A delineability-based method for computing critical sets of algebraic surfaces

Detection of Special Curves Via the Double Resultant

“…We remark that the name special values or critical values is also used in the literature to name the values where the corresponding double discriminants vanish. These values also play a significative role in the characterization of singular curves and they are used in many applications, see for instance [2,19,22].…”

“…We describe here two main differences between our approach and previous works dealing with similar questions, see for instance [1,2,3,19,22]. The first one is that we use resultants instead of discriminants.…”

“…The notion of offset is directly related to the concept of envelope. More precisely, the offset curve, at distance d, to an irreducible curve C over C is "essentially" the envelope of the system of circles centered at the points of C with fixed radius d. A more formal description of offset varieties can be found in [2].…”

“…Then, gcd(E 1 , E 2 ) = t 3 and in this case In [2] the double discriminant is used to study the topology of the offset curves, see also [1]. Our approach using the double resultant ∆ 2 * (F d ) is more precise in the sense that it discards some superfluous cases, because of the computation of both double resultants.…”

Exact geometric-topological analysis of algebraic surfaces

“…For topology computation of algebraic surfaces, two principle approaches can be distinguished: one is to consider level-curves of the surface for certain critical values and to connect the components of these levels in order to obtain a topological description of the surface; see the recent works of Mourrain and Técourt [37] (also in [11]), Fortuna et al [25], [26] (for non-singular curves) and Alcázar et al [1] (where the connection step is missing). The other approach is to project the critical points of the surface to the plane, obtaining the silhouette curve.…”