Can one measure the temperature of a curve? (Q1082725)

The entropy of a plane curve is defined in terms of the number of intersection points with a random line. The Gibbs distribution which maximizes the entropy enables one to define the temperature of the curve. At 0 temperature, the curve reduces to a straight segment. At high temperature, the curve is somewhat chaotic and ''behaves like a perfect gas''. We attempt to show that thermodynamic formalism can be used for the study of plane curves. The curves we discuss have finite length, unlike \textit{B. B. Mandelbrot}'s fractal curves [The fractal geometry of nature. (1982; Zbl 0504.28001)], yet we feel our approach to the mathematics is not far from his.