Minimal enclosing parallelepiped in 3D

Minimal enclosing parallelepiped in 3D

We investigate the problem of finding a minimal volume parallelepiped enclosing a given set of n threedimensional points. We give two mathematical properties of these parallelepipeds, from which we derive two algorithms of theoretical complexity O(n6). Experiments show that in practice our quickest algorithm runs in O(n2) (at least for n 105). We also present our application in structural biolo...

Minimal enclosing parallelepiped in 3D

Uniqueness results for minimal enclosing ellipsoids

We prove uniqueness of the minimal enclosing ellipsoid with respect to strictly eigenvalue convex size functions. Special examples include the classic case of minimal volume ellipsoids (Löwner ellipsoids), minimal surface area ellipsoids or, more generally, ellipsoids that are minimal with respect to quermass integrals.

Minimal Enclosing Hyperbolas of Line Sets

We prove the following theorem: If H is a slim hyperbola that contains a closed set S of lines in the Euclidean plane, there exists exactly one hyperbola Hmin of minimal volume that contains S and is contained in H. The precise concepts of “slim”, the “volume of a hyperbola” and “straight lines or hyperbolas being contained in a hyperbola” are defined in the text.

3D Modelling Using Geometric Constraints: A Parallelepiped Based Approach

In this paper, efficient and generic tools for calibration and 3D reconstruction are presented. These tools exploit geometric constraints frequently present in man-made environments and allow camera calibration as well as scene structure to be estimated with a small amount of user interactions and little a priori knowledge. The proposed approach is based on primitives that naturally characteriz...