Michal Bizzarri

A symbolic-numerical approach to approximate parameterizations of space curves using graphs of critical points

A simple algorithm for computing an approximate parameterization of real space algebraic curves using their graphs of critical points is designed and studied in this paper. The first step is determining a suitable space graph which contains all critical points of a real algebraic space curve C implicitly defined as the complete intersection of two surfaces. The construction of this graph is based on one projection of C in a general position onto an xy-plane and on an intentional choice of vertices. The second part of the designed method is a computation of a spline curve which replaces the edges of the constructed graph by segments of a chosen free-form curve. This step is formulated as an optimization problem when the objective function approximates the integral of the squared Euclidean distance of the constructed approximate curve to the intersection curve. The presented method, based on combining symbolic and numerical steps to the approximation problem, provides approximate parameterizations of space algebraic curves from a small number of approximating arcs. It may serve as a first step to several problems originating in technical practice where approximation curve parameterizations are needed.

Parameterizing rational offset canal surfaces via rational contour curves

A canal surface is the envelope of a 1-parameter set of spheres centered at the spine curve m(t) and with the radii described by the function r(t). Any canal surface given by rational m(t) and r(t) possesses a rational parameterization. However, an arbitrary rational canal surface does not have to fulfill the PN (Pythagorean normals) condition. Most (exact or approximate) parameterization methods are based on a construction of a rational unit normal vector field guaranteeing rational offsets. In this paper, we will study a condition which guarantees that a given canal surface has rational contour curves, which are later used for a straightforward computation of rational parameterizations of canal surfaces providing rational offsets. Using the contour curves in the parameterization algorithm brings another extra feature; the parameter lines do not unnecessarily wind around the canal surface. Our approach follows a construction of rational spatial MPH curves from the associated planar PH curves introduced in Kosinka and Lavicka (2010) [28] and gives it to the relation with the contour curves of canal surfaces given by their medial axis transforms. We also present simple methods for computing approximate PN parameterizations of given canal surfaces and rational offset blends between two canal surfaces.

C1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs

In this paper the C^1 Hermite interpolation problem by spatial Pythagorean-hodograph cubic biarcs is presented and a general algorithm to construct such interpolants is described. Each PH cubic segment interpolates C^1 data at one point and they are then joined together with a C^1 continuity at some unknown common point sharing some unknown tangent vector. Biarcs are expressed in a closed form with three shape parameters. Two of them are selected based on asymptotic approximation order, while the remaining one can be computed by minimizing the length of the biarc or by minimizing the elastic bending energy. The final interpolating spline curve is globally C^1 continuous, it can be constructed locally and it exists for arbitrary Hermite data configurations.

A symbolic-numerical method for computing approximate parameterizations of canal surfaces

A canal surface is the envelope of a one-parameter family of spheres centered at the spine curve m ( t ) and with the radii described by the function r ( t ) . It was proved in Peternell and Pottmann (1997)?9] that any canal surface to a rational spine curve and a rational radius function possesses a rational parameterization. Then a symbolic method for generating rational parameterizations of canal surfaces was developed in Landsmann et?al. (2001)?21]. Indeed, this method leads to the problem of decomposing a polynomial into a sum of two squares over reals, which is solved numerically in general. Hence, approximate techniques generating a parameterization within a certain region of interest are also worth studying. In this paper, we present a method for the computation of approximate rational parameterizations of canal surfaces. A main feature of our approach is a combination of symbolic and numerical techniques yielding approximate topology-based parameterizations of contour curves which are then applied to compute an approximate parameterization of the given canal surface. The algorithm is mainly suitable for implicit blend surfaces of the canal-surface-type. Highlights? We present new algorithms for computing approximate rational parameterizations of canal surfaces (given implicitly of by MAT). ? A main feature of our approach is a combination of symbolic and numerical techniques. ? An approximate parameterization of the given canal surface is based on approximate topology-based parameterizations of curves. ? A functionality of the method was demonstrated on chosen examples.

Canal surfaces with rational contour curves and blends bypassing the obstacles

In this paper, we will present an algebraic condition, see (20), which guarantees that a canal surface, given by its rational medial axis transform (MAT), possesses rational generalized contours (i.e., contour curves with respect to a given viewpoint). The remaining computational problem of this approach is how to find the right viewpoint. The canal surfaces fulfilling this distinguished property are suitable for being taken as modeling primitives when some rational approximations of canal surfaces are required. Mainly, we will focus on the low-degree cases such as quadratic and cubic MATs that are especially useful for applications. To document a practical usefulness of the presented approach, we designed and implemented two simple algorithms for computing rational offset blends between two canal surfaces based on the contour method which do not need any further advanced formalism (as e.g. interpolations with MPH curves). A main advantage of the designed blending technique is its simplicity and also an adaptivity to choose a suitable blend satisfying certain constrains (avoiding obstacles, bypassing other objects, etc.). Compared to other similar methods, our approach requires only one SOS decomposition for the whole family of rational canal surfaces sharing the same silhouette, which significantly simplifies the computational complexity. The rationality of generalized contours on rational canal surfaces is studied.The contour method is used for computing PN blends between two canal surfaces.The constructed blends can easily satisfy certain constrains, e.g. avoiding obstacles.Only one SOS decomposition for all canal surfaces with the same silhouette is needed.

Medial axis transforms yielding rational envelopes

Minkowski Pythagorean hodograph (MPH) curves provide a means for representing domains with rational boundaries via the medial axis transform. Based on the observation that MPH curves are not the only curves that yield rational envelopes, we define and study rational envelope (RE) curves that generalise MPH curves while maintaining the rationality of their associated envelopes.To demonstrate the utility of RE curves, we design a simple interpolation algorithm using RE curves, which is in turn used to produce rational surface blends between canal surfaces. Additionally, we initiate the study of rational envelope surfaces as a surface analogy to RE curves. We define rational envelope (RE) curves as a generalisation of MPH curves.RE curves are used in a simple canal surface blending scheme.We initiate the study of RE surfaces.

On modeling with rational ringed surfaces

A surface in Euclidean space is called ringed (or cyclic) if there exists a one-parameter family of planes that intersects this surface in circles. Well-known examples of ringed surfaces are the surfaces of revolution, (not only rotational) quadrics, canal surfaces, or Darboux cyclides. This paper focuses on modeling with rational ringed surfaces, mainly for blending purposes. We will deal with the question of rationality of ringed surfaces and discuss the usefulness of the so called P-curves for constructing rational ringed-surface-blends. The method of constructing blending surfaces that satisfy certain prescribed constraints, e.g. a necessity to avoid some obstacles, will be presented. The designed approach can be easily modified also for computing n -way blends. In addition, we will study the contour curves on ringed surfaces and use them for computing approximate parameterizations of implicitly given blends by ringed surfaces. The designed techniques and their implementations are verified on several examples. The paper focuses on modeling with rational ringed surfaces, mainly for blending purposes.We answer the question of their rationality and use P-curves for constructing rational ringed surfaces.The method for constructing blends that satisfy certain prescribed constraints is presented.The designed approach can be easily modified also for computing n -way blends.The contour curves are used for computing approximate parameterizations of implicitly given blends by ringed surfaces.

Approximation of Implicit Blends by Canal Surfaces of Low Parameterization Degree

In this paper, we present a modified method for the computation of approximate rational parameterizations of implicitly given canal surfaces. The designed algorithm, which improves and completes a recent approach from [1], is mainly suitable for implicit blend surfaces of the canal-surface-type. Its main advantage is that it produces rational parameterizations of low bidegree (7,2). A distinguished feature of our approach is a combination of symbolic and numerical techniques yielding approximate topology-based cubic parameterizations of contour curves which are then applied to compute an approximate parameterization of the given canal surface.

Hermite interpolation by piecewise polynomial surfaces with polynomial area element

This paper is devoted to the construction of polynomial 2-surfaces which possess a polynomial area element. In particular we study these surfaces in the Euclidean space

C 2 Hermite interpolation by Pythagorean-hodograph quintic triarcs

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