On the conditions for the coincidence of two cubic Bézier curves

AbstractIn a recent article, Wang et al. [2] derive a necessary and sufficient condition for the coincidence of two cubic Bézier curves with non-collinear control points. The condition reads that their control points must be either coincident or in reverse order. We point out that this uniqueness of the control points for polynomial cubics is a straightforward consequence of a previous and more general result of Barry and Patterson, namely the uniqueness of the control points for rational Bézier curves. Moreover, this uniqueness applies to properly parameterized polynomial curves of arbitrary degree

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

Cournot oligopoly interval games

In this paper we consider cooperative Cournot oligopoly games. Following Chander and Tulkens (1997) we assume that firms react to a deviating coalition by choosing individual best reply strategies. Lardon (2009) shows that if the inverse demand function is not differentiable, it is not always possible to define a Cournot oligopoly TU(Transferable Utility)-game. In this paper, we prove that we can always specify a Cournot oligopoly interval game. Furthermore, we deal with the problem of the non-emptiness of two induced cores: the interval gamma-core and the standard gamma-core. To this end, we use a decision theory criterion, the Hurwicz criterion (Hurwicz 1951), that consists in combining, for any coalition, the worst and the better worths that it can obtain in its worth interval. The first result states that the interval gamma-core is non-empty if and only if the oligopoly TU-game associated with the better worth of every coalition in its worth interval admits a non-empty gamma-core. However, we show that even for a very simple oligopoly situation, this condition fails to be satisfied. The second result states that the standard gamma-core is non-empty if and only if the oligopoly TU- game associated with the worst worth of every coalition in its worth interval admits a nonempty gamma-core. Moreover, we give some properties on every individual profit function and every cost function under which this condition always holds, what substantially extends the gamma-core existence results in Lardon (2009).Cournot oligopoly interval game; Interval gamma-core; Standard gamma-core; Hurwicz criterion;

A discrete methodology for controlling the sign of curvature and torsion for NURBS

This paper develops a discrete methodology for approximating the so-called convex domain of a NURBS curve, namely the domain in the ambient space, where a user-specified control point is free to move so that the curvature and torsion retains its sign along the NURBS parametric domain of definition. The methodology provides a monotonic sequence of convex polyhedra, converging from the interior to the convex domain. If the latter is non-empty, a simple algorithm is proposed, that yields a sequence of polytopes converging uniformly to the restriction of the convex domain to any user-specified bounding box. The algorithm is illustrated for a pair of planar and a spatial Bézier configuration

Discontinuities in numerical radiative transfer

Observations and magnetohydrodynamic simulations of solar and stellar atmospheres reveal an intermittent behavior or steep gradients in physical parameters, such as magnetic field, temperature, and bulk velocities. The numerical solution of the stationary radiative transfer equation is particularly challenging in such situations, because standard numerical methods may perform very inefficiently in the absence of local smoothness. However, a rigorous investigation of the numerical treatment of the radiative transfer equation in discontinuous media is still lacking. The aim of this work is to expose the limitations of standard convergence analyses for this problem and to identify the relevant issues. Moreover, specific numerical tests are performed. These show that discontinuities in the atmospheric physical parameters effectively induce first-order discontinuities in the radiative transfer equation, reducing the accuracy of the solution and thwarting high-order convergence. In addition, a survey of the existing numerical schemes for discontinuous ordinary differential systems and interpolation techniques for discontinuous discrete data is given, evaluating their applicability to the radiative transfer problem

Curves with rational chord-length parametrization

It has been recently proved that rational quadratic circles in standard Bezier form are parameterized by chord-length. If we consider that standard circles coincide with the isoparametric curves in a system of bipolar coordinates, this property comes as a straightforward consequence. General curves with chord-length parametrization are simply the analogue in bipolar coordinates of nonparametric curves. This interpretation furnishes a compact explicit expression for all planar curves with rational chord-length parametrization. In addition to straight lines and circles in standard form, they include remarkable curves, such as the equilateral hyperbola, Lemniscate of Bernoulli and Limacon of Pascal. The extension to 3D rational curves is also tackled

Constrained Interpolation And Shape Preserving Approximation By Space Curves [QA297.6. K82 2006 f rb].

Dua jenis masalah rekabentuk lengkung telah ipertimbangkan. Terlebih dahulu kami mempertimbangkan interpolasi satu set titik data ruang yang bertertib dengan satu lengkung licin tertakluk kepada satu set satah kekangan yang berbentuk terhingga atau tak terhingga di mana garis cebis demi cebis yang menyambung titik data secara berturutan tidak bersilang dengan satah kekangan. Two types of curve designing problem have been considered. We first consider the interpolation of a given set of ordered spatial data points by a smooth curve in the presence of a set of finite or infinite constraint planes, where the polyline joining consecutive data points does not intersect with the constraint planes