Jan Vršek

Determining surfaces of revolution from their implicit equations

Results of number of geometric operations are in many cases surfaces described implicitly. Then it is a challenging task to recognize the type of the obtained surface, find its characteristics and for the rational surfaces compute also their parameterizations. In?this contribution we will focus on surfaces of revolution. These objects, widely used in geometric modelling, are generated by rotating a generatrix around a given axis. If the generatrix is an algebraic curve then so is also the resulting surface, described uniquely by a polynomial which can be found by some well-established implicitation technique. However, starting from a polynomial it is not known how to decide if the corresponding algebraic surface is rotational or not. Motivated by this, our goal is to formulate a simple and efficient algorithm whose input is a?polynomial with the coefficients from some subfield of R and the output is the answer whether the shape is a surface of revolution. In the affirmative case we also find the equations of its axis and generatrix. Furthermore, we investigate the problem of rationality and unirationality of surfaces of revolution and show that this question can be efficiently answered discussing the rationality of a certain associated planar curve.

On a special class of polynomial surfaces with pythagorean normal vector fields

Rational shapes with rational offsets, especially Pythagorean hodograph (PH) curves and Pythagorean normal vector (PN) surfaces, have been thoroughly studied for many years. However compared to PH curves, Pythagorean normal vector surfaces were introduced using dual approach only in their rational version and a complete characterization of polynomial surfaces with rational offsets, i.e., a polynomial solution of the well-known surface Pythagorean condition, still remains an open and challenging problem. In this contribution, we study a remarkable family of cubic polynomial PN surfaces with birational Gauss mapping, which represent a surface counterpart to the planar Tschirnhausen cubic. A full description of these surfaces is presented and their properties are discussed.

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.

Surfaces with Pythagorean normals along rational curves

A rational curve on a rational surface such that the unit normal vector field of the surface along this curve is rational will be called a curve providing Pythagorean surface normals (or shortly a PSN curve). These curves represent rational paths on the surface along which the surface possesses rational offset curves. Our aim is to study rational surfaces containing enough PSN curves. The relation with PN surfaces will be also investigated and thoroughly discussed. The algebraic and geometric properties of PSN curves will be described using the theory of double planes. The main motivation for this contribution is to bring the theory of rational offsets of rational surfaces closer to the practical problems appearing in numerical-control machining where the milling cutter does not follow continuously the whole offset surface but only certain chosen trajectories on it. A special attention will be devoted to rational surfaces with pencils of PSN curves. An algebraic analysis of components of offsets to irreducible surfaces along rational curves is provided.A simple criterion for deciding if the curve on the surface is (proper/non-proper) PSN is formulated.Surfaces with PSN pencils such that a generic fiber is a proper/non-proper PSN curve are investigated.It is shown that all developable surfaces are PSN pencils (but not necessarily PN surfaces).For non-developable surfaces a method for generating all real parameterizations respecting PSN pencils is designed.

Reducibility of offsets to algebraic curves

Computing offset curves and surfaces is a fundamental operation in many technical applications. This paper discusses some issues that are encountered during the process of designing offsets, especially the problems of their reducibility and rationality (which are closely related). This study is crucial especially for formulating subsequent algorithms when the number and quality of offset components must be revealed. We will formulate new algebraic and geometric conditions on reducibility of offsets and demonstrate how they can be applied. In addition, we will present that our investigations can also serve to better understand the varieties fulfilling the Pythagorean conditions (PH curves/PN surfaces). A certain analogy of the PH condition for parameterized curves (or general parameterized hypersurfaces) will be presented also for implicitly given (not necessarily rational) curves (or hypersurfaces).

Recognizing implicitly given rational canal surfaces

It is still a challenging task of today to recognize the type of a given algebraic surface which is described only by its implicit representation. In~this paper we will investigate in more detail the case of canal surfaces that are often used in geometric modelling, Computer-Aided Design and technical practice (e.g. as blending surfaces smoothly joining two parts with circular ends). It is known that if the squared medial axis transform is a rational curve then so is also the corresponding surface. However, starting from a polynomial it is not known how to decide if the corresponding algebraic surface is rational canal surface or not. Our goal is to formulate a simple and efficient algorithm whose input is a~polynomial with the coefficients from some subfield of real numbers and the output is the answer whether the surface is a rational canal surface. In the affirmative case we also compute a rational parameterization of the squared medial axis transform which can be then used for finding a rational parameterization of the implicitly given canal surface.

Algebraic curves of low convolution degree

Studying convolutions of hypersurfaces (especially of curves and surfaces) has become an active research area in recent years. The main characterization from the point of view of convolutions is their convolution degree, which is an affine invariant associated to a hypersurface describing the complexity of the shape with respect to the operation of convolution. Extending the results from [1], we will focus on the two simplest classes of planar algebraic curves with respect to the operation of convolution, namely on the curves with the convolution degree one (so called LN curves) and two. We will present an algebraic analysis of these curves, provide their decomposition, and study their properties.

Translation Surfaces and Isotropic Transport Nets on Rational Minimal Surfaces

We will deal with the translation surfaces which are the shapes generated by translating one curve along another one. We focus on the geometry of translation surfaces generated by two algebraic curves in space and study their properties, especially those useful for geometric modelling purposes. It is a classical result that each minimal surface may be obtained as a translation surface generated by an isotropic curve and its complex conjugate. Thus, we can study the minimal surfaces as special instances of translation surfaces. All the results about translation surfaces will be directly applied also to minimal surfaces. Finally, we present a construction of rational isotropic curves with a prescribed tangent field which leads to the description of all rational minimal surfaces. A close relation to surfaces with Pythagorean normals will be also discussed.

Smooth surface interpolation using patches with rational offsets

We present a new method for the interpolation of given data points and associated normals with surface parametric patches with rational normal fields. We give some arguments why a dual approach is the most convenient for these surfaces, which are traditionally called Pythagorean normal vector (PN) surfaces. Our construction is based on the isotropic model of the dual space to which the original data are pushed. Then the bicubic Coons patches are constructed in the isotropic space and then pulled back to the standard three dimensional space. As a result we obtain the patch construction which is completely local and produces surfaces with the global G1~continuity.

Exploring hypersurfaces with offset-like convolutions

Offsetting is one of the fundamental operations in Computer Aided Design. Due to their high applicability, studying offsets of hypersurfaces has become a popular research area and many interesting problems related to this topic have arisen. In addition, various generalizations of classical offsets have been introduced and then investigated. In this paper we study a generalization which is based on considering offsets to (not only parameterized) hypersurfaces as convolutions with hyperspheres. In other words, we study hypersurfaces sharing the same convolution properties with hyperspheres and thus yielding offset-like convolutions. We will present an algebraic analysis of these hypersurfaces and study their properties suitable for subsequent applications, e.g. in geometric modelling. Moreover, our approach allows to derive distinguished properties of the well-known PH/PN parameterizations as special subcases of the introduced QN parameterizations.