  We propose a simple method of explicit description of families of closed
geodesics on a triaxial ellipsoid $Q$ that are cut out by algebraic surfaces in
${\mathbb R}^3$. Such geodesics are either connected components of spatial
elliptic curves or rational curves.
  Our approach is based on elements of the Weierstrass--Poncar\'e reduction
theory for hyperelliptic tangential covers of elliptic curves and the addition
law for elliptic functions.
  For the case of 3-fold and 4-fold coverings, explicit formulas for the
cutting algebraic surfaces are provided and some properties of the
corresponding geodesics are discussed.
