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We present a method for computing all the affine equivalences between two rational ruled surfaces defined by rational parametrizations that works directly in parametric rational form, i.e. without computing or making use of the implicit equation of the surface. The method proceeds by translating the problem into the parameter space, and relies on polynomial system solving. From a geometric point of view, an important observation is the fact that the problem is related to finding the projective equivalences between two projective curves (corresponding to the directions of the rulings of the surfaces). This problem was recently addressed by Hauer and Jüttler in [11], and the ideas by these authors are greatly exploited in the algorithm presented in this paper. The general idea is adapted to compute the isometries between two rational ruled surfaces, and the symmetries of a given rational ruled surface. The efficiency of the method is shown through several examples. and comprise translations, central symmetries, reflections in a plane, rotational symmetries (with axial symmetries as a special case), and their composites. A symmetry of a surface is a symmetry of 3-space that leaves the surface invariant. In particular, symmetries of 3-space are orthogonal transformations. Thus, two surfaces are isometric when one of them is the result of applying a rigid motion to the other one.Additionally, knowing the symmetries of a surface is useful in order to understand the geometry of the surface and to visualize the surface correctly. It is also useful in applications like image storage and medial axis computations, or, again, object detection and recognition. In the literature of applied fields like Computer Aided Geometric Design, Pattern Recognition or Computer Vision one can find many methods to detect symmetries (see for instance the Introduction to [1]), although these methods are usually applied to objects where no specific structure is assumed, and are more orientented towards finding approximate symmetries.The same thing can be said about methods to identify affine equivalences; see for example the paper [16] and the references provided therein. In fact, in applications the problem which has received more attention is the detection of affine equivalences between point clouds, since images and objects are often considered this way.In contrast, in this paper we address a type of surfaces with a strong structure, namely rational ruled algebraic surfaces, and we make use of the structure of the surfaces in order to compute affine equivalences, isometries or symmetries. Ruled surfaces consist of straight lines, and are classical in Differential and Algebraic Geometry. A complete account of many properties of these surfaces is given, for instance, in the books [13] and [15].Some recent publications address similar problems for curves and surfaces, too. Projective and affine equivalences between rational curves in arbitrary dimension are considered in [11]. The same problem for rational and polynomial surfaces is considered...

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We provide an algorithm to detect whether two bounded, planar parametrized curves are similar, i.e. whether there exists a similarity transforming one of the curves onto the other. The algorithm is valid for completely general parametrizations, and can be adapted to the case when the input is given with finite precision, using the notion of approximate [Formula: see text]. The algorithm is based on the computation of centers of gravity and inertia tensors of the considered curves or of the planar regions enclosed by the curves, which have nice properties when a similarity transformation is applied. In more detail, the centers of gravity are mapped onto each other, and the matrices representing the inertia tensors satisfy a simple relationship: when the similarity is a congruence (i.e. distances are preserved) the matrices are congruent, and in the more general case the relationship is analogous, but involves the square of the scaling constant. Using both properties, and except for certain pathological cases, the similarities can be found. Additional ideas are presented for the case of closed, i.e. compact, curves.