TY - JOUR

T1 - An Efficient Adaptive Algorithm for Constructing the Convex Differences Tree of a Simple Polygon

AU - Rappoport, Ari

PY - 1992/8

Y1 - 1992/8

N2 - The convex differences tree (CDT) representation of a simple polygon is useful in computer graphics, computer vision, computer aided design and robotics. The root of the tree contains the convex hull of the polygon and there is a child node recursively representing every connectivity component of the set difference between the convex hull and the polygon. We give an O(n log K + K log2 n) time algorithm for constructing the CDT, where n is the number of polygon vertices and K is the number of nodes in the CDT. The algorithm is adaptive to a complexity measure defined on its output while still being worst case efficient. For simply shaped polygons, where K is a constant, the algorithm is linear. In the worst case K = O(n) and the complexity is O(n log2 n). We also give an O(n log n) algorithm which is an application of the recently introduced compact interval tree data structure.

AB - The convex differences tree (CDT) representation of a simple polygon is useful in computer graphics, computer vision, computer aided design and robotics. The root of the tree contains the convex hull of the polygon and there is a child node recursively representing every connectivity component of the set difference between the convex hull and the polygon. We give an O(n log K + K log2 n) time algorithm for constructing the CDT, where n is the number of polygon vertices and K is the number of nodes in the CDT. The algorithm is adaptive to a complexity measure defined on its output while still being worst case efficient. For simply shaped polygons, where K is a constant, the algorithm is linear. In the worst case K = O(n) and the complexity is O(n log2 n). We also give an O(n log n) algorithm which is an application of the recently introduced compact interval tree data structure.

KW - Computational geometry

KW - adaptive algorithms

KW - convex differences tree (CDT)

KW - convexity

KW - polygon

UR - http://www.scopus.com/inward/record.url?scp=0026929737&partnerID=8YFLogxK

U2 - 10.1111/1467-8659.1140235

DO - 10.1111/1467-8659.1140235

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AN - SCOPUS:0026929737

SN - 0167-7055

VL - 11

SP - 235

EP - 240

JO - Computer Graphics Forum

JF - Computer Graphics Forum

IS - 4

ER -