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The choice of an appropriate representation of geometric objects (explicit, parametric, or implicit one) is a fundamental issue for the development of efficient algorithms. Whereas for example Computer Graphics seem to use all the above mentioned representations, CAD focuses on a few of them – mainly on the parametric one. Among all parameterizatio...
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Citations
... A simplified version of Algorithm 1 (only for ordinary curves) can be found in [1] or [20]. In this case, it is not necessary to construct the neighbouring graph N of C -the system of adjoint curves is given only by the singularities and by suitable simple point(s). ...
A rational parameterization of an algebraic curve yields a rational correspondence between this curve and the affine or projective line. One of the parameterization methods is based on finding all singular points and d-3 simple points of an implicitly given curve of degree d (see [17]). In this paper, we study some modifications of this well-known algorithm, which are then verified on several examples.
... In this section, we recall the algorithm for computing exact rational parameterizations of algebraic curves, which is based on the theory from [15] and then studied in more detail and modified in [3,4,11,12,13,14]. In the next section we will use this classical algorithm for finding an approximate parameterization of a given rational curve. ...
It is well known that an irreducible algebraic curve is rational if and only if its genus is zero. In this paper, we provide a simple symbolic algorithm to parameterize approximately affine rational plane curves by means of linear systems of adjoint curves and one chosen rational point with the guaranty that the coefficients of the obtained parameterization are rational. The designed technique is suitable especially for curves not containing rational points and for curves for which it is too difficult to find these points.
A simple algorithm for computing an approximate parameterization of real space algebraic curves using their graphs of critical points is designed and studied in this paper. The first step is determining a suitable space graph which contains all critical points of a real algebraic space curve CC implicitly defined as the complete intersection of two surfaces. The construction of this graph is based on one projection of CC in a general position onto an xyxy-plane and on an intentional choice of vertices. The second part of the designed method is a computation of a spline curve which replaces the edges of the constructed graph by segments of a chosen free-form curve. This step is formulated as an optimization problem when the objective function approximates the integral of the squared Euclidean distance of the constructed approximate curve to the intersection curve. The presented method, based on combining symbolic and numerical steps to the approximation problem, provides approximate parameterizations of space algebraic curves from a small number of approximating arcs. It may serve as a first step to several problems originating in technical practice where approximation curve parameterizations are needed.
