ABSTRACT
Following the definition of developable surface in differential ge-
ometry, the flattenable mesh surface, a special type of piecewise-
linear surface, inherits the good property of developable surface
abouthavinganisometricmapfromits3Dshapetoacorresponding
planar region. Differentfromthe developable surfaces, aflattenable
mesh surface is more flexible to model objects with complex shapes
(e.g., cramped paper or warped leather with wrinkles). Modelling
a flattenable mesh from a given input mesh surface can be com-
pleted under a constrained nonlinear optimization framework. In
this paper, we reformulate the problem in terms of estimation er-
ror. Therefore, the shape of a flattenable mesh can be computed by
the least-norm solutions faster. Moreover, the method for adding
shape constraints to the modelling of flattenable mesh surfaces has
been exploited. We show that the proposed method can compute
flattenable mesh surfaces from input piecewise linear surfaces suc-
cessfully and efficiently.
Index Terms: I.3.5 [Computational Geometry and Object Model-
ing]: Curve, surface, solid, and object representations—Physically
based modeling; J.6 [COMPUTER-AIDED ENGINEERING]:
Computer-aided design (CAD)—Computer-aided design (CAD)
1 INTRODUCTION
In sheet manufacturing industries, the products are fabricated from
two-dimensional patterns of sheet materials (e.g., metal in ship in-
dustry, fabric in apparel industry and toy industry, and leather in
shoe industry and furniture industry). The final products are fabri-
cated by warping and stitching 2D patterns together. The traditional
design process in these industries is conducted in a trial-and-error
manner. A designer will draft 2D pieces on a paper and then make
a prototype to check whether the fitting is good. If the result is not
satisfactory, the designer needs to modify the patterns by his expe-
rience and make another prototype. The prototyping and the mod-
ification steps will be applied repeatedly, which is very inefficient.
Designers in these industries wish to have a geometric modelling
tool to model the products by surfaces that can be flattened into 2D
pieces without stretching, i.e., holding an isometric map to some
2D regions. In the rest of the paper, we simply call it the stretch-
free flattening property. The well-known developable surface in
differential geometry [10] inherits such an elegant property. The
form. of developable surfaces could be planes, generalized cylin-
ders, conical surfaces (away from the apex), or tangent developable
surfaces. However, the shape of products in practice could be more
complex (e.g., as shown in Fig.1), which can hardly be modelled
by the conventional developable surfaces. The approach presented
in this paper provides an efficient method to deform. a user-defined
3D piecewise linear surface S into a new mesh surface M that will
induce minimized distortion error in flattening.
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Figure 1: Products made from 2D sheets. Their shape can hardly be
modelled by the conventional developable surfaces.
1.1 Problem definition
Without lose of generality, we assume that the user-defined 3D
piecewise linear surface S is with the disk-like topology. This is
because only the surface patches with such a topology can easily
be fabricated from 2D pieces. For example, without adding a cut
between the two loops, generalized cylinders cannot be made from
a piece of sheet material. Here, there are two general requirements
for the mesh surface M deformed from S:
• M is a mesh surface that shows the stretch-free flattening
property;
• The difference between M and S is minimized.
A mesh surface satisfying the stretch-free flattening property is
named as a flattenable mesh surface. The difference between two
mesh surface M and S can be measured by some shape error (e.g.,
[21]). Therefore, the problem we are going to solve is to find a flat-
tenable mesh surface M in ℜ3 to approximate the input piecewise
linear surface S. Note that this is different from mesh surface pa-
rameterization [11] in computer graphics, where the mesh is com-
putedinℜ2 sothattheshapedeformationin3Dcannotbeexplicitly
controlled.
1.2 Related work
The mesh processing method for flattenable mesh surface relates to
the developable surface in differential geometry [10]. In general, a
surface is developable if and only if the Gaussian curvature of ev-
ery surface point is zero. To satisfy this, there are many approaches
modelling [7, 16, 28] or approximating [6, 26, 27] a model with de-
velopable ruled surfaces (or ruled surfaces in other representations
– e.g., B-spline or B´ezier patches). However, it is difficult to use
these approaches to model freeform. surfaces.
Julius et al. developed an algorithm in [12] to separate a given
model into quasi-conical proxies. Based on a similar idea, in [8]
the authors processed a given mesh surface instead of segmenting
it, where a deformation process is applied to let the surface locally
approximate a conical surface. It is a sufficient (but not necessary)
condition for a conical mesh surface to be flattenable. In other