Discussion on Minimal Curvature Variation in Cubic Hermite Curve Construction

Abstract

In the fields of computer aided geometric design, computer graphics and so on, curvature variation minimization has been widely used for constructing curve and surface. This paper investigates the minimal curvature variation in constructing the cubic Hermite curve that interpolates the given positions and unit tangent vectors at two points, while the magnitudes of the tangent vectors are unknown. The computation of this problem is very hard to handle and a very time-consuming task. To reduce the computing cost, simpler models are used to approximate it, but the existing simpler models can't give a good approximation, and hence make the curves constructed have unsatisfactory shapes. So a new model is presented in this paper. In the new model, the magnitude of each tangent vector is expressed as polynomial function of the tangent vector angles, which is easy to compute, and the shapes of constructed curves are visually similar to the ones constructed by minimizing the accurate curvature variation.

Study on improving the winding forming uniformity of high-speed silk reeling machine with large package

“…Wang, Zhou, Fu and Chen, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.16, No.1 (2022) The linear reciprocating motion of traverse rod in a motion period is divided into 6 parts, and the boundary conditions of axial ideal kinematic model are only related to two parameters. In order to construct a smooth curve, this paper adopts 6-segment cubic Hermite interpolation polynomials to fit the ideal velocity curve (Li et al, 2012). In addition, from Fig 2, it can be seen that the ideal velocity curve is central symmetric relative to point ( ) 2, 0 T/ .…”

A class of quintic Hermite interpolation curve and the free parameters selection

“…Although the smoothness of a curve is difficult to be expressed in a quantitative way, the strain energy (also called bending energy) minimization or the curvature variation energy minimization is adopted to construct fair curves in most cases (see e.g. Ahn et al, 2014;Žagar, 2011a, 2011b;Li et al, 2012;Li, 2018;Lu, 2015aLu, , 2015bLu et al, 2017;Xu et al, 2011). Here, we determine the values of the free parameters by minimizing curvature variation energy to obtain the fairest αβ-Hermite curve.…”

Planar T-Bézier curve with approximate minimum curvature variation

“…In recent years, a lot of works on planar G 1 or G 2 Hermite interpolation via strain energy minimization or curvature variation minimization have been proposed (see e.g. Žagar, 2011a, 2011b;Li et al, 2012;Lu et al, 2015aLu et al, , 2015bLu et al, , 2017c.…”