We consider the inverse curvature flow of strictly convex hypersurfaces in the space form N of constant sectional curvature K_N with speed given by F^{−α}, where α∈(0,1] for K_N=0,−1 and α=1 for K_N=1, F is a smooth, symmetric homogeneous of degree one function which is inverse concave and has dual F_∗ approaching zero on the boundary of the positive cone Γ_+. We show that the ratio of the largest principal curvature to the smallest principal curvature of the flow hypersurface is controlled by its initial value. This can be used to prove the smooth convergence of the flows.

We study the rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvaturelike quantities for polyhedral surfaces are introduced and are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law. They can be considered as 2-dimensional counterparts of the Schlaefli formula.

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In this paper the Orlicz鈥揗inkowski problem, a generalization of the classical Minkowski problem, is studied. Using the variational method, we obtain a new existence result of solutions to this problem for general measures.

Laser scanning is an effective tool for acquiring geometric attributes of trees and vegetation, which lays a solid foundation for 3-dimensional tree modelling. Existing studies on tree modelling from laser scanning data are vast. However, some works cannot guarantee sufficient modelling accuracy, while some other works are mainly rule-based and therefore highly depend on user inputs. In this paper, we propose a novel method to accurately and automatically reconstruct detailed 3D tree models from laser scans. We first extract an initial tree skeleton from the input point cloud by establishing a minimum spanning tree using the Dijkstra shortest-path algorithm. Then, the initial tree skeleton is pruned by iteratively removing redundant components. After that, an optimization-based approach is performed to fit a sequence of cylinders to approximate the geometry of the tree branches. Experiments on various types of trees from different data sources demonstrate the effectiveness and robustness of our method. The overall fitting error (i.e., the distance between the input points and the output model) is less than 10 cm. The reconstructed tree models can be further applied in the precise estimation of tree attributes, urban landscape visualization, etc. The source code of this work is freely available at https://github.com/tudelft3d/adtree