Determining surfaces of revolution from their implicit equations

摘要

Results of number of geometric operations (often used in technical practise,as e.g. the operation of blending) are in many cases surfaces describedimplicitly. Then it is a challenging task to recognize the type of the obtainedsurface, find its characteristics and for the rational surfaces compute alsotheir parameterizations. In this contribution we will focus on surfaces ofrevolution. These objects, widely used in geometric modelling, are generated byrotating a generatrix around a given axis. If the generatrix is an algebraiccurve then so is also the resulting surface, described uniquely by a polynomialwhich can be found by some well-established implicitation technique. However,starting from a polynomial it is not known how to decide if the correspondingalgebraic surface is rotational or not. Motivated by this, our goal is toformulate a simple and efficient algorithm whose input is a polynomial with thecoefficients from some subfield of $\mathbb{R}$ and the output is the answerwhether the shape is a surface of revolution. In the affirmative case we alsofind the equations of its axis and generatrix. Furthermore, we investigate theproblem of rationality and unirationality of surfaces of revolution and showthat this question can be efficiently answered discussing the rationality of acertain associated planar curve.