Determining surfaces of revolution from their implicit equations

Abstract

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.

Recognizing implicitly given rational canal surfaces

“…Is X rotational? X is a canal surfaces, see [6] for more details c(X ) > 0 X is not a canal suraface by Proposition 2.1 g(H) = 0 X is not a canal surface by Theorem 2.3 g(H) = 1 p:= regular point on X q:= corresponding point, see (8) ∃q?…”

“…The simplest instances of canal surfaces are surfaces of revolution -in this case the spine curve is just a straight line. An algorithm for recognizing surfaces of revolution and deciding on their rationality has been recently presented in [6]. So, throughout this paper it is assumed that that X is not a surface of revolution.…”

“…However our goal is different -we start with an implicit representation (i.e., with some polynomial in x, y, z) and want to decide if the corresponding algebraic surface is a rational canal surface or not. This is still a challenging problem, only partially solved in the recent past for a subfamily of canal surfaces, namely for the surfaces of revolution -see [6]. Moreover, in case of the positive answer we want to compute the equation of the spine curve and also the radius function.…”

Symmetries of canal surfaces and Dupin cyclides

“…If S has a linear spine curve c, then it is a surface of revolution. Surfaces of revolution can be detected by using the methods in [1,27], and their symmetries essentially follow from those of the directrix curve [2, §2.2.4]. Hence from now on we will assume that c is non-linear.…”

Automatic fitting of conical envelopes to free-form surfaces for flank CNC machining

“…On the contrary, our strategy explores the space of axes globally to find the best tangentially-movable line segments. An algorithm that recognizes a surface of revolution from its implicit equation, and computes its axis and meridian, has been presented recently [31].…”