Minimum bounding box algorithms
- See "Minimum bounding box" for the box specified by minimal and maximal coordinates
In computational geometry, the smallest enclosing box problem is that of finding the box (hyperrectangle) of smallest measure (volume, area, perimeter, etc.) enclosing a given object or set of objects. It is a type of bounding volume.
It is sufficient to find the smallest enclosing box for the convex hull of the objects in question.
Two dimensions
For the convex polygon, a linear time algorithm for the minimum-area enclosing rectangle is known.It is based on the observation that a side of a minimum-area enclosing box must be collinear with a side of the convex polygon.[1] It is possible to enumerate of the boxes of this kind in linear time with the approach called rotating calipers by Godfried Toussaint in 1983 [2] The same approact is applicable for finding the minimum-perimeter enclosing rectangle. [2]
Three dimensions
In 1985, Joseph O'Rourke published an cubic-time algorithm to find the minimum-volume enclosing box of a 3-dimensional point set.[3] O'Rourke's approach uses a 3-dimensional rotating calipers technique. This algorithm has not been improved on as of August 2008, although heuristic methods for tackling the same problem have been developed.
Preparatory theorems in O'Rourke's work were proved to the effect that:
- There must exist two neighbouring faces of smallest-volume enclosing box which both contain an edge of the convex hull of the point set. This criterion is satisfied by a single convex hull edge collinear with an edge of the box, or by two distinct hull edges lying in adjacent box faces.
- The other four faces need only contain a point of the convex hull. Again, the points which they contain need not be distinct: a single hull point lying in the corner of the box already satisfies three of these four criteria.
It follows in the most general case where no convex hull vertices lie in edges of the minimal enclosing box, that at least 8 convex hull points must lie within faces of the box: two endpoints of each of the two edges, and four more points, one for each of the remaining four box faces. Conversely, if the convex hull consists of 7 or fewer vertices, at least one of them must lie within an edge of the hull's minimal enclosing box.
The minimal enclosing box of the regular tetrahedron is a cube, with the tetrahedron's vertices lying at (0,0,0), (0,1,1), (1,0,1) and (0,1,1).
See also
References
- ^ H. Freeman and R. Shapira, "Freeman Determining the Minimum-Area Encasing Rectangle for an Arbitrary Closed Curve", Comm. ACM, 1975, pp.409-413.
- ^ a b Toussaint, G. T (1983). "Solving geometric problems with the rotating calipers". Proc. MELECON '83, Athens.
{{cite journal}}: Cite journal requires|journal=(help) - ^ Joseph O'Rourke (1985), "Finding minimal enclosing boxes", Parallel Programming, Springer Netherlands