Maria Lucia Sampoli

Rational surfaces with linear normals and their convolutions with rational surfaces

It is shown that polynomial (or rational) parametric surfaces with a linear field of normal vectors are dual to graphs bivariate polynomials (or rational functions). We discuss the geometric properties of these surfaces. In particular, using the dual representation it is shown that the convolution with general rational surfaces yields again rational surfaces. Similar results hold in the case of curves.

Hermite interpolation by piecewise polynomial surfaces with rational offsets

Abstract We present a construction for polynomial spline surfaces with a piecewise linear field of normal vectors. As main advantageous feature these surfaces possess exact rational offsets. The spline surface is composed of quartic Clough–Tocher-type macro elements. Each element is capable of matching boundary data consisting of three points with associated normal vectors. The collection of the macro elements forms a G1 continuous spline surface. With the help of a reparamaterization technique we obtain an exact rational representation of the offset surfaces by rational triangular spline surfaces of degree 10.

Isogemetric analysis and symmetric Galerkin BEM

Isogeometric approach applied to Boundary Element Methods is an emerging research area (see e.g. Simpson et?al. (2012) 33). In this context, the aim of the present contribution is that of investigating, from a numerical point of view, the Symmetric Galerkin Boundary Element Method (SGBEM) devoted to the solution of 2D boundary value problems for the Laplace equation, where the boundary and the unknowns on it are both represented by B-splines (de Boor (2001) 9). We mainly compare this approach, which we call IGA-SGBEM, with a curvilinear SGBEM (Aimi et?al. (1999) 2), which operates on any boundary given by explicit parametric representation and where the approximate solution is obtained using Lagrangian basis. Both techniques are further compared with a standard (conventional) SGBEM approach (Aimi et?al. (1997) 1), where the boundary of the assigned problem is approximated by linear elements and the numerical solution is expressed in terms of Lagrangian basis. Several examples will be presented and discussed, underlying benefits and drawbacks of all the above-mentioned approaches.

On quadratic two-parameter families of spheres and their envelopes

In the present paper we investigate rational two-parameter families of spheres and their envelope surfaces in Euclidean R^3. The four dimensional cyclographic model of the set of spheres in R^3 is an appropriate framework to show that a quadratic triangular Bezier patch in R^4 corresponds to a two-parameter family of spheres with rational envelope surface. The construction shows also that the envelope has rational offsets. Further we outline how to generalize the construction to obtain a much larger class of surfaces with similar properties.

Computing the convolution and the Minkowski sum of surfaces

In many applications, such as NC tool path generation and robot motion planning, it is required to compute the Minkowski sum of two objects. Generally the Minkowski sum of two rational surfaces cannot be expressed in rational form. In this paper we show that for LN spline surfaces (surfaces with a linear field of normal vectors) a closed form representation is available.

Computing Isophotes on Free-Form Surfaces Based on Support Function Approximation

The support function of a free-form-surface is closely related to the implicit equation of the dual surface, and the process of computing both the dual surface and the support function can be seen as dual implicitization. The support function can be used to parameterize a surface by its inverse Gauss map. This map makes it relatively simple to study isophotes (which are simply images of spherical circles) and offset surfaces (which are obtained by adding the offsetting distance to the support function). We present several classes of surfaces which admit a particularly simple computation of the dual surfaces and of the support function. These include quadratic polynomial surfaces, ruled surfaces with direction vectors of low degree and polynomial translational surfaces of bidegree (3,2). In addition, we use a quasi-interpolation scheme for bivariate quadratic splines over criss-cross triangulations in order to formulate a method for approximating the support function. The inverse Gauss maps of the bivariate quadratic spline surfaces are computed and used for approximate isophote computation. The approximation order of the isophote approximation is shown to be 2.

A general scheme for shape preserving planar interpolating curves

This paper describes the application of the so-called Abstract Schemes (AS) for the construction of shape preserving interpolating planar curves. The basic idea behind AS is given by observing that when we interpolate some data points by a spline, we can dispose of several free parameters d0,d1,...,dN (di∈Rq), which are associated with the knots. If we now express shape constraints as conditions relative to each interval between two knots, they can be rewritten as a sequences of inclusion conditions: ({d}i,di+1)∈Di⊂R2q, where the sets Di are the corresponding feasible domains. In this setting the problems of existence, construction and selection of an optimal solution can be studied with the help of Set Theory in a general way. The method is then applied for the construction of shape preserving, planar interpolating curves.

C1 rational interpolation of spherical motions with rational rotation-minimizing directed frames

An interpolation method for constructing rational curves on the unit sphere with rational directed rotation-minimizing frames is presented. This type of curves is useful, for instance, in the description of smoothly varying camera motions. The proposed scheme uses as input data the initial and the final curve positions and tangents on the sphere, together with the associated end frame orientations. Both the curve and the rotation-minimizing directed frame produced by the scheme are C^1 continuous and are rational of degree 8, which, under suitable mild assumptions on angular velocity directions, can be reduced to 6 if only G^1 continuity is required.

Efficient assembly based on B-spline tailored quadrature rules for the IgA-SGBEM

Abstract This paper deals with the discrete counterpart of 2D elliptic model problems rewritten in terms of Boundary Integral Equations. The study is done within the framework of Isogeometric Analysis based on B-splines. In such a context, the problem of constructing appropriate, accurate and efficient quadrature rules for the Symmetric Galerkin Boundary Element Method is here investigated. The new integration schemes, together with row assembly and sum factorization, are used to build a more efficient strategy to derive the final linear system of equations. Key ingredients are weighted quadrature rules tailored for B-splines, that are constructed to be exact in the whole test space, also with respect to the singular kernel. Several simulations are presented and discussed, showing accurate evaluation of the involved integrals and outlining the superiority of the new approach in terms of computational cost and elapsed time with respect to the standard element-by-element assembly.

Boolean surfaces with shape constraints

In this paper, we present a new method for the construction of parametric surfaces reproducing an object from a set of spatial data. We adopt a hybrid scheme, based on the Boolean sum of variable degree spline operators, which both interpolate a set of grid lines and approximate the data. As usual the variable degrees can be chosen to satisfy proper shape constraints.