Generalized plane offsets and rational parameterizations

In the first part of the paper a planar generalization of offset curves is introduced and some properties are derived. In particular, it is seen that these curves exhibit good regularity properties and a study on self-intersection avoidance is performed. The representation of a rational curve as the envelope of its tangent lines, following the approach of Pottmann, is revisited to give the explicit expression of all rational generalized offsets. Other famous shapes, such as constant width curves, bicycle tire-tracks curves and Zindler curves are related to these generalized offsets. This gives rise to the second part of the paper, where the particular case of rational parameterizations by a support function is considered and explicit families of rational constant width curves, rational bicycle tire-track curves and rational Zindler curves are generated and some examples are shown

Interpolation of G1 Hermite data by C1 cubic-like sparse Pythagorean hodograph splines

open3siProvided that they are in appropriate configurations (tight data), given planar G1 Hermite data generate a unique cubic Pythagorean hodograph (PH) spline curve interpolant. On a given associated knot-vector, the corresponding spline function cannot be C1, save for exceptional cases. By contrast, we show that replacing cubic spaces by cubic-like sparse spaces makes it possible to produce infinitely many C1 PH spline functions interpolating any given tight G1 Hermite data. Such cubic-like sparse spaces involve the constants and monomials of consecutive degrees, and they have long been used for design purposes. Only lately they were investigated in view of producing PH curves and associated G1 PH spline interpolants with some flexibility. The present work strongly relies on these recent results.embargoed_20220415Ait-Haddou R.; Beccari C.V.; Mazure M.-L.Ait-Haddou R.; Beccari C.V.; Mazure M.-L

Geometric Hermite interpolation by rational curves of constant width

A constructive characterization of the support function for a rationally parameterized curve of constant width is given. In addition, a Hermite interpolation problem for such kind of curves is solved, which yields a method to determine a rational curve of constant width that passes through a set of free points with the corresponding tangent directions. Finally, the case of piecewise rational support functions is considered, which increases the design freedom. The procedure is presented in the general case of hedgehogs of constant width taking the advantage of projective hedgehogs, so that some constraints must be taken to ensure convexity of the desired curve.Funding for the other authors not affiliated with BCAM: Grant PID2021-124577NBI00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. Project PID2019-104927GB-C21 funded by MCIN/AEI/10.13039/501100011033. Project UJI-B2022-19 funded by Universitat Jaume I. Project CIAICO/2021/180 funded by Generalitat Valenciana

On Holditch's theorem and related kinematics

El teorema de Holditch es un resultado clásico sobre áreas de curvas planas generadas por el movimiento de segmentos. Esta construcción está estrechamente relacionada con otros tipos de curvas como, por ejemplo, curvas paralelas, curvas de anchura constante o curvas de bicicletas. Se compilan las propiedades básicas de este tipo de curvas y se da una revisión histórica sobre el teorema de Holditch y teoremas relacionados de cinemática. Primero, la situación plana de Holditch se define rigurosamente y se consideran ciertos problemas como la existencia de dicha construcción o el modo de evitar movimientos retrógrados en el segmento que se va moviendo. En el enunciado del teorema de Holditch aparece el área de una elipse oculta. En este trabajo se utiliza una aproximación poligonal a dicho teorema para mostrar geométricamente de dónde proviene esta elipse. También es posible dar generalizaciones inmediatas de este teorema y de otros relacionados con la cinemática en otros contextos y situaciones. Así, en la segunda parte, se da una introducción a la geometría no euclidiana y se presenta la extensión de los resultados mencionados a superficies de curvatura constante. Además, se encuentran curvas cerradas ocultas en la superficie de curvatura constante relacionadas con el enunciado generalizado de Holditch. Finalmente, se obtiene una nueva extensión natural del teorema de Holditch para curvas espaciales, lo cual nos lleva al concepto de superficie de Holditch.Holditch's theorem is a classical result on areas of planar curves generated by moving chords. The construction is closely related to other kinds of curves such as parallel curves, constant width curves or bicycle curves. The basic properties of these curves are compiled and a historical review on Holditch's theorem and related theorems in kinematics is given. First, the Holditch planar setting is rigorously defined and problems such as the existence of that construction or the avoidance of retrograde movements of the moving chord are considered. In the statement of Holditch's theorem, the area of a hidden ellipse appears. A polygonal approach to the theorem is used to show geometrically where this ellipse comes from. Moreover, immediate generalizations of Holditch's theorem and related results to other contexts are possible. So, in the second part, an introduction to non-Euclidean geometry is given and the extension of such results to constant curvature surfaces is presented. In addition, hidden closed curves in the constant curvature manifold related to the generalized statement of Holditch's theorem are found. Finally, a new extension of Holditch's theorem to space curves is derived in a natural way leading to the concept of Holditch surface

Pythagorean-hodograph ovals of constant width

Special Issue: Pythagorean-Hodograph Curves and Related TopicsInternational audienceA constructive geometric approach to rational ovals and rosettes of constant width formed by piecewise rational PH curves is presented. We propose two main constructions. The first construction, models with rational PH curves of algebraic class 3 (T-quartics) and is based on the fact that T-quartics are exactly the involutes of T-cubic curves. The second construction, models with rational PH curves of algebraic class m>4 and is based on the dual control structure of offsets of rational PH curves