1,184 research outputs found
Quintic trigonometric Bézier curve with two shape parameters
The fifth degree of trigonometric Bézier curve called quintic with two shapes parameter is presented in this paper. Shape parameters provide more control on the shape of the curve compared to the ordinary Bézier curve. This technique is one of the crucial parts in constructing curves and surfaces because the presence of shape parameters will allow the curve to be more flexible without changing its control points. Furthermore, by changing the value of shape parameters, the curve still preserves its geometrical features thus makes it more convenient rather than altering the control points. But, to interpolate curves from one point to another or surface patches, we need to satisfy certain continuity constraints to ensure the smoothness not just parametrically but also geometrically
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
Preserving Positivity And Monotonicity Of Real Data Using Bézier-Ball Function And Radial Basis Function
In this thesis, a rational cubic Bézier-Ball function which refers to a rational
cubic Bézier function expressed in terms of Ball control points and weights are used
to preserve positivity and monotonicity of real data sets. Four shape parameters are
proposed to preserve the characteristics of the data. A rational Bi-Cubic Bézier-Ball
function is introduced to preserve the positivity of surface generated from real data set
and from known functions. Eight shape parameters proposed can be modified to
preserve the positivity of the surface. Interpolating 2D and 3D real data using radial
basis function (RBF) is proposed as an alternative method to preserve the positivity of
the data. Two types of RBF which are Multiquadric (MQ) function and Gaussian
function, which contains a shape parameter are used. The boundaries (lower and
upper limit) of the shape parameter which preserves the positivity of real data are
proposed. Comparisons are made using the root-mean-square (RMS) error between
the proposed interpolation methods with existing works in literature. It was found that
MQ function and rational cubic Bézier-Ball is comparable with existing literature in
preserving positivity for both curves and surfaces. For preserving monotonicity, the
rational cubic Bézier-Ball is comparable but the MQ quasi-interpolation introduced
can only linearly interpolate the curve and the RMS values are big. Gaussian function
is able to preserve positivity of curves and surfaces but with unwanted oscillations
which result to unsmooth curves
B\'ezier curves that are close to elastica
We study the problem of identifying those cubic B\'ezier curves that are
close in the L2 norm to planar elastic curves. The problem arises in design
situations where the manufacturing process produces elastic curves; these are
difficult to work with in a digital environment. We seek a sub-class of special
B\'ezier curves as a proxy. We identify an easily computable quantity, which we
call the lambda-residual, that accurately predicts a small L2 distance. We then
identify geometric criteria on the control polygon that guarantee that a
B\'ezier curve has lambda-residual below 0.4, which effectively implies that
the curve is within 1 percent of its arc-length to an elastic curve in the L2
norm. Finally we give two projection algorithms that take an input B\'ezier
curve and adjust its length and shape, whilst keeping the end-points and
end-tangent angles fixed, until it is close to an elastic curve.Comment: 13 pages, 15 figure
Optimization of the Collection Efficiency of a Hexagonal Light Collector using Quadratic and Cubic B\'ezier Curves
Reflective light collectors with hexagonal entrance and exit apertures are
frequently used in front of the focal-plane camera of a very-high-energy
gamma-ray telescope to increase the collection efficiency of atmospheric
Cherenkov photons and reduce the night-sky background entering at large
incident angles. The shape of a hexagonal light collector is usually based on
Winston's design, which is optimized for only two-dimensional optical systems.
However, it is not known whether a hexagonal Winston cone is optimal for the
real three-dimensional optical systems of gamma-ray telescopes. For the first
time we optimize the shape of a hexagonal light collector using quadratic and
cubic B\'ezier curves. We demonstrate that our optimized designs simultaneously
achieve a higher collection efficiency and background reduction rate than
traditional designs.Comment: 9 pages, 9 figure
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