Shape Preserving Interpolation Using C

This paper discusses the construction of new C2 rational cubic spline interpolant with cubic numerator and quadratic denominator. The idea has been extended to shape preserving interpolation for positive data using the constructed rational cubic spline interpolation. The rational cubic spline has three parameters αi, βi, and γi. The sufficient conditions for the positivity are derived on one parameter γi while the other two parameters αi and βi are free parameters that can be used to change the final shape of the resulting interpolating curves. This will enable the user to produce many varieties of the positive interpolating curves. Cubic spline interpolation with C2 continuity is not able to preserve the shape of the positive data. Notably our scheme is easy to use and does not require knots insertion and C2 continuity can be achieved by solving tridiagonal systems of linear equations for the unknown first derivatives di, i=1,…,n-1. Comparisons with existing schemes also have been done in detail. From all presented numerical results the new C2 rational cubic spline gives very smooth interpolating curves compared to some established rational cubic schemes. An error analysis when the function to be interpolated is ft∈C3t0,tn is also investigated in detail

Görbék és felületek a geometriai modellezésben = Curves and surfaces in geometric modelling

B-spline görbék/felületek pontjai által, az alakzat két csomóértékének szimmetrikus változtatásakor leírt pályagörbéket vizsgáltuk, és olyan alakmódosítási eljárást adtunk, amivel a felület adott pontját/paramétervonalát előre megadott helyre mozgathatjuk a csomóértékek változtatásával. A C-Bézier, C-B-spline és F-B-spline görbék pályagörbéinek geometriai tulajdonságait írtuk le, és erre alapozva geometriai kényszereket kielégítő alakmódosításokat vizsgáltuk. Olyan általános leírási módot (linear blending) adtunk, mely egységesen kezeli az alakparaméterekkel rendelkező görbék széles osztályát, továbbá konkrét esetekben e paraméterek geometriai hatását írtuk le és kényszeres alakmódosításokra adtunk megoldást. A csomóértékeknek az interpoláló görbére gyakorolt hatását vizsgáltuk, mely alapján a harmadfokú interpoláció esetére interaktív alakmódosító eljárást dolgoztunk ki. Kontrollpontokkal adott görbék szingularitásainak detektálására a kontrollpontok helyzetén alapuló megoldást adtunk. Kontrollpont alapú szükséges és elégséges feltételt adtunk arra, hogy a Bézier-felület paramétervonalai egyenesek legyenek. Olyan Monte Carlo módszert dolgoztunk ki, amely rendezetlen ponthalmaz felülettel való interpolálásához négyszöghálót hoz létre a pontfelhő (mely elágazásokat és hurkokat is tartalmazhat) és annak topológikus gráfja ismeretében. A csonkolt Fourier-sorok terében olyan ciklikus bázist adtunk meg, amellyel végtelen simaságú zárt görbéket és felületeket írhatunk 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-Bézier, 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 Bézier 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

Anatomical shape reconstruction and manufacturing: solving topological changes of lumen vessel trough geometric approach

Over the last years there has been an increasing growth of interest in Rapid Prototyping (RP) techniques applied to various fields of medicine. RP makes it possible, in vascular surgery, to produce accurate anatomic replicas of patient vessels. These replicas can help the customization of surgical invasive interventions such as in situ stent-graft insertion in carotid region. The main goal of this work is to obtain high quality in lumen reconstruction and manufacturing replicas by RP technique. This goal is achieved through the complete control of each phase of the generating process. We present a semi-automatic method for carotid lumen reconstruction based on Boundary Representation (BRep). All parameters influencing the quality of the shape reconstruction are presented and discussed: shape acquisition, shape reconstruction and shape manufacturing. The shape acquisition starts by extracting the points belonging to the boundary of the lumen vessel, from Computer Tomography (CT) images. These points, parameterised in a vector, are the input data of the shape reconstruction algorithm based on B-Spline interpolation. The B-Spline type for representing curves and surfaces were chosen because of their properties of continuity and local control. In the shape reconstruction stage we had to face problems due to the topological change on the vessel structure. For vessel regions where there are not changes of topology, we use the closed B-Spline curves (belonging to adjacent acquisition planes) as generating curves to build a B-Spline surface. For vessel regions with at least a change of topology (ex. bifurcation region) our algorithm split automatically the involved curves to obtain three rectangular B-Spline patches. Such patches are joined together to obtain the bifurcation vessel lumen. The set of lumen surfaces is then inserted in a Boundary Representation in order to get a valid solid. To analyse the quality of the reconstructed shapes, the final object is compared with the acquisition image. This solid is correctly tessellated in triangles to produce the data format used by the RP devices (STL)

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

Morphing the CMB: a technique for interpolating power spectra

The confrontation of the Cosmic Microwave Background (CMB) theoretical angular power spectrum with available data often requires the calculation of large numbers of power spectra. The standard practice is to use a fast code to compute the CMB power spectra over some large parameter space, in order to estimate likelihoods and constrain these parameters. But as the dimensionality of the space under study increases, then even with relatively fast anisotropy codes, the computation can become prohibitive. This paper describes the employment of a "morphing" strategy to interpolate new power spectra based on previously calculated ones. We simply present the basic idea here, and illustrate with a few examples; optimization of interpolation schemes will depend on the specific application. In addition to facilitating the exploration of large parameter spaces, this morphing technique may be helpful for Fisher matrix calculations involving derivatives.Comment: 18 pages, including 6 figures, uses elsart.cls, accepted for publication in New Astronomy, changes to match published versio

A rational cubic spline with tension

A rational cubic spline curve is described which has tension control parameters for manipulating the shape of the curve. The spline is presented in both interpolatory and rational B-spline forms, and the behaviour of the resulting representations is analysed with respect to variation of the control parameters