Zbyněk Šír

Approximating curves and their offsets using biarcs and Pythagorean hodograph quintics

This paper compares two techniques for the approximation of the offsets to a given planar curve. The two methods are based on approximate conversion of the planar curve into circular splines and Pythagorean hodograph (PH) splines, respectively. The circular splines are obtained using a novel variant of biarc interpolation, while the PH splines are constructed via Hermite interpolation of C^1 boundary data. We analyze the approximation order of both conversion procedures. As a new result, the C^1 Hermite interpolation with PH quintics is shown to have approximation order 4 with respect to the original curve, and 3 with respect to its offsets. In addition, we study the resulting data volume, both for the original curve and for its offsets. It is shown that PH splines outperform the circular splines for increasing accuracy, due to the higher approximation order.

Hermite interpolation by hypocycloids and epicycloids with rational offsets

We show that all rational hypocycloids and epicycloids are curves with Pythagorean normals and thus have rational offsets. Then, exploiting the convolution properties and (implicit) support function representation of these curves, we design an efficient algorithm for G^1 Hermite interpolation with their arcs. We show that for all regular data, there is a unique interpolating hypocycloidal or epicycloidal arc of the given canonical type.

Spatial pythagorean hodograph quintics and the approximation of pipe surfaces

As observed by Farouki et al.[9], any set of C1 space boundary data (two points with associated first derivatives) can be interpolated by a Pythagorean hodograph (PH) curve of degree 5. In general there exists a two dimensional family of interpolants. In this paper we study the properties of this family in more detail. We introduce a geometrically invariant parameterization of the family of interpolants. This parameterization is used to identify a particular solution, which has the following properties. Firstly, it preserves planarity, i.e., the interpolant to planar data is a planar PH curve. Secondly, it has the best possible approximation order (4). Thirdly, it is symmetric in the sense that the interpolant of the “reversed” set of boundary data is simply the “reversed” original interpolant. These observations lead to a fast and precise algorithm for converting any (possibly piecewise) analytical curve into a piecewise PH curve of degree 5 which is globally C1. Finally we exploit the rational frames associated with any space PH curve (the Euler-Rodrigues frame) in order to obtain a simple rational approximation of pipe surfaces with a piecewise analytical spine curve and we analyze its approximation order.

On rationally supported surfaces

We analyze the class of surfaces which are equipped with rational support functions. Any rational support function can be decomposed into a symmetric (even) and an antisymmetric (odd) part. We analyze certain geometric properties of surfaces with odd and even rational support functions. In particular it is shown that odd rational support functions correspond to those rational surfaces which can be equipped with a linear field of normal vectors, which were discussed by Sampoli et al. (Sampoli, M.L., Peternell, M., Juttler, B., 2006. Rational surfaces with linear normals and their convolutions with rational surfaces. Comput. Aided Geom. Design 23, 179-192). As shown recently, this class of surfaces includes non-developable quadratic triangular Bezier surface patches (Lavicka, M., Bastl, B., 2007. Rational hypersurfaces with rational convolutions. Comput. Aided Geom. Design 24, 410-426; Peternell, M., Odehnal, B., 2008. Convolution surfaces of quadratic triangular Bezier surfaces. Comput. Aided Geom. Design 25, 116-129).

Rational Pythagorean-hodograph space curves

A method for constructing rational Pythagorean-hodograph (PH) curves in R^3 is proposed, based on prescribing a field of rational unit tangent vectors. This tangent field, together with its first derivative, defines the orientation of the curve osculating planes. Augmenting this orientation information with a rational support function, that specifies the distance of each osculating plane from the origin, then completely defines a one-parameter family of osculating planes, whose envelope is a developable ruled surface. The rational PH space curve is identified as the edge of regression (or cuspidal edge) of this developable surface. Such curves have rational parametric speed, and also rational adapted frames that satisfy the same conditions as polynomial PH curves in order to be rotation-minimizing with respect to the tangent. The key properties of such rational PH space curves are derived and illustrated by examples, and simple algorithms for their practical construction by geometric Hermite interpolation are also proposed.

Reparameterization of curves and surfaces with respect to their convolution

Given two parametric planar curves or surfaces we find their new parameterizations (which we call coherent) permitting to compute their convolution by simply adding the points with the same parameter values. Several approaches based on rational reparameterization of one or both input objects or direct computation of new parameterizations are shown. Using the Grobner basis theory we decide the simplest possible way for obtaining coherent parametrizations. We also show that coherent parameterizations exist whenever the convolution hypersurface is rational.

Computational and structural advantages of circular boundary representation

Boundary approximation of planar shapes by circular arcs has quantitive and qualitative advantages compared to using straightline segments. We demonstrate this by way of three basic and frequent computations on shapes - convex hull, decomposition, and medial axis. In particular, we propose a novel medial axis algorithm that beats existing methods in simplicity and practicality, and at the same time guarantees convergence to the medial axis of the original shape.

Euclidean and Minkowski Pythagorean hodograph curves over planar cubics

Starting with a given planar cubic curve [x(t),y(t)]^T, we construct Pythagorean hodograph (PH) space curves of the form [x(t),y(t),z(t)]^T in Euclidean and in Minkowski space, which interpolate the tangent vector at a given point. We prove the existence of these curves for any regular planar cubic and we express all solutions explicitly. It is shown that the constructed curves provide upper and lower polynomial bounds on the parametrical speed and the arc-length function of the given cubic. We analyze the approximation order and derive an explicit formula for the gap between the bounds. In addition, we discuss the approximation of the offset curves. Finally we define an invariant which measures the deviation of a given planar cubic from being a PH curve.

C2 Hermite interpolation by Minkowski Pythagorean hodograph curves and medial axis transform approximation

We describe and fully analyze an algorithm for C^2 Hermite interpolation by Pythagorean hodograph curves of degree 9 in Minkowski space R^2^,^1. We show that for any data there exists a four-parameter system of interpolants and we identify the one which preserves symmetry and planarity of the input data and which has the optimal approximation degree. The new algorithm is applied to an efficient approximation of segments of the medial axis transform of a planar domain leading to rational parameterizations of the offsets of the domain boundaries with a high order of approximation.

G 2 hermite interpolation with curves represented by multi-valued trigonometric support functions

It was recently proved in [27] that all rational hypocycloids and epicycloids are Pythagorean hodograph curves, i.e., rational curves with rational offsets. In this paper, we extend the discussion to a more general class of curves represented by trigonometric polynomial support functions. We show that these curves are offsets to translated convolutions of scaled and rotated hypocycloids and epicycloids. Using this result, we formulate a new and very simple G 2 Hermite interpolation algorithm based on solving a small system of linear equations. The efficiency of the designed method is then presented on several examples. In particular, we show how to approximate general trochoids, which, as we prove, are not Pythagorean hodograph curves in general.