We provide a method to compute the topology of the image of a parametric curve under a birational mapping of the plane. The method proceeds by exploiting as much as possible the initial parametrization in order to reduce the computational cost. The self-intersections of the image curve are derived from points in the image where the inverse of the birational mapping is not defined. In order to compute these points, we prove a result characterizing birational planar mappings, together with an algorithm to compute the inverse of a birational mapping. We apply the method when the original curve is rational, in which case the image of the curve is also rational but with a higher degree, and when the original curve is an exp-log-arctan function. In this last case the image is a non-rational curve admitting an analytic parametrization, a problem not treated in the literature so far.

We present a complete algorithm to compute the rotational, axial, reflectional and central symmetries of an algebraic surface defined by means of its implicit equation. The algorithms rely on the fact that the symmetries of the surface satisfy an algebraic condition verified by the polynomial defining the surface, and use a variety of ideas, some of them extensions to space of ideas already used to compute the symmetries of planar implicit curves, and some of them new. We flesh out our algorithms with several examples.

We provide a simple and efficient algorithm for recognizing whether or not a given rational surface is ruled, and, in the affirmative case, for computing a rational standard parametrization, i.e. a rational parametrization of the form x(t,s)=u(t)+sv(t), where u(t), v(t) are rational vector functions. The results are based on the fact, proved in the paper, that the asymptotic directions of a ruled rational surface are rational.

Given an implicit polynomial equation or a rational parametrization, we develop algorithms to determine whether the set of real and complex points defined by the equation, i.e., the surface defined by the equation, in the sense of Algebraic Geometry, is a cylindrical surface, a conical surface, or a surface of revolution. The algorithms are directly applicable to, and formulated in terms of, the implicit equation or the rational parametrization. When the surface is cylindrical, we show how to compute the direction of its rulings; when the surface is conical, we show how to compute its vertex; and when the surface is a surface of revolution, we show how to compute its axis of rotation directly from the defining equations.

We provide an algorithm to check whether two rational space curves are related by a similarity, i.e., whether they are equal up to position, orientation and scale. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra setting. Helical curves, where curvature and torsion are proportional, need to be distinguished as a special case. The algorithm is easy to implement, as it involves only standard computer algebra techniques, such as greatest common divisors and resultants, and Grobner bases for the special case of helical curves.

We present an algorithm for extracting the axis of revolution from the implicit equation of an algebraic surface of revolution based on three distinct computational methods: factoring the highest order form into quadrics, contracting the tensor of the highest order form, and using univariate resultants and gcds. We compare and contrast the advantages and disadvantages of each of these three techniques and we derive conditions under which each technique is most appropriate. In addition, we provide several necessary conditions for an implicit algebraic equation to represent a surface of revolution.