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

Construction and evaluation of PH curves in exponential-polynomial spaces

In the past few decades polynomial curves with Pythagorean Hodograph (for short PH curves) have received considerable attention due to their usefulness in various CAD/CAM areas, manufacturing, numerical control machining and robotics. This work deals with classes of PH curves built-upon exponential-polynomial spaces (for short EPH curves). In particular, for the two most frequently encountered exponential-polynomial spaces, we first provide necessary and sufficient conditions to be satisfied by the control polygon of the B\'{e}zier-like curve in order to fulfill the PH property. Then, for such EPH curves, fundamental characteristics like parametric speed or cumulative and total arc length are discussed to show the interesting analogies with their well-known polynomial counterparts. Differences and advantages with respect to ordinary PH curves become commendable when discussing the solutions to application problems like the interpolation of first-order Hermite data. Finally, a new evaluation algorithm for EPH curves is proposed and shown to compare favorably with the celebrated de Casteljau-like algorithm and two recently proposed methods: Wo\'zny and Chudy's algorithm and the dynamic evaluation procedure by Yang and Hong

A practical method for computing with piecewise Chebyshevian splines

A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under knot insertion. The interest in such kind of spaces is justified by the fact that, similarly as for polynomial splines, the related parametric curves exhibit the desired properties of convex hull inclusion, variation diminution and intuitive relation between the curve shape and the location of the control points. For a good-for-design space, in this paper we construct a set of functions, called transition functions, which allow for efficient computation of the B-spline basis, even in the case of nonuniform and multiple knots. Moreover, we show how the spline coefficients of the representations associated with a refined knot partition and with a raised order can conveniently be expressed by means of transition functions. This result allows us to provide effective procedures that generalize the classical knot insertion and degree raising algorithms for polynomial splines. We further discuss how the approach can straightforwardly be generalized to deal with geometrically continuous piecewise Chebyshevian splines as well as with splines having section spaces of different dimensions. From a numerical point of view, we show that the proposed evaluation method is easier to implement and has higher accuracy than other existing algorithms

O algebarskim minimalnim plohama

We give an overiew on various constructions of algebraic minimal surfaces in Euclidean three-space. Especially low degree examples shall be studied. For that purpose, we use the different representations given by WEIERSTRASS including the so-called Bjorling formula. An old result by LIE dealing with the evolutes of space curves can also be used to construct minimal surfaces with rational parametrizations. We describe a one-parameter family of rational minimal surfaces which touch orthogonal hyperbolic paraboloids along their curves of constant Gaussian curvature. Furthermore, we find a new class of algebraic and even rationally parametrizable minimal surfaces and call them cycloidal minimal surfaces.Dajemo pregled različitih konstrukcija algebarskih minimalnih ploha u euklidskom trodimenzionalnom prostoru. Posebice se promatraju primjeri niskog stupnja. U tu svrhu koristimo različite prikaze koje daje WEIERSTRASS, uključujući takozvanu Bjorlingovu formulu. LIJEV stari rezultat pokazuje da se evolute prostornih krivulja mogu koristiti za konstruiranje minimalnih ploha s racionalnim parametrizacijama. Mi opisujemo jednoparametarsku familiju racionalnih minimalnih ploha koje diraju ortogonalne hiperboličke paraboloide duž njihovih krivulja s konstantnom Gaussovom zakrivljenosću. Štoviše, nalazimo novu klasu algebarskih i čak racionalno parametrizirajućih minimalnih ploha i nazivamo ih cikloidnim minimalnim plohama

O algebarskim minimalnim plohama

We give an overiew on various constructions of algebraic minimal surfaces in Euclidean three-space. Especially low degree examples shall be studied. For that purpose, we use the different representations given by WEIERSTRASS including the so-called Bjorling formula. An old result by LIE dealing with the evolutes of space curves can also be used to construct minimal surfaces with rational parametrizations. We describe a one-parameter family of rational minimal surfaces which touch orthogonal hyperbolic paraboloids along their curves of constant Gaussian curvature. Furthermore, we find a new class of algebraic and even rationally parametrizable minimal surfaces and call them cycloidal minimal surfaces.Dajemo pregled različitih konstrukcija algebarskih minimalnih ploha u euklidskom trodimenzionalnom prostoru. Posebice se promatraju primjeri niskog stupnja. U tu svrhu koristimo različite prikaze koje daje WEIERSTRASS, uključujući takozvanu Bjorlingovu formulu. LIJEV stari rezultat pokazuje da se evolute prostornih krivulja mogu koristiti za konstruiranje minimalnih ploha s racionalnim parametrizacijama. Mi opisujemo jednoparametarsku familiju racionalnih minimalnih ploha koje diraju ortogonalne hiperboličke paraboloide duž njihovih krivulja s konstantnom Gaussovom zakrivljenosću. Štoviše, nalazimo novu klasu algebarskih i čak racionalno parametrizirajućih minimalnih ploha i nazivamo ih cikloidnim minimalnim plohama

Aspects of the design of a circular warp knitting machine

The warp knitting machine market has long been dominated by large-scale flat models, which have been steadily developed. Tubular fabrics are generally made in a special version of flat warp knitting machines containing two needle bars, one for each side of the tube, joined on the sides by yarns knitting alternatively on each bar. Warp knitting technology has failed to enter the circular knitting industry, dominated by weft knitting, due to its complexity in achieving warp knit structures in circular form. This thesis presents the design, synthesis, manufacture and test of an innovative method of producing tubular warp knitting fabrics, using a circular format rather than flat needle bars. This novel concept opens up many industrial applications from medical textiles to fruit packaging. [Continues.

Pythagorean-hodograph cycloidal curves

In the paper, Pythagorean-hodograph cycloidal curves as an extension of PH cubics are introduced. Their properties are examined and a constructive geometric characterization is established. Further, PHC curves are applied in the Hermite interpolation, with closed form solutions been determined. The asymptotic approximation order analysis carried out indicates clearly which interpolatory curve solution should be selected in practice. This makes the curves introduced here a useful practical tool, in particular in algorithms that guide CNC machines