Characterization and computation of closed geodesics on toroidal surfaces. (English) Zbl 1228.53006

Mourad, Ayman; Moukadem, Imad

J. Geom. Symmetry Phys. 16, 23-37 (2009).

Summary: The aim of the present study is to characterize and compute closed geodesics on toroidal surfaces. We show that a closed geodesic must make a number of rotations about the equatorial part (\(k\) rotations) and the axis of revolution (\(k'\) rotations) of the surface. We give the relation that exists between the numbers \(k\) and \(k'\), and Clairaut’s constant \(C\) corresponding to the geodesic. Moreover, we prove that the numbers \(k\) and \(k'\) are relatively prime. We validate our findings by constructing closed geodesics on some examples of toroidal surfaces using MAPLE. Finally, using experimental data on cardiac fiber direction, we show that the fibers run as geodesies in the left ventricle whose geometrical shape looks like a toroidal surface.

MSC:

53A05	Surfaces in Euclidean and related spaces
53C22	Geodesics in global differential geometry
68U05	Computer graphics; computational geometry (digital and algorithmic aspects)
92C05	Biophysics

Keywords:

toroidal surfaces; Clairaut’s constant; cardiac fiber; ventricle

Software:

Maple