Return of the plane evolute
[Retour de la développée plane]
Piene, Ragni1 ; Riener, Cordian2 ; Shapiro, Boris3
1 Department of Mathematics, University of Oslo, NO-0316 Oslo (Norway)
2 Department of Mathematics and Statistics, UiT The Arctic University of Norway, NO-9037 Tromsø(Norway)
3 Department of Mathematics, Stockholm University, SE-106 91, Stockholm (Sweden)
Annales de l'Institut Fourier, Online first, 67 p.

Nous considérons les développées des courbes algébriques réelles planes et discutons certaines de leurs propriétés dans le cas complexe et réel. En particulier, pour un degré donné d≥2, nous fournissons des bornes inférieures pour les quatre invariants numériques suivants :

En particulier, pour un degré donné d≥2, nous fournissons des bornes inférieures pour les quatre invariants numériques suivants :

  • (1) le nombre maximal de fois qu’une droite réelle peut intersecter la développée d’une courbe algébrique réelle de degré d ;
  • (2) le nombre maximal de points de rebroussement réels pouvant survenir sur la développée d’une courbe algébrique réelle de degré d ;
  • (3) le nombre maximal de points double ordinaires pouvant survenir sur la courbe duale à la développée d’une courbe algébrique réelle de degré d ;
  • (4) le nombre maximal de points double ordinaires pouvant survenir sur la développée d’une courbe algébrique réelle de degré d.

We consider the evolutes of plane real-algebraic curves and discuss some of their complex and real-algebraic properties. In particular, for a given degree d≥2, we provide lower bounds for the following four numerical invariants:

  • (1) the maximal number of times a real line can intersect the evolute of a real-algebraic curve of degree d;
  • (2) the maximal number of real cusps which can occur on the evolute of a real-algebraic curve of degree d;
  • (3) the maximal number of crunodes which can occur on the dual curve to the evolute of a real-algebraic curve of degree d;
  • (4) the maximal number of crunodes which can occur on the evolute of a real-algebraic curve of degree d.
Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3703
Classification : 14H50, 51A50, 51M05
Keywords: evolute, plane real algebraic curve
Mots-clés : développée, courbe algébrique réelle plane

Piene, Ragni 1 ; Riener, Cordian 2 ; Shapiro, Boris 3

1 Department of Mathematics, University of Oslo, NO-0316 Oslo (Norway)
2 Department of Mathematics and Statistics, UiT The Arctic University of Norway, NO-9037 Tromsø(Norway)
3 Department of Mathematics, Stockholm University, SE-106 91, Stockholm (Sweden) 
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Piene, Ragni; Riener, Cordian; Shapiro, Boris. Return of the plane evolute. Annales de l'Institut Fourier, Online first, 67 p.

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