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The Blaschke-Petkantschin formula is a geometric measure decomposition of the q-fold product of Lebesgue measure on ℝ n. Here we discuss another decomposition called polar decomposition by considering ℝ n× ×ℝ n as n×k and using its polar decomposition. This is a generalisation of the Blaschke-Petkantschin formula and may be useful when one needs to integrate a function g:ℝ n××ℝ n→ℝ with rotational symmetry, that is, for each orthogonal transformation O,g(O(x1),O(xk))=g(x1,xk) . As an application we compute the moments of a Gaussian determinant. Copyright  

A pair of bodies rolling on each other is an interesting example of nonholonomic systems in control theory. There is a geometric condition equivalent to the rolling constraint which enables us to generalize the rolling motions for any two-dimensional Riemannian manifolds. This system has a five-dimensional phase space. In order to study the controllability of the rolling surfaces, we lift the system to a six-dimensional space and show that the lifted system is controllable unless the two surfaces have isometric universal covering spaces. In the non-controllable case there are some three-dimensional orbits each of which corresponds to an isometry of the universal covering spaces  

A pair of bodies rolling on each other is an interesting example of nonholonomic systems in control theory. Here the controllability of rolling bodies is investigated with a global approach. By using simple geometric facts, this problem has been completely solved in the special case where one of them is a plane or a sphere  

This paper proposes distributed optimal attitude consensus control for single-integrator multi rigid bodies with undirected network evolving on Special Orthogonal Group SO(3) while simultaneously guarantees the connectivity preservation property for agents using descent gradient algorithm. Since by Use of the Euclidean distance on Lie group as a measure of the energy of the state does not define and preserve the topology of SO(3); besides, solving the Hamilton–Jacobi–Bellman equation in optimal control problems shows difficulty implementing Euclidean distances and limits the results for SO(3) configuration state spaces. As a result, in this paper, the generic distance on SO(3) associated to... 

Shape analysis plays an important role in many medical imaging studies. One of the recent shape analysis methods uses the Laplace Beltrami eigenvalues which is also used in this paper for global shape comparison of hippocampus of normal subjects and epileptic patients. Popularity of the Laplace Beltrami operator in this field is due to its isometry-invariance which avoids pre-processing steps like mapping, registration, and alignment. In addition, it is capable of revealing fine details in shapes that makes this method a good choice for deformation detecting purposes like epilepsy diagnosis. To examine capability of the proposed method, statistical analysis and two ways of classification,...