Reshaping Convex Polyhedra
O'Rourke, Joseph, Vîlcu, Costin
- 出版商: Springer
- 出版日期: 2024-02-29
- 售價: $5,030
- 貴賓價: 9.5 折 $4,779
- 語言: 英文
- 頁數: 243
- 裝訂: Hardcover - also called cloth, retail trade, or trade
- ISBN: 3031475100
- ISBN-13: 9783031475108
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the="" study="" of="" convex="" polyhedra="" in="" ordinary="" space="" is="" a="" central="" piece="" classical="" and="" modern="" geometry="" that="" has="" had="" significant="" impact="" on="" many="" areas="" mathematics="" also="" computer="" science.="" present="" book="" project="" by="" joseph="" o'rourke="" costin="" vîlcu="" brings="" together="" two="" important="" strands="" subject="" --="" combinatorics="" polyhedra, ="" intrinsic="" underlying="" surface.="" this="" leads="" to="" remarkable="" interplay="" concepts="" come="" life="" wide="" range="" very="" attractive="" topics="" concerning="" polyhedra.="" gets="" message="" across="" thetheory="" although="" with="" roots, ="" still="" much="" alive="" today="" continues="" be="" inspiration="" basis="" lot="" current="" research="" activity.="" work="" presented="" manuscript="" interesting="" applications="" discrete="" computational="" geometry, ="" as="" well="" other="" mathematics.="" treated="" detail="" include="" unfolding="" onto="" surfaces, ="" continuous="" flattening="" convexity="" theory="" minimal="" length="" enclosing="" polygons.="" along="" way, ="" open="" problems="" suitable="" for="" graduate="" students="" are="" raised, ="" both="" 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 tailoringallows 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 studiedin 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.
商品描述(中文翻譯)
這本專著的焦點是通過一種作者稱之為“裁剪”的操作,將一個三維凸多面體轉換成另一個。凸多面體是由平面面組成的寶石狀形狀,自柏拉圖和歐幾里得以來一直是研究的重點。裁剪操作是剪掉多面體的一個角(一個“頂點”),並將孔洞縫合起來。這類似於約翰內斯·開普勒的“頂點截斷”,但不同之處在於截斷頂點留下的孔洞被填充了新的表面,而裁剪則將孔洞封閉。1950年,A.D.亞歷山德羅夫的一個強大的“黏合”定理保證,在封閉孔洞後,結果是一個新的凸多面體。給定兩個凸多面體P和Q,位於P內部的Q,重複的裁剪操作可以將P重塑為Q。將任何Q重新調整大小以適應P,結果是通用的:任何P都可以重塑為任何Q。這是第一部分的主要定理之一,具有意想不到的理論結果。
第二部分對“頂點合併”進行了系統研究,這種技術可以看作是裁剪的一種逆操作。在這裡,起點是P,逐漸通過插入新的表面沿著裂縫進行最大程度的擴大。在某種意義上,重複的頂點合併將P重塑為更接近平面的形狀。這個過程的一個端點導致P被切割並“粘貼”在一個圓柱體內。然後,在平面上滾動圓柱體可以實現P的展開。其中隱含的基本問題是由杰弗里·謝帕德在1975年提出的,並且已經由15世紀阿爾布雷希特·杜勒爾的繪畫所暗示:是否每個凸多面體都可以展開成一個平面“網”。為了達到這個目標,作者們開始探索凸多面體上的凸性,這是一個在文獻中很少研究但具有相當潛力的主題。
這本專著揭示了幾何學中幾個看似相距甚遠的主題之間的新研究方向和聯繫:亞歷山德羅夫的黏合定理、最短路徑和切割位置、柯西的臂引理、圓頂、拟地线、凸性以及整本書中的算法。這些主題之間的相互作用以及主要思想在整本書中的發展方式,即使對於不直接對特定主題感興趣的幾何學生和研究人員來說,也可能使“旅程”變得有價值。部分材料將對本科生感興趣並且易於理解。儘管證明的難度從簡單到相當複雜不等,一些證明跨越了幾章,但許多例子和125幅圖片有助於簡化說明並說明概念。
作者簡介
Joseph O'Rourke is Professor at Smith College. Prior to joining Smith in 1988 to found and chair the computer science department, Joseph O'Rourke was an assistant and then associate professor at Johns Hopkins University. His research is in the field of computational geometry. In 2001, he was awarded the NSF Director's Award for Distinguished Teaching Scholars. He is also a professor of mathematics.
Costin Vîlcu is affiliated with the Simion Stoilow Institute of Mathematics of the Romanian Academy. His research interests include geometry of Alexandrov surfaces and intrinsic geometry of convex surfaces, including polyhedral convex surfaces.
作者簡介(中文翻譯)
Joseph O'Rourke是史密斯學院的教授。在1988年加入史密斯學院之前,他曾在約翰霍普金斯大學擔任助理教授和副教授,並創立並擔任計算機科學系主任。他的研究領域是計算幾何學。2001年,他獲得了美國國家科學基金會(NSF)頒發的傑出教學學者獎。他也是數學教授。
Costin Vîlcu隸屬於羅馬尼亞科學院的Simion Stoilow數學研究所。他的研究興趣包括亞歷山大夫曲面的幾何學和凸曲面的內在幾何學,包括多面體凸曲面。