Evolutes and isoperimetric deficit in two-dimensional spaces of constant curvature

Julià Cufí and Agustí Reventós

Address: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193-Bellaterra (Barcelona), Catalonia, Spain

E-mail:
jcufi@mat.uab.cat
agusti@mat.uab.cat

Abstract: We relate the total curvature and the isoperimetric deficit of a curve $\gamma $ in a two-dimensional space of constant curvature with the area enclosed by the evolute of $\gamma $. We provide also a Gauss-Bonnet theorem for a special class of evolutes.

AMSclassification: primary 53A04; secondary 52A55, 52A10.

Keywords: curvature, evolutes, isoperimetric deficit, Gauss-Bonnet.

DOI: 10.5817/AM2014-4-219