A new method to compute the singularities of offsets to rational plane curves

Juan Gerardo Alcázar ^[1] ; Jorge Caravantes ^[2] ; Gema M. Diaz-Toca ^[3]
1. [1] Universidad de Alcalá
  
  Universidad de Alcalá
  
  Alcalá de Henares, España
2. [2] Universidad Complutense de Madrid
  
  Universidad Complutense de Madrid
  
  Madrid, España
3. [3] Universidad de Murcia
  
  Universidad de Murcia
  
  Murcia, España
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Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 290, Nº 1, 2015, págs. 385-402
Idioma: inglés
DOI: 10.1016/j.cam.2015.06.001
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Resumen
- Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from the parametrization of the original curve, without computing or making use of the implicit equation of the offset. By using this result, a finite set containing all the real singularities of the offset, and in particular all the real self-intersections of the offset, can be computed. We also report on experiments carried out in the computer algebra system Maple, showing the efficiency of the algorithm for moderate degrees.