國立高雄大學圖資館 |

Reshaping convex polyhedra

Record Type:	Electronic resources : Monograph/item
Title/Author:	Reshaping convex polyhedraby Joseph O'Rourke, Costin Vilcu.
Author:	O'Rourke, Joseph.
other author:	Vilcu, Costin.
Published:	Cham :Springer Nature Switzerland :2024.
Description:	xiv, 243 p. :ill. (some col.), digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	Convex geometry.
Online resource:	https://doi.org/10.1007/978-3-031-47511-5
ISBN:	9783031475115$q(electronic bk.)

Reshaping convex polyhedra
O'Rourke, Joseph.

Reshaping convex polyhedra[electronic resource] /by Joseph O'Rourke, Costin Vilcu. - Cham :Springer Nature Switzerland :2024. - xiv, 243 p. :ill. (some col.), digital ;24 cm.

I. 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.

The 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.

ISBN: 9783031475115$q(electronic bk.)

Standard No.: 10.1007/978-3-031-47511-5doiSubjects--Topical Terms:

187435
Convex geometry.

LC Class. No.: QA639.5

Dewey Class. No.: 516.08

Reshaping convex polyhedra
LDR:04242nmm a2200349 a 4500 001 663438
003 DE-He213
005 20240228191129.0
006 m d
007 cr nn 008maaau
008 241203s2024 sz s 0 eng d
020 $a 9783031475115$q(electronic bk.)
020 $a 9783031475108$q(paper)
024 7 $a 10.1007/978-3-031-47511-5 $2 doi
035 $a 978-3-031-47511-5
040 $a GP $c GP
041 0 $a eng
050 4 $a QA639.5
072 7 $a PBM $2 bicssc
072 7 $a PBD $2 bicssc
072 7 $a MAT012020 $2 bisacsh
072 7 $a PBM $2 thema
072 7 $a PBD $2 thema
082 0 4 $a 516.08 $2 23
090 $a QA639.5 $b .O74 2024
100 1 $a O'Rourke, Joseph. $3 544876
245 1 0 $a Reshaping convex polyhedra $h [electronic resource] / $c by Joseph O'Rourke, Costin Vilcu.
260 $a Cham : $b Springer Nature Switzerland : $b Imprint: Springer, $c 2024.
300 $a xiv, 243 p. : $b ill. (some col.), digital ; $c 24 cm.
505 0 $a I. 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 $a The 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 $a Convex geometry. $3 187435
650 0 $a Polyhedra. $3 189600
650 1 4 $a Convex and Discrete Geometry. $3 277230
700 1 $a Vilcu, Costin. $3 747899
710 2 $a SpringerLink (Online service) $3 273601
773 0 $t Springer Nature eBook
856 4 0 $u https://doi.org/10.1007/978-3-031-47511-5
950 $a Mathematics and Statistics (SpringerNature-11649)