000 05145nam a22005295i 4500
001 978-3-031-47511-5
003 DE-He213
005 20251030111449.0
007 cr nn 008mamaa
008 240228s2024 sz | s |||| 0|eng d
020 _a9783031475115
_9978-3-031-47511-5
024 7 _a10.1007/978-3-031-47511-5
_2doi
050 4 _aQA639.5-640.7
050 4 _aQA640.7-640.77
072 7 _aPBM
_2bicssc
072 7 _aPBD
_2bicssc
072 7 _aMAT012020
_2bisacsh
072 7 _aPBM
_2thema
072 7 _aPBD
_2thema
100 1 _aO'Rourke, Joseph.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_971928
245 1 0 _aReshaping Convex Polyhedra
_h[electronic resource] /
_cby Joseph O'Rourke, Costin Vîlcu.
250 _a1st ed. 2024.
264 1 _aCham :
_bSpringer Nature Switzerland :
_bImprint: Springer,
_c2024.
300 _aXIV, 243 p. 117 illus., 116 illus. in color.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
505 0 _aI. Tailoring for Every Body -- 1. Introduction to Part I -- 2. Preliminaries -- 3. Domes and Pyramids -- 4. Tailoring via Sculpting -- 5. Pyramid Seal Graph -- 6. Algorithms for Tailoring via Sculpting -- 7. Crests -- 8. Tailoring via Flattening -- 9. Applications of Tailoring -- II. Vertex-Merging and Convexity -- 10. Introduction to Part II -- 11. Vertex-Merging Reductions and Slit Graphs -- 12. Planar Spiral Slit Tree -- 13. Convexity on Convex Polyhedra -- 14. Minimal-length Enclosing Polygon -- 15. Spiral Tree on Polyhedron -- 16. Unfolding via Slit Trees -- 17. Vertices on Quasigeodesics -- 18. Conclusions -- Bibliography -- References -- Index.
520 _aThe focus of this monograph is converting—reshaping—one 3D convex polyhedron to another via an operation the authors call “tailoring.” A convex polyhedron is a gem-like shape composed of flat facets, the focus of study since Plato and Euclid. The tailoring operation snips off a corner (a “vertex”) of a polyhedron and sutures closed the hole. This is akin to Johannes Kepler’s “vertex truncation,” but differs in that the hole left by a truncated vertex is filled with new surface, whereas tailoring zips the hole closed. A powerful “gluing” theorem of A.D. Alexandrov from 1950 guarantees that, after closing the hole, the result is a new convex polyhedron. Given two convex polyhedra P, and Q inside P, repeated tailoring allows P to be reshaped to Q. Rescaling any Q to fit inside P, the result is universal: any P can be reshaped to any Q. This is one of the main theorems in Part I, with unexpected theoretical consequences. Part II carries out a systematic study of “vertex-merging,” a technique that can be viewed as a type of inverse operation to tailoring. Here the start is P which is gradually enlarged as much as possible, by inserting new surface along slits. In a sense, repeated vertex-merging reshapes P to be closer to planarity. One endpoint of such a process leads to P being cut up and “pasted” inside a cylinder. Then rolling the cylinder on a plane achieves an unfolding of P. The underlying subtext is a question posed by Geoffrey Shephard in 1975 and already implied by drawings by Albrecht Dürer in the 15th century: whether every convex polyhedron can be unfolded to a planar “net.” Toward this end, the authors initiate an exploration of convexity on convex polyhedra, a topic rarely studied in the literature but with considerable promise for future development. This monograph uncovers new research directions and reveals connections among several, apparently distant, topics in geometry: Alexandrov’s Gluing Theorem, shortest paths and cut loci, Cauchy’s Arm Lemma, domes, quasigeodesics, convexity, and algorithms throughout. The interplay between these topics and the way the main ideas develop throughout the book could make the “journey” worthwhile for students and researchers in geometry, even if not directly interested in specific topics. Parts of the material will be of interest and accessible even to undergraduates. Although the proof difficulty varies from simple to quite intricate, with some proofs spanning several chapters, many examples and 125 figures help ease the exposition and illustrate the concepts.
650 0 _aConvex geometry .
_971929
650 0 _aDiscrete geometry.
_971930
650 1 4 _aConvex and Discrete Geometry.
_971931
700 1 _aVîlcu, Costin.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_971932
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9783031475108
776 0 8 _iPrinted edition:
_z9783031475122
776 0 8 _iPrinted edition:
_z9783031475139
856 4 0 _uhttps://doi.org/10.1007/978-3-031-47511-5
912 _aZDB-2-SMA
912 _aZDB-2-SXMS
999 _c57948
_d57948