The Robotics Institute

Carnegie Mellon Robotics Institute

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Abstract

This paper introduces the notion of space-ﬁlling trees, which are analogous to space-ﬁlling curves, but have a branching, tree-like structure and are rooted. A space-ﬁlling tree is deﬁned by an incremental process that results in a tree for which every point in the space has a ﬁnite-length path that converges to it. In contrast to space-ﬁlling curves, individual paths in the tree are short, allowing any part of the space to be quickly reached from the root. These structures have interesting parallels in nature, including ﬂuid distribution systems, vascular networks, and fractal plant growth, and may have many uses in engineering and computer science. We provide basic examples, general deﬁnitions and constructions, characterizations of some tree properties, and leave many open mathematical questions.

Notes

Number of pages: 8

Text Reference

James Kuffner and Steven M. LaValle, "Space-Filling Trees," tech. report CMU-RI-TR-09-47, Robotics Institute, Carnegie Mellon University, December, 2009

BibTeX Reference

@techreport{Kuffner_2009_6539,
   author = "James Kuffner and Steven M. LaValle",
   title = "Space-Filling Trees",
   booktitle = "",
   institution = "Robotics Institute",
   month = "December",
   year = "2009",
   number= "CMU-RI-TR-09-47",
   address= "Pittsburgh, PA",
}

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