In this paper we show that Poisson brackets linked to geometric flows of curves on flat Riemannian manifolds are Poisson reductions of the Kac–Moody bracket of SO(n). The bracket is reduced to submanifolds defined by either the Riemannian or the natural curvatures of the curves. We show that these two cases are (formally) Poisson equivalent and we give explicit conditions on the coefficients of...

In this paper, we propose a new approach to estimate curvature information of point-sampled surfaces. We estimate curvatures in terms of the extremal points of a one-dimensional energy function for discrete surfels (points equipped with normals) and a multi-dimensional energy function for discrete unstructured point clouds. Experimental results indicate that our approaches can estimate curvatur...

In this work, the notion of a slant helix is extended to the space E. First, we introduce the type-2 harmonic curvatures of a regular curve. Thereafter, by using this, we present some necessary and sufficient conditions for a curve to be a slant helix in Euclidean n-space. We also express some integral characterizations of such curves in terms of the curvature functions. Finally, we give some c...

Symbolic computation of NURBS plays an important role in many areas of NURBSbased geometric computation and design. However, any nontrivial symbolic computation, especially when rational B-splines are involved, would typically result in B-splines with high degrees. In this paper we develop degree reduction strategies for NURBS symbolic computation on curves. The specific topics we consider incl...

A classical theorem due to M. Berger [2] and W. Klingenberg [11] states that a simply connected Riemannian manifold whose sectional curvatures all lie in the interval [1, 4] is either isometric to a symmetric space or homeomorphic to Sn (see also [12], Theorems 2.8.7 and 2.8.10). In this paper, we provide a classification, up to diffeomorphism, of all Riemannian manifolds whose sectional curvat...

The shape of a liquid surface bounded by an acute or obtuse planar angular sector is considered by using classical analysis methods. For acute angular sectors the two principal curvatures are of the order of the (fixed) mean curvature. But for obtuse sectors, the principal curvatures both diverge as the vertex is approached. The powerlaw divergence becomes stronger with increasing opening angle...