Circular Arc Approximation by Quartic H-Bézier Curve

The quartic H-Bézier curve is used for the approximation of circular arcs. It has five control points and one positive real free parameter. The four control points are carried out b

Circular Arc Approximation by Quartic H-Bézier Curve

The quartic H-Bézier curve is used for the approximation of circular arcs. It has five control points and one positive real free parameter. The four control points are carried out b

On geometric Hermite arcs

A geometric Hermite arc is a cubic curve in the plane that is specified by its endpoints along with unit tangent vectors and signed curvatures at them. This problem has already been solved by means of numerical procedures. Based on projective geometric considerations, we deduce the problem to finding the base points of a pencil of conics, that reduces the original quartic problem to a cubic one that easier can exactly be solved. A simple solvability criterion is also provided

Control of Curvature Extrema in Curve Modeling

We present a method for constructing almost-everywhere curvature-continuous curves that interpolate a list of control points and have local maxima of curvature only at the control points. Our premise is that salient features of the curve should occur only at control points to avoid the creation of features unintended by the artist. While many artists prefer to use interpolated control points, the creation of artifacts, such as loops and cusps, away from control points has limited the use of these types of curves. By enforcing the maximum curvature property, loops and cusps cannot be created unless the artist intends to create such features. To create these curves, we analyze the curvature monotonicity of quadratic, rational quadratic and cubic curves and develop a framework to connect such curve primitives with curvature continuity. We formulate an energy to encode the desired properties in a boxed constrained optimization and provide a fast method of estimating the solution through a numerical optimization. The optimized curve can serve as a real-time curve modeling tool in art design applications

A NURBS-based Discontinuous Galerkin method for conservation laws with high-order moving meshes

International audienceThe objective of the present work is to develop a new numerical framework for simulations including moving bodies, in the specific context of high-order meshes consistent with Computer-Aided Design (CAD) representations. Thus, the proposed approach combines ideas from isogeometric analysis, able to handle exactly CAD-based geometries, and Discontinuous Galerkin (DG) methods with an Arbitrary Lagrangian-Eulerian (ALE) formulation , able to solve complex problems with moving grids. The resulting approach is a DG method based on rational Bézier elements, that can be easily constructed from Non-Uniform Rational B-Splines (NURBS), formulated in a general ALE setting. We focus here on applications in compressible aerodynamics, but the method could be applied to other models. Two verification exercises are conducted, to assess rigorously the properties of the method and the convergence rates for representations up to sixth order. Finally, two problems are analysed in depth, involving compressible Euler and Navier-Stokes equations , for an oscillating cylinder and a pitching airfoil. In particular, the convergence of flow characteristics is investigated, as well as the impact of using curved boundaries in the context of deformable domains

On automatic tuning of basis functions in Bezier method

A transition from the fixed basis in Bezier's method to some class of base functions is proposed. A parameter vector of a basis function is introduced as additional information. This achieves a more universal form of presentation and analytical description of geometric objects as compared to the non-uniform rational B-splines (NURBS). This enables control of basis function parameters including control points, their weights and node vectors. This approach can be useful at the final stage of constructing and especially local modification of compound curves and surfaces with required differential and shape properties; it also simplifies solution of geometric problems. In particular, a simple elimination of discontinuities along local spline curves due to automatic tuning of basis functions is demonstrated