SS seamlessly integrated in our system
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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SS seamlessly integrated in our system
Meancurvature
<−0.2035 > 0.07465<−0.2053
> 0.07465
Catmull-Clark Local correction
Isophotes
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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SS seamlessly integrated in our system
Meancurvature
<−0.2035 > 0.07465<−0.2053
> 0.07465
Catmull-Clark Local correction
Isophotes
Meancurvature
<−0.2053
> 0.07465
Catmull-Clark Local correction
Isophotes
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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.
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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
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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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Surface tuningObjective: local correction of quality issues
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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.
Idea: local correction through polynomial blending (A. Levin, 2006; Zorin, 2006)
S∗i := w Si +(1−w) P
Catmull-Clark Local correction
blendingregion
S∗i w Si P
blended surface
weight function
Catmull-Clark surface
approximating polynomial
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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.
Idea: local correction through polynomial blending (A. Levin, 2006; Zorin, 2006)
S∗i := w Si +(1−w) P
Catmull-Clark Local correction
blendingregion
S∗i w Si P
blended surface
weight function
Catmull-Clark surface
approximating polynomial
• 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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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.
Idea: local correction through polynomial blending (A. Levin, 2006; Zorin, 2006)
S∗i := w Si +(1−w) P
Catmull-Clark Local correction
blendingregion
S∗i w Si P
blended surface
weight function
Catmull-Clark surface
approximating polynomial
• 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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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A common parameterization domain
• Si,P,w must be parameterized over a common domain
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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A common parameterization 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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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A common parameterization 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
• 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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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A common parameterization 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
• 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
0 1u
v
Q
Si
ψSi
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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A common parameterization 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
• 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
0 01 hu
v
Q Qh
Si
σh ◦φ
ψSi
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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A common parameterization 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
• 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
0 01 1 h
0 1
u
v
s
t
Q Qh
Qh∣∣
Q
Si
K[n]0
σh ◦φ
ψSi
ψK[n]0
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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A common parameterization 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
• 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
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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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A common parameterization 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
• 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
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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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A common parameterization 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
• 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
• 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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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A common parameterization 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
• 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
• Star-shaped domain
Kn :=n−1⋃
i=0
{
κi (u,v)∣
∣
∣(u,v) ∈ Q and σh(φ(u,v)) ∈ Q
}
• 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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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)
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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)
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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
• V TV c =V T (p−pev) =⇒ precompute and store(
V TV)−1
V T
for each valence n
12 uniformly distributedapproximation pointsper sector in D̃n,i ⊂ Q
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
![Page 41: Subdivision surfaces for CAD: integration through ...antonelm/talks/Milano2012.pdf · difficulty of integration into the modeling workflow Seamless integration • The desired accuracy](https://reader034.fdocuments.net/reader034/viewer/2022051608/603e42f00d596675cb63cddf/html5/thumbnails/41.jpg)
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
• 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)))
∣
∣
∣(u,v) ∈ Q and σh(φ(u,v)) ∈ Q
}
12 uniformly distributedapproximation pointsper sector in D̃n,i ⊂ Q
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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Handling of heterogeneous rep.• Workflow for the creation and editing of a Subd-B-rep
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Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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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
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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Validation fromIntegration between conceptual design and engineering phase
Subd-B-rep solid
Model with thickness Division of the object in two parts
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi
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Thankyou!
Subdivision surfaces integrated in a CAD system M. Antonelli, C. Beccari, G. Casciola, S. Morigi