Abstract
This paper presents algorithms for estimating the minimum volume bounding box based on a three-dimensional point set measured by a coordinate measuring machine. A new algorithm, which calculates the minimum volume with high accuracy and reduced number of computations, is developed. The algorithm is based on the convex hull operation and established theories about a minimum bounding box circumscribing a convex polyhedron. The new algorithm includes a pre-processing operation that removes convex polyhedron faces located near the edges of the measured object. As showed in the paper, the solution of the minimum bonding box is not based on faces located near the edges; therefore, we can save computation time by excluding them from the convex polyhedron data set. The algorithms have been demonstrated on physical objects measured by a coordinate measuring machine, and on theoretical 3D models. The results show that the algorithm can be used when high accuracy is required, for example in calibration of reference standards.
Highlights
In various applications, it can be useful to circumscribe a given set of three-dimension coordinate points by an ideal shape rectangular parallelepiped
The term face is mainly used for the inscribed convex polyhedron faces, which are the product of 3D convex hull operation
The minimum volume bounding box edge (MVBBE) method completely satisfies to both theorems given in Sections 2.1 and 2.2, and it can be used as the reference for the other alternative methods
Summary
It can be useful to circumscribe a given set of three-dimension coordinate points by an ideal shape rectangular parallelepiped. It was suggested by Dupuis [1] to use the term cuboid when referring to a rectangular parallelepiped. In the literature of the computational geometry, the term box is commonly associated with the rectangular parallelepiped. In this text, we use the term side for the bounding box face. All six sides (faces) of the box are rectangles and each side is parallel with the opposite side and orthogonal with the other four adjacent sides These four adjacent sides comprise a “closed loop”. The opposite, Bottom side has the same “closed loop” of adjacent sides as the Top side
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