Results of number of geometric operations (often used in technical practise, as e.g. the operation of blending) 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.

This paper is devoted to the investigation of selected situations when the computation of projective (and other) equivalences of algebraic varieties can be efficiently solved with the help of finding projective equivalences of finite sets on the projective line. In particular, we design a unifying approach that finds for two algebraic varieties X, Y from special classes an associated set of automorphisms of the projective line (the so called good candidate set) consisting of candidates for the construction of possible mappings X → Y . The functionality of the designed method is presented on computing projective equivalences of rational curves, on determining projective equivalences of rational ruled surfaces, on the detection of affine transformations between planar curves, and on computing similarities between two implicitly given algebraic surfaces. When possible, symmetries of given shapes are also discussed as special cases.

We present novel, deterministic, efficient algorithms to compute the symmetries of a planar algebraic curve, implicitly defined, and to check whether or not two given implicit planar algebraic curves are similar, i.e. equal up to a similarity transformation. Both algorithms are based on the fact, well-known in Harmonic Analysis, that the Laplacian operator commutes with orthogonal transformations, and on efficient algorithms to find the symmetries/similarities of a harmonic algebraic curve/two given harmonic algebraic curves. In fact, we show that in general the problem can be reduced to the harmonic case, except for some special cases, easy to treat.

We present a simple functional 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 especially 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 G 1 continuity.

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