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

Real-time interpolation of streaming data

One of the key elements of real-time $C^1$-continuous cubic spline interpolation of streaming data is an estimator of the first derivative of the interpolated function that is more accurate than the ones based on finite difference schemas.Two such greedy look-ahead heuristic estimators (denoted as MinBE and MinAJ2) based on Calculus of Variations are formally defined (in closed form) together with the corresponding cubic splines they generate, and then comparatively evaluated in a series of numerical experiments involving different types of performance measures. The results presented show that the cubic Hermite splines generated by heuristic MinAJ2 significantly outperformed these based on finite difference schemas in terms of all tested performance measures (including convergence).The proposed approach is quite general. It can be directly applied to streams of univariate functional data like time-series. Multidimensional curves defined parametrically, after splitting, can be handled as well. The streaming character of the algorithm means that it can also be useful in processing data sets that are too large to fit in memory (e.g., edge computing devices, embedded time-series databases)

Approximating Generalised Cornu Spiral With Low Energy Curve

The computations of visual pleasing and mathematically fair curve are an ongoing process. In the earlier literature, researchers have been studying the smooth curve, or in a more technically term, the fair curve through the physical approach known as elastica, Elastica means a thin strip of elastic material. Daniel Bernoulli stated the elastica as variational problem in terms of strain energy and Malcom (1977) mentioned that the simplest way to characterize a spline mathematically is with the fact that a spline assumes a shape which minimizes its elastic strain energy

Curvature Variation Minimizing Cardinal Spline Curves

Since the free parameter and the two auxiliary points have effect on the shape of the Cardinal spline curves, a natural idea arises to find the optimal parameter and auxiliary points to obtain the smoothest curves. We use curvature variation minimization to achieve this goal in this paper. By minimizing an appropriate approximation of the curvature variation energy, the unique solution can be easily obtained. Some examples show that the Cardinal spline curves with minimum curvature variation are better than the Catmull-Rom spline curves in dealing with interpolation problems

Novel Discretization Schemes for the Numerical Simulation of Membrane Dynamics

Motivated by the demands of simulating flapping wings of Micro Air Vehicles, novel numerical methods were developed and evaluated for the dynamic simulation of membranes. For linear membranes, a mixed-form time-continuous Galerkin method was employed using trilinear space-time elements, and the entire space-time domain was discretized and solved simultaneously. For geometrically nonlinear membranes, the model incorporated two new schemes that were independently developed and evaluated. Time marching was performed using quintic Hermite polynomials uniquely determined by end-point jerk constraints. The single-step, implicit scheme was significantly more accurate than the most common Newmark schemes. For a simple harmonic oscillator, the scheme was found to be symplectic, frequency-preserving, and conditionally stable. Time step size was limited by accuracy requirements rather than stability. The spatial discretization scheme employed a staggered grid, grouping of nonlinear terms, and polygon shape functions in a strong-form point collocation formulation. Validation against existing experimental data showed the method to be accurate until hyperelastic effects dominate

Parametric Spiral And Its Application As Transition Curve

Lengkung Bezier merupakan suatu perwakilan lengkungan yang paling popular digunakan di dalam applikasi Rekabentuk Berbantukan Komputer (RBK) dan Rekabentuk Geometrik Berbantukan Komputer (RGBK). The Bezier curve representation is frequently utilized in computer-aided design (CAD) and computer-aided geometric design (CAGD) applications. The curve is defined geometrically, which means that the parameters have geometric meaning; they are just points in three-dimensional space