29,244 research outputs found
An Intuitive Approach to Geometric Continuity for Parametric Curves and Surfaces (Extended Abstract)
The notion of geometric continuity is extended to an arbitrary order for curves and surfaces, and an intuitive development of constraints equations is presented that are necessary for it. The constraints result from a direct application of the univariate chain rule for curves, and the bivariate chain rule for surfaces. The constraints provide for the introduction of quantities known as shape parameters. The approach taken is important for several reasons: First, it generalizes geometric continuity to arbitrary order for both curves and surfaces. Second, it shows the fundamental connection between geometric continuity of curves and geometric continuity of surfaces. Third, due to the chain rule derivation, constraints of any order can be determined more easily than derivations based exclusively on geometric measures
Recommended from our members
Smooth parametric surfaces and n-sided patches
The theory of 'geometric continuity' within the subject of CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an n-sided hole occurring within a smooth rectangular patch complex. A number of solutions to this problem are surveyed
Degenerate polynomial patches of degree 4 and 5 used for geometrically smooth interpolation in R3
The problem of interpolating scattered 3D data by a geometrically smooth surface is considered. A completely local method is proposed, based on employing degenerate triangular Bernstein-Bézier patches. An analysis of these patches is given and some numerical experiments with quartic and quintic patches are presented
Circle and sphere blending with conformal geometric algebra
Blending schemes based on circles provide smooth `fair' interpolations
between series of points. Here we demonstrate a simple, robust set of
algorithms for performing circle blends for a range of cases. An arbitrary
level of G-continuity can be achieved by simple alterations to the underlying
parameterisation. Our method exploits the computational framework provided by
conformal geometric algebra. This employs a five-dimensional representation of
points in space, in contrast to the four-dimensional representation typically
used in projective geometry. The advantage of the conformal scheme is that
straight lines and circles are treated in a single, unified framework. As a
further illustration of the power of the conformal framework, the basic idea is
extended to the case of sphere blending to interpolate over a surface.Comment: 20 pages, 13 figure
Various Types of Aesthetic Curves
The research on developing planar curves to produce visually pleasing
products (ranges from electric appliances to car body design) and
indentifying/modifying planar curves for special purposes namely for railway
design, highway design and robot trajectories have been progressing since
1970s. The pattern of research in this field of study has branched to five
major groups namely curve synthesis, fairing process, improvement in control of
natural spiral, construction of new type of planar curves and, natural spiral
fitting & approximation techniques. The purpose of is this paper is to briefly
review recent progresses in Computer Aided Geometric Design (CAGD) focusing on
the topics states above
Constructing Trivariate B-splines with Positive Jacobian by Pillow Operation and Geometric Iterative Fitting
The advent of isogeometric analysis has prompted a need for methods to
generate Trivariate B-spline Solids (TBS) with positive Jacobian. However, it
is difficult to guarantee a positive Jacobian of a TBS since the geometric
pre-condition for ensuring the positive Jacobian is very complicated. In this
paper, we propose a method for generating TBSs with guaranteed positive
Jacobian. For the study, we used a tetrahedral (tet) mesh model and segmented
it into sub-volumes using the pillow operation. Then, to reduce the difficulty
in ensuring a positive Jacobian, we separately fitted the boundary curves and
surfaces and the sub-volumes using a geometric iterative fitting algorithm.
Finally, the smoothness between adjacent TBSs is improved. The experimental
examples presented in this paper demonstrate the effectiveness and efficiency
of the developed algorithm
Additive continuity of the renormalized volume under geometric limits
We study the infimum of the renormalized volume for convex-cocompact
hyperbolic manifolds, as well as describing how a sequence converging to such
values behaves. In particular, we show that the renormalized volume is
continuous under the appropriate notion of limit. This result generalizes
previous work in the subject.Comment: 24 pages, 5 figure
TiGL - An Open Source Computational Geometry Library for Parametric Aircraft Design
This paper introduces the software TiGL: TiGL is an open source high-fidelity
geometry modeler that is used in the conceptual and preliminary aircraft and
helicopter design phase. It creates full three-dimensional models of aircraft
from their parametric CPACS description. Due to its parametric nature, it is
typically used for aircraft design analysis and optimization. First, we present
the use-case and architecture of TiGL. Then, we discuss it's geometry module,
which is used to generate the B-spline based surfaces of the aircraft. The
backbone of TiGL is its surface generator for curve network interpolation,
based on Gordon surfaces. One major part of this paper explains the
mathematical foundation of Gordon surfaces on B-splines and how we achieve the
required curve network compatibility. Finally, TiGL's aircraft component module
is introduced, which is used to create the external and internal parts of
aircraft, such as wings, flaps, fuselages, engines or structural elements
Continuity of the renormalized volume under geometric limits
We extend the concept of renormalized volume for geometrically finite
hyperbolic -manifolds, and show that is continuous for geometrically
convergent sequences of hyperbolic structures over an acylindrical 3-manifold
with geometrically finite limit. This allows us to show that the
renormalized volume attains its minimum (in terms of the conformal class at
) at the geodesic class, the conformal class for which the
boundary of the convex core is totally geodesic.Comment: 13 page
On the Complexity of Smooth Spline Surfaces from Quad Meshes
This paper derives strong relations that boundary curves of a smooth complex
of patches have to obey when the patches are computed by local averaging. These
relations restrict the choice of reparameterizations for geometric continuity.
In particular, when one bicubic tensor-product B-spline patch is associated
with each facet of a quadrilateral mesh with n-valent vertices and we do not
want segments of the boundary curves forced to be linear, then the relations
dictate the minimal number and multiplicity of knots: For general data, the
tensor-product spline patches must have at least two internal double knots per
edge to be able to model a G^1-conneced complex of C^1 splines. This lower
bound on the complexity of any construction is proven to be sharp by suitably
interpreting an existing surface construction. That is, we have a tight bound
on the complexity of smoothing quad meshes with bicubic tensor-product B-spline
patches
- …