The cut loci and the conjugate loci on ellipsoids

Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces

“…Arc length geodesic curves γ(t) = (θ 1 (t), θ 2 (t)) can be seen as a function of the parameter but we use the one from [7], introducing the unit tangent vector:…”

“…In the analytic case, the cut locus is a finite tree and the extremity of each branch is a cusp point. But the explicit computation of the number of branches and cusps points is a very complicated problem and only very recently was proved the four cusps Jacobi conjecture on ellipsoids [7], [12].…”

Cut loci and conjugate loci on Liouville surfaces

“…In the paper [4], we investigated the cut loci and the conjugate loci of general points on two-dimensional (tri-axial) ellipsoids. In particular, we showed that: 1) the cut locus of a general point is a segment of the curvature line which passes through the antipodal point of the initial point; 2) the conjugate locus of a general point contains just four singularities, which are cusps and located on the curvature lines passing through the antipodal point of the initial point.…”

“…The proof is divided into two steps: First we assume a stronger condition, under which the proof goes completely parallel to the case of ellipsoid [4], and we give only an outline; secondly we use a technique of "projectively equivalent metrics" to weaken the condition. Although projective equivalence does not preserve distance in general, it preserves natural coordinate lines as well as the geodesic orbits in the present case, which is enough for the determination of cut loci.…”

“…In the earlier paper [4], we showed that the cut locus of a general point on two-dimensional ellipsoid is a segment of a curvature line and proved "Jacobi's last geometric statement" on the singularities of the conjugate locus. In the present paper, we show that a wider class of Liouville surfaces possess such simple cut loci and conjugate loci.…”