26,381 research outputs found
Urn Models and Beta-splines
Some insight into the properties of beta-splines is gained by applying the techniques of urn models. Urn models are used to construct beta-spline basis functions and to derive the basic properties of these blending functions and the corresponding beta-spline curves. Only the simple notion of linear geometric continuity and with the most elementary beta parameter are outlined. Non-linear geometric continuity leads to additional beta parameters and to more complicated basis functions. Whether urn models can give us any insight into these higher order concepts still remains to be investigated
Query processing of geometric objects with free form boundarie sin spatial databases
The increasing demand for the use of database systems as an integrating
factor in CAD/CAM applications has necessitated the development of database
systems with appropriate modelling and retrieval capabilities. One essential
problem is the treatment of geometric data which has led to the development of
spatial databases. Unfortunately, most proposals only deal with simple geometric
objects like multidimensional points and rectangles. On the other hand, there has
been a rapid development in the field of representing geometric objects with free
form curves or surfaces, initiated by engineering applications such as mechanical
engineering, aviation or astronautics. Therefore, we propose a concept for the realization
of spatial retrieval operations on geometric objects with free form
boundaries, such as B-spline or Bezier curves, which can easily be integrated in
a database management system. The key concept is the encapsulation of geometric
operations in a so-called query processor. First, this enables the definition of
an interface allowing the integration into the data model and the definition of the
query language of a database system for complex objects. Second, the approach
allows the use of an arbitrary representation of the geometric objects. After a
short description of the query processor, we propose some representations for free
form objects determined by B-spline or Bezier curves. The goal of efficient query
processing in a database environment is achieved using a combination of decomposition
techniques and spatial access methods. Finally, we present some experimental
results indicating that the performance of decomposition techniques is
clearly superior to traditional query processing strategies for geometric objects
with free form boundaries
Enhancement of surface definition and gridding in the EAGLE code
Algorithms for smoothing of curves and surfaces for the EAGLE grid generation program are presented. The method uses an existing automated technique which detects undesirable geometric characteristics by using a local fairness criterion. The geometry entity is then smoothed by repeated removal and insertion of spline knots in the vicinity of the geometric irregularity. The smoothing algorithm is formulated for use with curves in Beta spline form and tensor product B-spline surfaces
Computing a Compact Spline Representation of the Medial Axis Transform of a 2D Shape
We present a full pipeline for computing the medial axis transform of an
arbitrary 2D shape. The instability of the medial axis transform is overcome by
a pruning algorithm guided by a user-defined Hausdorff distance threshold. The
stable medial axis transform is then approximated by spline curves in 3D to
produce a smooth and compact representation. These spline curves are computed
by minimizing the approximation error between the input shape and the shape
represented by the medial axis transform. Our results on various 2D shapes
suggest that our method is practical and effective, and yields faithful and
compact representations of medial axis transforms of 2D shapes.Comment: GMP14 (Geometric Modeling and Processing
Adaptive resolution of 1D mechanical B-spline
International audienceThis article presents an adaptive approach to B-spline curve physical simulation. We combine geometric refinement and coarsening techniques with an appropriate continuous mechanical model. We thus deal with the (temporal and geometric) continuity issues implied when mechanical adaptive resolution is used. To achieve real-time local adaptation of spline curves, some criteria and optimizations are shown. Among application examples, real-time knot tying is presented, and curve cutting is also pointed out as a nice sideeffect of the adaptive resolution animation framework
Görbék és felületek a geometriai modellezésben = Curves and surfaces in geometric modelling
B-spline görbĂ©k/felĂĽletek pontjai által, az alakzat kĂ©t csomóértĂ©kĂ©nek szimmetrikus változtatásakor leĂrt pályagörbĂ©ket vizsgáltuk, Ă©s olyan alakmĂłdosĂtási eljárást adtunk, amivel a felĂĽlet adott pontját/paramĂ©tervonalát elĹ‘re megadott helyre mozgathatjuk a csomóértĂ©kek változtatásával. A C-BĂ©zier, C-B-spline Ă©s F-B-spline görbĂ©k pályagörbĂ©inek geometriai tulajdonságait Ărtuk le, Ă©s erre alapozva geometriai kĂ©nyszereket kielĂ©gĂtĹ‘ alakmĂłdosĂtásokat vizsgáltuk. Olyan általános leĂrási mĂłdot (linear blending) adtunk, mely egysĂ©gesen kezeli az alakparamĂ©terekkel rendelkezĹ‘ görbĂ©k szĂ©les osztályát, továbbá konkrĂ©t esetekben e paramĂ©terek geometriai hatását Ărtuk le Ă©s kĂ©nyszeres alakmĂłdosĂtásokra adtunk megoldást. A csomóértĂ©keknek az interpolálĂł görbĂ©re gyakorolt hatását vizsgáltuk, mely alapján a harmadfokĂş interpoláciĂł esetĂ©re interaktĂv alakmĂłdosĂtĂł eljárást dolgoztunk ki. Kontrollpontokkal adott görbĂ©k szingularitásainak detektálására a kontrollpontok helyzetĂ©n alapulĂł megoldást adtunk. Kontrollpont alapĂş szĂĽksĂ©ges Ă©s elĂ©gsĂ©ges feltĂ©telt adtunk arra, hogy a BĂ©zier-felĂĽlet paramĂ©tervonalai egyenesek legyenek. Olyan Monte Carlo mĂłdszert dolgoztunk ki, amely rendezetlen ponthalmaz felĂĽlettel valĂł interpolálásához nĂ©gyszöghálĂłt hoz lĂ©tre a pontfelhĹ‘ (mely elágazásokat Ă©s hurkokat is tartalmazhat) Ă©s annak topolĂłgikus gráfja ismeretĂ©ben. A csonkolt Fourier-sorok terĂ©ben olyan ciklikus bázist adtunk meg, amellyel vĂ©gtelen simaságĂş zárt görbĂ©ket Ă©s felĂĽleteket Ărhatunk le. | We studied paths of points of B-spline curves/surfaces obtained by the symmetric alteration of two knot values and provided a constrained shape modification method that is capable of moving a point/isoparametric line of the surface to a user specified position. We described the geometric properties of paths of C-BĂ©zier, C-B-spline and F-B-spline curves and on this basis we studied shape modifications subject to geometric constraints. We developed the general linear blending method that treats a wide class of curves with shape parameters in a uniform way; in special cases we described the geometric effects of shape parameters and provided constrained shape modification methods. We examined the impact of knots on the shape of interpolating curves, based on which we developed an interactive shape modification method for cubic interpolation. We proposed a control point based solution to the problem of singularity detection of curves described by control points. We provided control point based necessary and sufficient conditions for BĂ©zier surfaces to have linear isoparametric lines. We developed a Monte Carlo method to generate a quadrilateral mesh (for surface interpolation) from point clouds (with possible junctions and loops) and their topological graph. We specified a cyclic basis in the space of truncated Fourier series by means of which we can describe closed curves and surfaces with C^infinity
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Automatic, computer aided geometric design of free-knot, regression splines
A new algorithm for Computer Aided Geometric Design of least squares (LS) splines with variable knots, named GeDS, is presented. It is based on interpreting functional spline regression as a parametric B-spline curve, and on using the shape preserving property of its control polygon. The GeDS algorithm includes two major stages. For the first stage, an automatic adaptive, knot location algorithm is developed. By adding knots, one at a time, it sequentially "breaks" a straight line segment into pieces in order to construct a linear LS B-spline fit, which captures the "shape" of the data. A stopping rule is applied which avoids both over and under fitting and selects the number of knots for the second stage of GeDS, in which smoother, higher order (quadratic, cubic, etc.) fits are generated. The knots appropriate for the second stage are determined, according to a new knot location method, called the averaging method. It approximately preserves the linear precision property of B-spline curves and allows the attachment of smooth higher order LS B-spline fits to a control polygon, so that the shape of the linear polygon of stage one is followed. The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline regression functions. The GeDS algorithm is very fast, since no deterministic or stochastic knot insertion/deletion and relocation search strategies are involved, neither in the first nor the second stage. Extensive numerical examples are provided, illustrating the performance of GeDS and the quality of the resulting LS spline fits. The GeDS procedure is compared with other existing variable knot spline methods and smoothing techniques, such as SARS, HAS, MDL, AGS methods and is shown to produce models with fewer parameters but with similar goodness of fit characteristics, and visual quality
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