Computational Geometry

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AMS/IP

Studies in Advanced Mathematics Volume 34

Computational Geometry

Lectures at the Morningsid e Center of Mathematic s

Ren-Hong Wang, Editor

American Mathematical Society • International Press

"SBE D P

https://doi.org/10.1090/amsip/034

Shing-Tung Yau , Genera l Edito r

2000 Mathematics Subject Classification. Primar y 35-02 , 35L67 ; Secondar y 35L60 , 35L80, 76N10 , 46N20 .

Library o f Congres s Cataloging-in-Publicatio n D a t a

Computational Geometr y : Lecture s a t th e Morningsid e Cente r o f Mathematic s / Ren-Hon g Wang, editor .

p. cm . — (AMS/I P Studie s i n Advance d Mathematics ; v . 34 ) Includes bibliographica l references . ISBN 0-8218-2044- 3 (alk . paper ) 1. Singularitie s (Mathematics ) 2 . Differentia l equations , Hyperbolic . I . Wang , Renhong .

II. Series .

QA614.58.C66 200 3 516.3'5—dc21 200304538 1

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Contents

Series Prefac e vi i

Acknowledgment i x

Credits fo r th e Quote s fro m "Quantu m Teleportatio n an d Spi n Echo : A

Unitary Symplecti c Spino r Approach " x

List o f Attendent s x i

On Computationa l Geometr y REN-HONG WAN G 1

Geometry fo r Analysi s o f Cornea l Shap e BRIAN A . BARSK Y 3 3

Approximate Implicitizatio n o f Rationa l Surface s CHEN FALA I 5 7

A Geometri c Approac h t o dim S%(AMS) HONG D U 6 7

Subdivision fo r C 1 Surfac e Interpolatio n NIRA DYN , SIGALI T H E D , AN D DAVI D LEVI N 7 1

A Permanence Principl e fo r Shap e Contro l GERALD FARI N AN D DIANN E HANSFOR D 8 3

Blending Severa l Implici t Algebrai c Surface s wit h Rule d Surface s GUOCHEN FENG , TIER U W U , KA I Y U , SHUGON G ZHANG , AN D YUNSH I

ZHOU 8 9

Lagrange Interpolatio n b y Spline s o n Triangulation s GUNTHER NURNBERGE R AN D FRAN K ZEILFELDE R 11 3

Quantum Teleportatio n an d Spi n Echo : A Unitar y Symplecti c Spino r Approach ERNST BIN Z AN D WALTE R SCHEMP P 13 3

The Generalizatio n o f Pascal' s Theore m an d Morgan-Scott' s Partitio n XIQUAN SH I AN D R E N - H O N G WAN G 17 9

'Optimal' Triangulatio n o f Surface s an d Bodie s C. R . TRAA S 18 9

vi C O N T E N T S

Multivariate Spline s an d Geometr y REN-HONG WAN G 19 5

Geometric Continuou s B-splin e — A Generalizatio n o f th e Approac h of 7- splin e ZONGMIN W u 20 1

Adaptive an d Smoot h Surfac e Constructio n b y Triangula r A-Patche s GUOLIANG X u 21 3

A B-splin e functio n i n S 3 (i?3, A£) GUOHUI ZHA O AN D REN-HON G WAN G 22 7

Series Prefac e

This is a subseries of AMS-IP publications dedicated to the research activities of the Morningsid e Cente r o f Mathematics , Chines e Academ y o f Science s i n Beijing , China. Th e Morningsid e Cente r support s researc h activitie s b y invitin g leadin g figures i n differen t discipline s o f mathematics t o giv e lectures . Thes e leader s direc t programs i n thei r disciplines . Th e participant s ar e youn g Chines e mathematician s or some visitors from overseas . A t the end of the programs, the lectures are collected and organize d b y the program directors . W e believe these programs hav e been ver y successful an d helpfu l i n promotin g mathematic s a s a whole .

We wis h t o than k th e Morningsid e Foundatio n an d th e Chines e Academ y o f Sciences fo r providin g th e funding . W e als o wis h t o than k th e staf f a t ou r Morn -ingside Cente r wh o helpe d ru n ou r programs . W e woul d als o lik e t o tak e thi s opportunity t o than k al l o f the universitie s i n Chin a wh o hav e cooperate d wit h u s on thes e activities , especiall y t o th e mathematician s al l ove r th e worl d wh o hav e provided unselfis h help . W e than k the m fo r thei r efforts . W e woul d als o lik e t o thank Internationa l Pres s an d th e America n Mathematica l Societ y fo r assistin g i n the publicatio n o f this volume .

YANG L o and S.T . Ya u

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Acknowledgement

The progra m o n Computationa l Geometr y wa s held a t th e Morningsid e Cen -ter o f Mathematics , Chines e Academ y o f Science , Beijing , China , fro m Marc h t o August o f 1998.

During thi s program , ther e wa s als o a Symposiu m o f Computationa l Geom -etry organize d b y th e Morningsid e Cente r o f Mathematics , Insitut e o f Softwar e (Prof. J.C . Sun), and Institute o f Computing Technolog y (Prof . H . Li), held a t the Morningside Cente r o f Mathematics o n June 11-15 , 1998.

The progra m successfull y brough t togethe r th e participants , includin g youn g researchers, t o exchange idea s an d to generat e interdisciplinar y results .

Many expert s wer e invite d t o attend th e program an d give lectures . The proceeding s i s based o n the invited lecture s an d som e contribute d paper s

presented b y researchers workin g in the Morningside Cente r o f Mathematics durin g the program .

I woul d lik e t o tak e thi s opportunit y t o than k th e invite d lecturers , youn g researchers an d al l the participants .

I woul d lik e als o t o than k thos e wh o assisted i n variou s way s wit h th e prepa -ration for , and running of , thi s program . I would especiall y lik e to thank Professo r Shing-Tung Ya u an d Professo r L o Yang fo r givin g u s thi s ver y goo d opportunit y to organiz e th e progra m an d givin g u s stron g support . Withou t thei r help , th e program, a s well a s these proceedings , wouldn' t hav e bee n possible .

Ren-Hong Wan g

Inst, o f Math . Sci. , Dalia n Univ . o f Tech. , Dalian 116024 , Chin a

Morningside Cente r o f Mathematics , Chines e Academy o f Sciences , Beijin g 100080 , Chin a

Credits fo r th e Quote s i n th e articl e "Quantu m Teleportation an d Spi n Echo : A Unitar y Symplecti c Spino r

Approach"

A quotatio n fro m J . S . Waugh , "Spi n Echoe s an d Thermodynamics, " i n Pulsed Magnetic Resonance: NMR, ESR and Optics —A Recognition of E. L. Hahn, D. M . S . Bagguley , ed. , Oxfor d Universit y Pres s (1992) , © Copyrigh t 199 2 b y Oxford Universit y Press .

A quotatio n fro m M . Wege r an d J . L . Birman , Laser Physics 12 , pp . 1208-1219 , 2002.

A quotation fro m Anatol e Abragan , Pulsed Magnetic Resonance: NMR, ESR and Optics—A Recognition of E. L. Hahn, D . M . S . Bagguley , ed. , Oxfor d Universit y Press (1992) . © Copyrigh t 199 2 by Oxfor d Universit y Press .

A quotatio n fro m Ra y Freeman , "Spi n Choreography . Basi c Step s i n Hig h Resolution NMR, " publishe d b y Oxfor d Universit y Pres s (1997) , © Copyrigh t 1997 Oxford Universit y Press .

A quotatio n fro m Gusta v vo n Schulthess , "Clinica l M R i n th e yea r 2010, " MAGMA 8 (3) , pp. 133-145 , 1999 .

A quotatio n fro m Mathia s Fink , "Tim e Reverse d Acoustics, " Physics Today 20 , pp. 34-40 , Mar s (1997) , © Copyrigh t 199 7 by th e America n Institut e o f Physics .

A quotation fro m C . H . Bennett , G . Brassard , C . Crepeau , J . Jozsa , A . Peres , an d W. Wootters , "Teleportin g a n Unknow n Quantu m Stat e vi a Dua l Classica l an d Einstein-Podolsky-Rosen Channels, " Phys. Rev. Lett 70 , pp . 1895-189 9 (1993) . © Copyrigh t 199 3 by th e America n Physica l Societ y

A quotatio n fro m Ton y Sudbery , Nature 390 , pp . 551-552 , 1997 .

LIST O F ATTENDENT S

Brian A . Barsky , Deparmen t o f Compute r Science , Universit y o f California , Berkeley, Californi a 94720-1776 , U.S.A .

Gengzhe Chang , Departmen t o f Mathematics , Universit y o f Scienc e an d Technology o f China , Hefei , 230026 , P . R . Chin a

Falai Chen , Departmen t o f Mathematics , Universit y o f Scienc e an d Technolog y o f China, Hefei , 230026 , P . R . Chin a

Youdong Ding , Departmen t o f Mathematics , Fuda n University , Shanghai , 200433 , P. R . China ;

Hong Du , Institut e o f System s Science , Chines e Academ y o f Sciences , Beijing , 100080,R R . Chin a

Gerald Farin , Departmen t o f Compute r Scienc e an d Engineering , Arizon a Stat e University, Tempe , A Z 85287-5406 , U . S . A .

Yuyu Feng , Departmen t o f Mathematics , Universit y o f Scienc e an d Technolog y o f China, Hefei , 230026 , P . R . Chin a

Xuanji Hua , Departmen t o f Mathematics , Fuda n University , Shanghai , 200433 , P R . China ;

Lei Jiang , Institut e o f Mathematica l Sciences , Dalia n Universit y o f Technology , Dalian, 116024 , P . R . Chin a

Tongguang Jin , Departmen t o f Mathematics , Zhejian g University , Hangzhou , 310027, P . R . Chin a

David Levin , Sackle r Facult y o f Exac t Sciences , Schoo l o f Mathematica l Sciences , Tel-Aviv University , Tel-Avi v 69978 , Israe l

Li Li , Institut e o f Mathematica l Sciences , Dalia n Universit y o f Technology , Dalian, 116024 , P . R . Chin a

Hua Li , Institute o f Software , Chines e Academ y o f Sciences , Beijing , 100080 , P . R . China

Youdong Liang , Departmen t o f Compute r Science , Zhejian g University , Hangzhou, 310027 , P . R . Chin a

xii LIS T O F A T T E N D E N T S

Guixian Lin , Institut e o f Mathematica l Sciences , Dalia n Universit y o f Technology , Dalian, 116024 , P . R . Chin a

Lizhuang Ma , Stat e Ke y Lab . o f CA D & CG, Institut e o f Artificia l Intelligence , Zhejiang University , Hangzhou , 310027 , P . R . Chin a

Giinter Niirnberger , Fakulta t fir r Mathemati k un d Informatik , Universita t Mannheim, D-6813 1 Mannheim , German y

Dongxu Qi , CA D Center , Norther n Polytechnica l University , Beijing , P . R . Chin a

Walter Schempp , Lehrstuh l fu r Mathemati k I , Universit y o f Siegen , D-57068 , Siegen, German y

Xiquan Shi , Institut e o f Mathematica l Sciences , Dalia n Universit y o f Technology , Dalian, 116024 , P . R . Chin a

Jiachang Sun , Institut e o f Software , Chines e Academ y o f Sciences , Beijing , 100080, P . R . Chin a

Rongxi Tang , Roo m 703 , Beijing Universit y o f Aeronautic s & Astronautics , Beijing, P . R . Chin a

Cornells R . Traas , Departmen t o f Mathematics , Universit y o f Twente , Ensdude , The Netherland s

Jiaye Wang , Departmen t o f Compute r Science , Shandon g University , Jinan , P . R . China

Jing Wang , Institut e o f Mathematica l Sciences , Dalia n Universit y o f Technology , Dalian, 116024 , P . R . Chin a

Ren-Hong Wang , Institut e o f Mathematica l Sciences , Dalia n Universit y o f Technology, Dalian , 116024 , P . R . Chin a

Zongmin Wu , Departmen t o f Mathematics , Fuda n University , Shanghai , 200433 , P. R . Chin a

Guoliang Xu , Stat e Ke y Laborator y o f Scientifi c an d Engineerin g Computing , ICMSEC, Chines e Academ y o f Sciences , Beijing , 100080 , P . R . Chin a

Yinwei Zhan , Departmen t o f Mathematics , Shanto u University , Shantou , 515063 , P. R . Chin a

Guohui Zhao , Departmen t o f Mathematics , Sicua n University , Chengdu , 610064 , P. R . Chin a

Quan Zheng , Institut e o f Mathematica l Sciences , Dalia n Universit y o f Technology , Dalian, 116024 , P . R . Chin a

Yunshi Zhou , Departmen t o f Mathematics , Jili n University , Changchun , 130023 , P. R . Chin a

Titles i n Thi s Serie s

34 Ren-Hon g Wang , Editor , Computationa l Geometry , 200 3

33 Eri c D'Hoker , Duon g Phong , an d Shing-Tun g Yau , Mirro r Symmetr y IV , 200 2

32 X i -P in g Zhu , Lecture s o n Mea n Curvatur e Flows , 200 2

31 Kiyosh i Igusa , Highe r Franz-Reidemeiste r Torsion , 200 2

30 W e i m a n Ha n an d Mirce a Sofonea , Quasistati c Contac t Problem s i n Viscoelasticit y

and Viscoplasticity , 200 2

29 S . T . Ya u an d Shuxin g Chen , Editors , Geometr y an d Nonlinea r Partia l Differentia l

Equations, 200 2

28 Valenti n Afraimovic h an d Sze-B i Hsu , Lecture s o n Chaoti c dynamica l Systems , 200 2

27 M . R a m Murty , Introductio n t o p-adi c Analyti c Numbe r Theory , 200 2

26 Raymon d Chan , Yue-Kue n Kwok , Davi d Yao , an d Qian g Zhang , Editors ,

Applied Probability , 200 2

25 Dongga o Deng , Dare n Huang , Rong-Qin g Jia , We i Lin , an d Jia n Zhon g Wong ,

Editors, Wavele t Analysi s an d Applications , 200 2

24 Jan e Gilman , Wil l ia m W . Menasco , an d Xiao-Son g Lin , Editors , Knots , Braids ,

and Mappin g Clas s Groups—Paper s Dedicate d t o Joa n S . Birman , 200 1

23 Cumru n Vaf a an d S.-T . Yau , Editors , Winte r Schoo l o n Mirro r Symmetry , Vecto r

Bundles an d Lagrangia n Submanifolds , 200 1

22 Carlo s Berenste in , Der -Che n Chang , an d Jingzh i Tie , Laguerr e Calculu s an d It s

Applications o n th e Heisenber g Group , 200 1

21 Jiirge n Jost , Bosoni c Strings : A Mathematica l Treatment , 200 1

20 L o Yan g an d S.-T . Yau , Editors , Firs t Internationa l Congres s o f Chines e

Mathematicians, 200 1

19 So-Chi n Che n an d Mei-Ch i Shaw , Partia l Differentia l Equation s i n Severa l Comple x

Variables, 200 1

18 Fangyan g Zheng , Comple x Differentia l Geometry , 200 0

17 Le i G u o an d Stephe n S.-T . Yau , Editors , Lecture s o n Systems , Control , an d

Information, 200 0

16 Rud i Weikar d an d Gilber t Weinstein , Editors , Differentia l Equation s an d

Mathematical Physics , 200 0

15 Lin g Hsia o an d Zhoupin g Xin , Editors , Som e Curren t Topic s o n Nonlinea r

Conservation Laws , 200 0

14 Jun-ich i Igusa , A n Introductio n t o th e Theor y o f Loca l Zet a Functions , 200 0

13 Vasilio s Alexiade s an d Georg e Siopsis , Editors , Trend s i n Mathematica l Physics ,

1999

12 Shen g Gong , Th e Bieberbac h Conjecture , 199 9

11 Shinich i Mochizuki , Foundation s o f p-adic Teichmiille r Theory , 199 9

10 Duon g H . Phong , Lu c Vinet , an d Shing-Tun g Yau , Editors , Mirro r Symmetr y III ,

1999

9 Shing-Tun g Yau , Editor , Mirro r Symmetr y I , 199 8

8 Jiirge n Jost , Wilfri d Kendall , U m b e r t o Mosco , Michae l Rockner ,

and Karl-Theodo r Sturm , Ne w Direction s i n Dirichle t Forms , 199 8

7 D . A . Buel l an d J . T . Teitelbaum , Editors , Computationa l Perspective s o n Numbe r

Theory, 199 8

6 Harol d Levine , Partia l Differentia l Equations , 199 7

For a complet e lis t o f tit le s i n thi s series , visi t t h e AMS Bookstor e a t w w w . a m s . o r g / b o o k s t o r e / .