The arc tree: An approximation scheme to represent arbitrary curved shapes

Computer Vision, Graphics, and Image Processing Pub Date : 1990-09-01 DOI:10.1016/0734-189X(90)90006-H

Oliver Günther, Eugene Wong

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Abstract

This paper introduces the arc tree, a hierarchical data structure to represent arbitrary curved shapes. The arc tree is a balanced binary tree that represents a curve of length l such that any subtree whose root is on the kth tree level is representing a subcurve of length $l 2^{k}$ . Each tree level is associated with an approximation of the curve; lower levels correspond to approximations of higher resolution. Based on this hierarchy of detail, queries such as point search or intersection detection and computation can be solved in a hierarchical manner. Algorithms start out near the root of the tree and try to solve the queries at a very coarse resolution. If that is not possible, the resolution is increased where necessary. We describe and analyze several such algorithms to compute a variety of set and search operators. Various related approximation schemes to represent curved shapes are also discussed.

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弧树:表示任意曲线形状的近似方案

本文介绍了一种表示任意曲线形状的分层数据结构——弧树。弧树是一棵平衡二叉树，它表示一条长度为l的曲线，使得任何根位于第k树层的子树都表示一条长度为l2k的子曲线。每个树级别都与曲线的近似值相关联;较低的级别对应于较高分辨率的近似值。基于这种细节层次结构，点搜索或交叉点检测和计算等查询可以以层次方式解决。算法从树的根附近开始，并尝试以非常粗糙的分辨率解决查询。如果不可能，则在必要时增加分辨率。我们描述和分析了几种这样的算法来计算各种集合和搜索运算符。还讨论了表示曲线形状的各种相关近似格式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Computer Vision, Graphics, and Image Processing

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