This paper explores an analytic, geometric approach to finding optimal routes for commercial formation flight. A weighted extension of the classical Fermat point problem is used to develop a scalable methodology for the formation routing problem, enabling quick calculation of formation costs. This rapid evaluation allows the large-scale fleet assignment problem to be solved via a mixed integer linear program in reasonable time. Weighting schemes for aircraft performance characteristics are first introduced and then extended to allow for differential rates of fuel burn. Finally, a case study for 210 transatlantic flight routes is presented, with results showing possible average fuel-burn savings against solo flight of around 8.7% for formations of two and 13.1% for formations of up to three.

Analytic Approach to Optimal Routing for Commercial Formation Flight

We extend the theoretical analysis of a recently proposed single subspace learning algorithm, called Dual Principal Component Pursuit (DPCP), to the case where the data are drawn from of a union of hyperplanes. To gain insight into the properties of the $\ell_1$ non-convex problem associated with DPCP, we develop a geometric analysis of a closely related continuous optimization problem. Then transferring this analysis to the discrete problem, our results state that as long as the hyperplanes are sufficiently separated, the dominant hyperplane is sufficiently dominant and the points are uniformly distributed inside the associated hyperplanes, then the non-convex DPCP problem has a unique global solution, equal to the normal vector of the dominant hyperplane. This suggests the correctness of a sequential hyperplane learning algorithm based on DPCP. A thorough experimental evaluation reveals that hyperplane learning schemes based on DPCP dramatically improve over the state-of-the-art methods for the case of synthetic data, while are competitive to the state-of-the-art in the case of 3D plane clustering for Kinect data.

Hyperplane Clustering Via Dual Principal Component Pursuit

A Geometric Approach to Optimal Routing for Commercial Formation Flight

The von Neumann-Mullins law for the area evolution of a cell in the plane describes how a dry foam coarsens in time Recent theory and experiment suggest that the dynamics are different on the surface of a three-dimensional object such as a sphere This work considers the dynamics of dry foams on the surface of a sphere Starting from first principles, we use computer simulation to show that curvature-driven motion of the cell boundaries leads to exponential growth and decay of the areas of cells, in contrast to the planar case where the growth is linear We describe the evolution and distribution of cells to the final stationary state

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Curvature-driven foam coarsening on a sphere: A computer simulation.

The Fermat-Torricelli problem of triangles on the sphere under Euclidean metric asks to find the optimal point P on the sphere $S^2$ for three given points A, B, C on $S^2$, so that the sum of the Euclidean distances $L=PA+PB+PC$ from that point P to the three vertices is minimal (or maximal). In this paper we introduce a solution to this problem done with help of the symbolic computation software Maple and interpolation of implicit function, where the minimal and the maximal sum of the distances are expressed by same polynomial f(L, a, b, c) of degree 12 with $a=BC, b=CA, c=AB$.

Khuloud Ghalieh

Papers

80.45 The Fermat point of a spherical triangle