Implicit variable-radius arc canal surfaces for solid modeling

In this paper we consider the problem of obtaining an implicit form for the canal surface whose spine is the arc and the radius changes linearly in respect to the angle. We present a number of different solutions to the problem including exact and approximated ones and discuss the scenarios where each of the solutions is appropriate to use in solid modeling with real functions

Using implicit equations of parametric curves and surfaces without computing them: Polynomial algebra by values

The availability of the implicit equation of a plane curve or of a 3D surface can be very useful in order to solve many geometric problems involving the considered curve or surface: for example, when dealing with the point position problem or answering intersection questions. On the other hand, it is well known that in most cases, even for moderate degrees, the implicit equation is either difficult to compute or, if computed, the high degree and the big size of the coefficients makes extremely difficult its use in practice. We will show that, for several problems involving plane curves, 3D surfaces and some of their constructions (for example, offsets), it is possible to use the implicit equation (or, more precisely, its properties) without needing to explicitly determine it. We replace the computation of the implicit equation with the evaluation of the considered parameterizations in a set of points. We then translate the geometric problem in hand, into one or several generalized eigenvalue problems on matrix pencils (depending again on several evaluations of the considered parameterizations). This is the so-called “polynomial algebra by values” approach where the huge polynomial equations coming from Elimination Theory (e.g., using resultants) are replaced by big structured and sparse numerical matrices. For these matrices there are well-known numerical techniques allowing to provide the results we need to answer the geometric questions on the considered curves and surfaces

Dupin Cyclides as a Subspace of Darboux Cyclides

Dupin cyclides are interesting algebraic surfaces used in geometric design and architecture to join canal surfaces smoothly and construct model surfaces. Dupin cyclides are special cases of Darboux cyclides, which in turn are rather general surfaces in R^3 of degree 3 or 4. This article derives the algebraic conditions (on the coefficients of the implicit equation) for recognition of Dupin cyclides among the general implicit form of Darboux cyclides. We aim at practicable sets of algebraic equations describing complete intersections inside the parameter space.Comment: 20 pages, 1 figur

INTERSECTIONS OF SURFACES OF REVOLUTION

In this paper, we deal with surfaces of revolution and their intersections. We start with the surfaces of revolution RS that have their axis along the x3–axis and find intersections with a line, a plane, and then intersection of two such RS. Furthermore, we apply formulas for the intersection with a line to determine the visibility of RS. Later we develop formulas for the intersection of two surfaces of revolution that have their axis along different arbitrary straight lines, and, as a special case, the intersections of two spheres and intersections of general surface of revolution with a sphere and a surface given by an equation. We apply our own software to the graphical representation of all the results we present

Fourier spectra from exoplanets with polar caps and ocean glint

The weak orbital-phase dependent reflection signal of an exoplanet contains information on the planet surface, such as the distribution of continents and oceans on terrestrial planets. This light curve is usually studied in the time domain, but because the signal from a stationary surface is (quasi)periodic, analysis of the Fourier series may provide an alternative, complementary approach. We study Fourier spectra from reflected light curves for geometrically simple configurations. Depending on its atmospheric properties, a rotating planet in the habitable zone could have circular polar ice caps. Tidally locked planets, on the other hand, may have symmetric circular oceans facing the star. These cases are interesting because the high-albedo contrast at the sharp edges of the ice-sheets and the glint from the host star in the ocean may produce recognizable light curves with orbital periodicity, which could also be interpreted in the Fourier domain. We derive a simple general expression for the Fourier coefficients of a quasiperiodic light curve in terms of the albedo map of a Lambertian planet surface. Analytic expressions for light curves and their spectra are calculated for idealized situations, and dependence of spectral peaks on the key parameters inclination, obliquity, and cap size is studied.Comment: 15 pages, 2 tables, 13 figure

Eliciting Expertise

Since the last edition of this book there have been rapid developments in the use and exploitation of formally elicited knowledge. Previously, (Shadbolt and Burton, 1995) the emphasis was on eliciting knowledge for the purpose of building expert or knowledge-based systems. These systems are computer programs intended to solve real-world problems, achieving the same level of accuracy as human experts. Knowledge engineering is the discipline that has evolved to support the whole process of specifying, developing and deploying knowledge-based systems (Schreiber et al., 2000) This chapter will discuss the problem of knowledge elicitation for knowledge intensive systems in general

Computing all parametric solutions for blending parametric surfaces

Publicado en OA (open archive)In this paper we prove that, for a given set of parametric primary surfaces and parametric clipping curves, all parametric blending solutions can be expressed as the addition of a particular parametric solution and a generic linear combination of the basis of a free module of rank 3. As a consequence, we present an algorithm that outputs a generic expression for all the parametric solutions for the blending problem. In addition, we also prove that the set of all polynomial parametric solutions (i.e. solutions that have polynomial parametrizations) for a parametric blending problem can also be expressed in terms of the basis of a free module of rank 3, and we prove an algorithmic criterion to decide whether there exist parametric polynomial solutions. As a consequence we also present an algorithm that decides the existence of polynomial solutions, and that outputs (if this type of solution exists) a generic expression for all polynomial parametric solutions for the problem