Rational quadratic Bézier spirals

A quadratic Bézier representation withholds a curve segment with free from loops, cusps and inflection points. Furthermore, this rational form provides extra freedom to generate visually pleasing curves due to the existence of weights. In this paper, we propose sufficient conditions for rational quadratic Bézier curves to possess monotonic increasing/decreasing curvatures by means of monotone curvature tests which are based on the derivative of curvature functions. We have derived a simple interval of the middle weight that assures the construction of a family of rational quadratic Bézier curves to be planar spirals, which is characterized by the turning angle, end curvatures and the chords of control polygon. The proposed formulation can be used by CAD systems for aesthetic product design, highway/railway design and robot trajectory design avoiding unwanted curvature oscillations

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

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

Intersecting biquadratic Bézier surface patches

International audienceWe present three symbolic–numeric techniques for computing the in- tersection and self–intersection curve(s) of two Bézier surface patches of bidegree (2,2). In particular, we discuss algorithms, implementation, illustrative examples and provide a comparison of the methods

Intersecting biquadratic Bézier surface patches

International audienceWe present three symbolic–numeric techniques for computing the in- tersection and self–intersection curve(s) of two Bézier surface patches of bidegree (2,2). In particular, we discuss algorithms, implementation, illustrative examples and provide a comparison of the methods