6 research outputs found
On 2-step, corank 2 nilpotent sub-Riemannian metrics
In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics
that are nilpotent approximations of general sub-Riemannian metrics. We exhibit
optimal syntheses for these problems. It turns out that in general the cut time
is not equal to the first conjugate time but has a simple explicit expression.
As a byproduct of this study we get some smoothness properties of the spherical
Hausdorff measure in the case of a generic 6 dimensional, 2-step corank 2
sub-Riemannian metric
Superintegrability of Sub-Riemannian Problems on Unimodular 3D Lie Groups
Left-invariant sub-Riemannian problems on unimodular 3D Lie groups are
considered. For the Hamiltonian system of Pontryagin maximum principle for
sub-Riemannian geodesics, the Liouville integrability and superintegrability
are proved
Geometrical optical illusion via sub-Riemannian geodesics in the roto-translation group
We present a neuro-mathematical model for geometrical optical illusions (GOIs), a class of illusory phenomena that consists in a mismatch of geometrical properties of the visual stimulus and its associated percept. They take place in the visual areas V1/V2 whose functional architecture have been modeled in previous works by Citti and Sarti as a Lie group equipped with a sub-Riemannian (SR) metric. Here we extend their model proposing that the metric responsible for the cortical connectivity is modulated by the modeled neuro-physiological response of simple cells to the visual stimulus, hence providing a more biologically plausible model that takes into account a presence of visual stimulus. Illusory contours in our model are described as geodesics in the new metric. The model is confirmed by numerical simulations, where we compute the geodesics via SR-Fast Marching
On the subRiemannian cut locus in a model of free two-step Carnot group
We characterize the subRiemannian cut locus of the origin in the free Carnot
group of step two with three generators. We also calculate explicitly the cut
time of any extremal path and the distance from the origin of all points of the
cut locus. Finally, by using the Hamiltonian approach, we show that the cut
time of strictly normal extremal paths is a smooth explicit function of the
initial velocity covector. Finally, using our previous results, we show that at
any cut point the distance has a corner-like singularity.Comment: Added Section 6. Final version, to appear on Calc. Va
On 2-step, corank 2, nilpotent sub-riemannian metrics
In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics that are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a by-product of this study we get some smoothness properties of the spherical Hausdorff measure in the case of a generic 6-dimensional, 2-step, corank 2 sub-Riemannian metric. \ua9 2012 Society for Industrial and Applied Mathematics