Rotating calipers

In computational geometry, the method of rotating calipers is an algorithm design technique that can be used to solve optimization problems including finding the width or diameter of a set of points. In computational geometry, the method of rotating calipers is an algorithm design technique that can be used to solve optimization problems including finding the width or diameter of a set of points. The method is so named because the idea is analogous to rotating a spring-loaded vernier caliper around the outside of a convex polygon. Every time one blade of the caliper lies flat against an edge of the polygon, it forms an antipodal pair with the point or edge touching the opposite blade. The complete 'rotation' of the caliper around the polygon detects all antipodal pairs; the set of all pairs, viewed as a graph, forms a thrackle. The method of rotating calipers can be interpreted as the projective dual of a sweep line algorithm in which the sweep is across slopes of lines rather than across x- or y-coordinates of points. The rotating calipers method was first used in the dissertation of Michael Shamos in 1978. Shamos uses this method to generate all antipodal pairs of points on a convex polygon and to compute the diameter of a convex polygon in O ( n ) {displaystyle O(n)} time. Godfried Toussaint coined the phrase 'rotating calipers' and also demonstrated that the method was applicable in solving many other computational geometry problems. Shamos gave following algorithm in his dissertation (pp 77–82) for the rotating calipers method that generated all antipodal pairs of vertices on convex polygon: Another version of this algorithm appeared in the text by Preparata and Shamos in 1985 that avoided calculation of angles: This method has several advantages including that it avoids calculation of area or angles as well as sorting by polar angles. The method is based on finding convex hull using Monotone chain method devised by A.M. Andrew which returns upper and lower portions of hull separately that then can be used naturally for rotating calipers analogy. Toussaint and Pirzadeh describes various applications of rotating calipers method.