5,820 research outputs found
Density-equalizing maps for simply-connected open surfaces
In this paper, we are concerned with the problem of creating flattening maps
of simply-connected open surfaces in . Using a natural principle
of density diffusion in physics, we propose an effective algorithm for
computing density-equalizing flattening maps with any prescribed density
distribution. By varying the initial density distribution, a large variety of
mappings with different properties can be achieved. For instance,
area-preserving parameterizations of simply-connected open surfaces can be
easily computed. Experimental results are presented to demonstrate the
effectiveness of our proposed method. Applications to data visualization and
surface remeshing are explored
Explosive rigidity percolation in kirigami
Controlling the connectivity and rigidity of kirigami, i.e. the process of
cutting paper to deploy it into an articulated system, is critical in the
manifestations of kirigami in art, science and technology, as it provides the
resulting metamaterial with a range of mechanical and geometric properties.
Here we combine deterministic and stochastic approaches for the control of
rigidity in kirigami using the power of choices, an approach borrowed from
the statistical mechanics of explosive percolation transitions. We show that
several methods for rigidifying a kirigami system by incrementally changing
either the connectivity or the rigidity of individual components allow us to
control the nature of the explosive transition by a choice of selection rules.
Our results suggest simple lessons for the design and control of mechanical
metamaterials
Efficient conformal parameterization of multiply-connected surfaces using quasi-conformal theory
Conformal mapping, a classical topic in complex analysis and differential
geometry, has become a subject of great interest in the area of surface
parameterization in recent decades with various applications in science and
engineering. However, most of the existing conformal parameterization
algorithms only focus on simply-connected surfaces and cannot be directly
applied to surfaces with holes. In this work, we propose two novel algorithms
for computing the conformal parameterization of multiply-connected surfaces. We
first develop an efficient method for conformally parameterizing an open
surface with one hole to an annulus on the plane. Based on this method, we then
develop an efficient method for conformally parameterizing an open surface with
holes onto a unit disk with circular holes. The conformality and
bijectivity of the mappings are ensured by quasi-conformal theory. Numerical
experiments and applications are presented to demonstrate the effectiveness of
the proposed methods
Free-boundary conformal parameterization of point clouds
With the advancement in 3D scanning technology, there has been a surge of
interest in the use of point clouds in science and engineering. To facilitate
the computations and analyses of point clouds, prior works have considered
parameterizing them onto some simple planar domains with a fixed boundary shape
such as a unit circle or a rectangle. However, the geometry of the fixed shape
may lead to some undesirable distortion in the parameterization. It is
therefore more natural to consider free-boundary conformal parameterizations of
point clouds, which minimize the local geometric distortion of the mapping
without constraining the overall shape. In this work, we develop a
free-boundary conformal parameterization method for disk-type point clouds,
which involves a novel approximation scheme of the point cloud Laplacian with
accumulated cotangent weights together with a special treatment at the boundary
points. With the aid of the free-boundary conformal parameterization,
high-quality point cloud meshing can be easily achieved. Furthermore, we show
that using the idea of conformal welding in complex analysis, the point cloud
conformal parameterization can be computed in a divide-and-conquer manner.
Experimental results are presented to demonstrate the effectiveness of the
proposed method
- …