This paper addresses the problem of fitting flattenable mesh surfaces in R3onto piecewise linear boundary curves, where a flattenable mesh surface inherits the isometric mapping to a planar region in R2. The developable surface in differential geometry shows the nice property. However, it is difficult to fit developable surfaces to a boundary with complex shape. The technique presented in this paper can model a piecewise linear flattenable surface that interpolates the given boundary curve and approximates the cross-tangent normal vectors on the boundary. At first, an optimal planar polygonal region is computed from the given boundary curve BR3, triangulated into a planar mesh surface, and warped into a mesh surface in R3, satisfying the continuities defined on B. Then, the fitted mesh surface is further optimized into a flattenable Laplacian (FL) mesh, which preserves the positional continuity and minimizes the variation of cross-tangential normals. Assembled set of such FL mesh patches can be employed to model complex products fabricated from sheets without stretching.

1.
Wang
,
C.
, 2008, “
Towards Flattenable Mesh Surfaces
,”
CAD
0010-4485,
40
(
1
), pp.
109
122
.
2.
Meyer
,
M.
,
Desbrun
,
M.
,
Schroder
,
P.
, and
Barr
,
A.
, 2002, “
Discrete Differential-Geometry Operators for Triangulated 2-Manifolds
,”
Proceeding of Visualization and Mathematics
.
3.
Julius
,
D.
,
Kraevoy
,
V.
, and
Sheffer
,
A.
, 2005, “
D-Charts: Quasi-Developable Mesh Segmentation
,”
Comput. Graph. Forum
1067-7055,
24
, pp.
581
590
.
4.
Liu
,
Y.
,
Pottmann
,
H.
,
Wallner
,
J.
,
Yang
,
Y.-L.
, and
Wang
,
W.
, 2006, “
Geometric Modeling With Conical Meshes and Developable Surfaces
,”
ACM Trans. Graphics
0730-0301,
25
(
3
), pp.
681
689
.
5.
Wang
,
C.
, and
Tang
,
K.
, 2004, “
Achieving Developability of a Polygonal Surface by Minimum Deformation: A Study of Global and Local Optimization Approaches
,”
Visual Comput.
0178-2789,
20
, pp.
521
539
.
6.
Desbrun
,
M.
,
Meyer
,
M.
, and
Alliez
,
P.
, 2002, “
Intrinsic Parameterizations of Surface Meshes
,”
Comput. Graph. Forum
1067-7055,
21
(
3
), pp.
209
218
.
7.
Karni
,
Z.
,
Gotsman
,
C.
, and
Gortler
,
S.
, 2005, “
Free-Boundary Linear Parameterization of 3d Meshes in the Presence of Constraints
,”
Proceedings of Shape Modeling International
, pp.
266
275
.
8.
Lee
,
Y.
,
Kim
,
H.-S.
, and
Lee
,
S.
, 2002, “
Mesh Parameterization With a Virtual Boundary
,”
Comput. Graphics
0097-8493,
26
, pp.
677
686
.
9.
Lévy
,
B.
,
Petitjean
,
S.
,
Ray
,
N.
, and
Maillot
,
J.
, 2002, “
Least Squares Conformal Maps for Automatic Texture Atlas Generation
,”
ACM Trans. Graphics
0730-0301,
21
(
3
), pp.
362
371
.
10.
Sheffer
,
A.
,
Lévy
,
B.
,
Mogilnitsky
,
M.
, and
Bogomjakov
,
A.
, 2005, “
Abf++: Fast and Robust Angle Based Flattening
,”
ACM Trans. Graphics
0730-0301,
24
(
2
), pp.
311
330
.
11.
Sheffer
,
A.
, and
de Sturler
,
E.
, 2001, “
Parameterization of Faceted Surfaces for Meshing Using Angle Based Flattening
,”
Eng. Comput.
0177-0667,
17
(
3
), pp.
326
337
.
12.
Azariadis
,
P.
, and
Aspragathos
,
N.
, 1997, “
Design of Plane Development of Doubly Curved Surface
,”
Comput.-Aided Des.
0010-4485,
29
, pp.
675
685
.
13.
Aono
,
M.
,
Breen
,
D.
, and
Wozny
,
M.
, 2001, “
Modeling Methods for the Design of 3d Broadcloth Composite Parts
,”
Comput.-Aided Des.
0010-4485,
33
, pp.
989
1007
.
14.
Aono
,
M.
,
Breen
,
D.
, and
Wozny
,
M.
, 1994, “
Fitting a Woven-Cloth Model to a Curved Surface: Mapping Algorithms
,”
Comput.-Aided Des.
0010-4485,
26
, pp.
278
292
.
15.
McCartney
,
J.
,
Hinds
,
B.
, and
Seow
,
B.
, 1999, “
The Flattening of Triangulated Surfaces Incorporating Darts and Gussets
,”
Comput.-Aided Des.
0010-4485,
31
, pp.
249
260
.
16.
Wang
,
C.
,
Smith
,
S.
, and
Yuen
,
M.
, 2002, “
Surface Flattening Based on Energy Model
,”
Comput.-Aided Des.
0010-4485,
34
(
11
), pp.
823
833
.
17.
Wang
,
C.
,
Tang
,
K.
, and
Yeung
,
B.
, 2005, “
Freeform Surface Flattening by Fitting a Woven Mesh Model
,”
Comput.-Aided Des.
0010-4485,
37
, pp.
799
814
.
18.
Decaudin
,
P.
,
Julius
,
D.
,
Wither
,
J.
,
Boissieux
,
L.
,
Sheffer
,
A.
, and
Cani
,
M.-P.
, 2005, “
Virtual Garments: A Fully Geometric Approach for Clothing Design
,”
Comput. Graph. Forum
1067-7055,
25
(
3
), pp.
625
634
.
19.
do Carmo
,
M.
, 1976,
Differential Geometry of Curves and Surfaces
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
20.
Leopoldseder
,
S.
, and
Pottmann
,
H.
, 1998, “
Approximation of Developable Surfaces With Cone Spline Surfaces
,”
Comput.-Aided Des.
0010-4485,
30
, pp.
571
582
.
21.
Pottmann
,
H.
, and
Wallner
,
J.
, 1999, “
Approximation Algorithms for Developable Surfaces
,”
Comput. Aided Geom. Des.
0167-8396,
16
, pp.
539
5562
.
22.
Chu
,
C.
, and
Séquin
,
C.
, 2002, “
Developable Bézier Patches: Properties and Design
,”
Coron. Artery Dis.
0954-6928,
34
(
7
), pp.
511
527
.
23.
Chen
,
H.-Y.
,
Lee
,
I.-K.
,
Leopoldseder
,
S.
,
Pottmann
,
H.
,
Randrup
,
T.
, and
Wallner
,
J.
, 1999, “
On Surface Approximation Using Developable Surfaces
,”
Graph. Models Image Process.
1077-3169,
61
, pp.
110
124
.
24.
Peternell
,
M.
, 2004, “
Recognition and Reconstruction of Developable Surfaces From Point Clouds
,”
Proceedings of Geometric Modeling and Processing 2004
, pp.
301
310
.
25.
Peternell
,
M.
, and
Steiner
,
T.
, 2004, “
Reconstruction of Piecewise Planar Objects From Point Clouds
,”
Comput.-Aided Des.
0010-4485,
36
, pp.
333
342
.
26.
Wang
,
C.
,
Wang
,
Y.
, and
Yuen
,
M.
, 2004, “
On Increasing the Developability of a Trimmed Nurbs Surface
,”
Eng. Comput.
0177-0667,
20
(
1
), pp.
54
64
.
27.
Mortenson
,
M.
, 1997,
Geometric Modeling
,
2nd ed.
Wiley
,
New York
.
28.
Madsen
,
K.
,
Nielsen
,
H.
, and
Tingleff
,
O.
, 2004,
Optimization With Constraints, Lecture Notes
.
29.
Chew
,
L.
, 1987, “
Constrained Delaunay Triangulations
,”
Proceedings of the Third Annual Symposium on Computational Geometry Table of Contents
, pp.
215
222
.
30.
Botsch
,
M.
, and
Kobbelt
,
L.
, 2005, “
Real-Time Shape Editing Using Radial Basis Functions
,”
Comput. Graph. Forum
1067-7055,
24
(
3
), pp.
611
621
.
31.
Turk
,
G.
, and
O’Brien
,
J.
, 2002, “
Modelling With Implicit Surfaces that Interpolate
,”
ACM Trans. Graphics
0730-0301,
21
(
4
), pp.
855
873
.
32.
Yngve
,
G.
, and
Turk
,
G.
, 2002, “
Robust Creation of Implicit Surfaces from Polygonal Meshes
,”
IEEE Trans. Vis. Comput. Graph.
1077-2626,
8
(
4
), pp.
346
359
.
33.
Sorkine
,
O.
, 2006, “
Differential Representations for Mesh Processing
,”
Comput. Graph. Forum
1067-7055,
25
(
4
), pp.
789
807
.
34.
Chen
,
D.
,
Cohen-Or
,
D.
,
Sorkine
,
O.
, and
Toledo
,
S.
, 2005, “
Algebraic Analysis of High-Pass Quantization
,”
ACM Trans. Graphics
0730-0301,
24
(
4
), pp.
1259
1282
.
35.
Sorkine
,
O.
,
Cohen-Or
,
D.
,
Irony
,
D.
, and
Toledo
,
S.
, 2005, “
Geometry-Aware Bases for Shape Approximation
,”
IEEE Trans. Vis. Comput. Graph.
1077-2626,
11
(
2
), pp.
171
180
.
36.
Sorkine
,
O.
,
Cohen-Or
,
D.
,
Lipman
,
Y.
,
Alexa
,
M.
,
Rossl
,
C.
, and
Seidel
,
H.-P.
, 2004, “
Laplacian Surface Editing
,”
Eurographics/ACM SIGGRAPH Symposium on Geometry Processing 2004
, pp.
179
188
.
37.
Sorkine
,
O.
, and
Cohen-Or
,
D.
, 2004, “
Least-Squares Meshes
,”
Proceedings of Shape Modeling International 2004
, pp.
191
199
.
38.
Nocedal
,
J.
, and
Wright
,
S.
, 1999,
Numerical Optimization
,
Springer-Verlag
,
Berlin
.
39.
Li
,
S.
,
Demmel
,
J.
, and
Gilbert
,
J.
, 2006, SuperLU, http://crd.lbl.gov/xiaoye/SuperLU/http://crd.lbl.gov/xiaoye/SuperLU/, Feb.
40.
Kobbelt
,
L.
, 2000, “
3-Subdivision
,”
Proceedings of SIGGRAPH 2000
, pp.
103
112
.
You do not currently have access to this content.