4,784 research outputs found
Smooth quasi-developable surfaces bounded by smooth curves
Computing a quasi-developable strip surface bounded by design curves finds
wide industrial applications. Existing methods compute discrete surfaces
composed of developable lines connecting sampling points on input curves which
are not adequate for generating smooth quasi-developable surfaces. We propose
the first method which is capable of exploring the full solution space of
continuous input curves to compute a smooth quasi-developable ruled surface
with as large developability as possible. The resulting surface is exactly
bounded by the input smooth curves and is guaranteed to have no
self-intersections. The main contribution is a variational approach to compute
a continuous mapping of parameters of input curves by minimizing a function
evaluating surface developability. Moreover, we also present an algorithm to
represent a resulting surface as a B-spline surface when input curves are
B-spline curves.Comment: 18 page
Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization
In this paper, we propose a general framework for constructing IGA-suitable
planar B-spline parameterizations from given complex CAD boundaries consisting
of a set of B-spline curves. Instead of forming the computational domain by a
simple boundary, planar domains with high genus and more complex boundary
curves are considered. Firstly, some pre-processing operations including
B\'ezier extraction and subdivision are performed on each boundary curve in
order to generate a high-quality planar parameterization; then a robust planar
domain partition framework is proposed to construct high-quality patch-meshing
results with few singularities from the discrete boundary formed by connecting
the end points of the resulting boundary segments. After the topology
information generation of quadrilateral decomposition, the optimal placement of
interior B\'ezier curves corresponding to the interior edges of the
quadrangulation is constructed by a global optimization method to achieve a
patch-partition with high quality. Finally, after the imposition of
C1=G1-continuity constraints on the interface of neighboring B\'ezier patches
with respect to each quad in the quadrangulation, the high-quality B\'ezier
patch parameterization is obtained by a C1-constrained local optimization
method to achieve uniform and orthogonal iso-parametric structures while
keeping the continuity conditions between patches. The efficiency and
robustness of the proposed method are demonstrated by several examples which
are compared to results obtained by the skeleton-based parameterization
approach
A Survey of the Differential Geometry of Discrete Curves
Discretization of curves is an ancient topic. Even discretization of curves
with an eye toward differential geometry is over a century old. However there
is no general theory or methodology in the literature, despite the ubiquitous
use of discrete curves in mathematics and science. There are conflicting
definitions of even basic concepts such as discrete curvature {\kappa},
discrete torsion {\tau}, or discrete Frenet frame.Comment: 19 pages, 16 figure
A survey of partial differential equations in geometric design
YesComputer aided geometric design is an area
where the improvement of surface generation techniques
is an everlasting demand since faster and more accurate
geometric models are required. Traditional methods
for generating surfaces were initially mainly based
upon interpolation algorithms. Recently, partial differential
equations (PDE) were introduced as a valuable
tool for geometric modelling since they offer a number
of features from which these areas can benefit. This work
summarises the uses given to PDE surfaces as a surface
generation technique togethe
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3D ball skinning using PDEs for generation of smooth tubular surfaces
We present an approach to compute a smooth, interpolating skin of an ordered set of 3D balls. By construction, the skin is constrained to be C-1 continuous, and for each ball, it is tangent to the ball along a circle of contact. Using an energy formulation, we derive differential equations that are designed to minimize the skin's surface area, mean curvature, or convex combination of both. Given an initial skin, we update the skin's parametric representation using the differential equations until convergence occurs. We demonstrate the method's usefulness in generating interpolating skins of balls of different sizes and in various configurations
3D ball skinning using PDEs for generation of smooth tubular surfaces
We present an approach to compute a smooth, interpolating skin of an ordered set of
3D balls. By construction, the skin is constrained to be C1 continuous, and for each
ball, it is tangent to the ball along a circle of contact. Using an energy formulation,
we derive differential equations that are designed to minimize the skinās surface area,
mean curvature, or convex combination of both. Given an initial skin, we update the
skinās parametric representation using the differential equations until convergence
occurs. We demonstrate the methodās usefulness in generating interpolating skins
of balls of different sizes and in various configurations
A variational model for data fitting on manifolds by minimizing the acceleration of a B\'ezier curve
We derive a variational model to fit a composite B\'ezier curve to a set of
data points on a Riemannian manifold. The resulting curve is obtained in such a
way that its mean squared acceleration is minimal in addition to remaining
close the data points. We approximate the acceleration by discretizing the
squared second order derivative along the curve. We derive a closed-form,
numerically stable and efficient algorithm to compute the gradient of a
B\'ezier curve on manifolds with respect to its control points, expressed as a
concatenation of so-called adjoint Jacobi fields. Several examples illustrate
the capabilites and validity of this approach both for interpolation and
approximation. The examples also illustrate that the approach outperforms
previous works tackling this problem
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