In the paper, the planar polynomial geometric interpolation of data points is revisited. Simple sufficient geometric conditions that imply the existence of the interpolant are derived in general. ...They require data points to be convex in a certain discrete sense. Since the geometric interpolation is based precisely on the known data only, one may consider it as the parametric counterpart to the polynomial function interpolation. The established result confirms the Höllig–Koch conjecture on the existence and the approximation order in the planar case for parametric polynomial curves of any degree stated quite a while ago.
In this paper the problem of geometric interpolation of planar data by parametric polynomial curves is revisited. The conjecture that a parametric polynomial curve of degree \le n can interpolate 2 n ...given points in \mathbb{R}^2 is confirmed for n \le 5 under certain natural restrictions. This conclusion also implies the optimal asymptotic approximation order. More generally, the optimal order 2 n can be achieved as soon as the interpolating curve exists.
In the paper, the uniform approximation of a circle arc (or a whole circle) by a parametric polynomial curve is considered. The approximant is obtained in a closed form. It depends on a parameter ...that should satisfy a particular equation, and it takes only a couple of tangent method steps to compute it. For low degree curves, the parameter is provided exactly. The distance between a circle arc and its approximant asymptotically decreases faster than exponentially as a function of polynomial degree. Additionally, it is shown that the approximant could be applied for a fast evaluation of trigonometric functions too.
In this paper a new approach for a construction of polynomial surfaces with rational field of unit normals (PN surfaces) is presented. It is based on bivariate polynomials with quaternion ...coefficients. Relations between these coefficients are derived that allow one to construct PN surfaces of general odd and even degrees. For low degree PN surfaces the theoretical results are supplemented with algorithms and illustrated with numerical examples.
•Construction of polynomial PN surfaces based on bivariate polynomials with quaternion coefficients is presented.•Particular polynomial PN surfaces of odd and even degrees are derived.•Algorithms for low degree PN surfaces are given.•Curvature properties are examined.•Simple interpolation scheme is proposed.
In this paper the G1 interpolation of two data points and two tangent directions with spatial cubic rational PH curves is considered. It is shown that interpolants exist for any true spatial data ...configuration. The equations that determine the interpolants are derived by combining a closed form representation of a ten parametric family of rational PH cubics given in Kozak et al. (2014), and the Gram matrix approach. The existence of a solution is proven by using a homotopy analysis, and numerical method to compute solutions is proposed. In contrast to polynomial PH cubics for which the range of G1 data admitting the existence of interpolants is limited, a switch to rationals provides an interpolation scheme with no restrictions.
•G1 interpolation scheme with spatial rational PH cubics is presented.•The existence of the interpolant is proven for any true spatial data configurations.•Homotopy analysis and a Gram matrix approach are used.•Numerical method to compute the solutions is proposed.
In this paper, the dual representation of spatial parametric curves and its properties are studied. In particular, rational curves have a polynomial dual representation, which turns out to be both ...theoretically and computationally appropriate to tackle the main goal of the paper: spatial rational Pythagorean-hodograph curves (PH curves). The dual representation of a rational PH curve is generated here by a quaternion polynomial which defines the Euler–Rodrigues frame of a curve. Conditions which imply low degree dual form representation are considered in detail. In particular, a linear quaternion polynomial leads to cubic or reparameterized cubic polynomial PH curves. A quadratic quaternion polynomial generates a wider class of rational PH curves, and perhaps the most useful is the ten-parameter family of cubic rational PH curves, determined here in the closed form.
•Dual representation of spatial rational PH curves is presented.•Connection between the degrees of a dual and a point representation of rational curves is revealed.•It is proven that linear quaternion polynomials lead to reparameterized cubic PH curves.•Spatial rational PH curves of a class m=3,4,5,6 are derived in a closed form having 2m+4 degrees of freedom.
In this paper, a class of rational spatial curves that have a rational binormal is introduced. Such curves (called PB curves) play an important role in the derivation of rational rotation-minimizing ...osculating frames. The PB curve construction proposed is based upon the dual curve representation and the Euler-Rodrigues frame obtained from quaternion polynomials. The construction significantly simplifies if the curve is a polynomial one. Further, polynomial PB curves of the degree ≥ 7 and rational PB curves of the degree ≥ 6 that possess rational rotation-minimizing osculating frames are derived, and it is shown that no lower degree curves, constructed from quadratic quaternion polynomials, with such a property exist.
In the paper, the planar polynomial geometric interpolation of data points is revisited. Simple sufficient geometric conditions that imply the existence of the interpolant are derived in general. ...They require data points to be convex in a certain discrete sense. Since the geometric interpolation is based precisely on the known data only, one may consider it as the parametric counterpart to the polynomial function interpolation. The established result confirms the H\"{o}llig-Koch conjecture on the existence and the approximation order in the planar case for parametric polynomial curves of any degree stated quite a while ago.
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