We study symmetries of specific left-invariant sub-Riemannian structure with filtration (4, 7) and their impact on geodesics corresponding control problem. show that there are two very different types geodesics, they either do not intersect the fixed point set or contained in this for all times. use symmetry reduction to properties geodesics.

We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group SE(2) = R o S with a metric tensor depending on a smooth external cost C : SE(2) → [δ, 1], δ > 0, computed from image data. The method consists of a first step where geodesically equidistant surfaces are computed as a viscosity solution of a Ha...

We describe an implementation of a virtual environment for visualizing the geometry of curved spacetime by the display of interactive geodesics. This technique displays the paths of particles under the influence of gravity as described by the general theory of relativity, and is useful in the investigation of solutions to the field equations of that theory. A boom-mounted six degree of freedom ...

We study the structure of a family of static, spherically symmetric space-times generated by an anisotropic fluid and governed by a particular type of f(R) theory. We find that for a range of parameters with physical interest, such solutions represent black holes with the central singularity replaced by a finite size wormhole. We show that time-like geodesics and null geodesics with nonzero ang...

Conditions for the existence of repeatable light paths (RLPs) in the shearfree normal cosmological models are investigated. It is found that in the conformally nonflat models the only RLPs are radial null geodesics (in the spherical case) and their analogues in the plane and hyperbolically symmetric cases. In the conformally flat Stephani models, there exist special spherically, plane, and hype...

Flips of diagonals in colored triangle-free triangulations of a convex polygon are interpreted as moves between two adjacent chambers in a certain graphic hyperplane arrangement. Properties of geodesics in the associated flip graph are deduced. In particular, it is shown that: (1) every diagonal is flipped exactly once in a geodesic between distinguished pairs of antipodes; (2) the number of ge...

In Riemannian metrics that are only Hölder continuous of order α, 0 α 1, let minimal geodesics be the continuous curves realizing the shortest distance between two points. It is shown that for 0 < α 1, minimal geodesics are differentiable with a derivative which is at least Hölder continuous of order α=2 for 0 < α < 1 and which is Hölder continuous of order 1 for α= 1.