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Hans-Christian HegeKonrad Polthier (Eds.)

MathematicalVisualizationAlgorithms, Applicationsand Numerics

With 187 Figures, 46 in Colorand 12 Tables

Jfl Springer

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Tab le o f Con ten t s

Pr e f a c e VL i st of C o n t r i b u t o r s X V

I M e s h e s , M u l t i l e v e l A p p r o x i m a t i o n , a n d V i s u a l i z a t i o nT e t r a h e d r a B a s e d V o l u m e V i s u a l i z a t i o n 3Paolo Cignoni, Claudio Montani, Roberto Scopigno1 I n t r o d u c t i o n 32 V o lum e M od e l ing B a s e d on S im pl i c i a l C o m p le xe s 43 V i s u a l i z a t i o n o f S im pl i c i a l C o m p le xe s 54 I s os u r f a c e F i t t i ng 65 D i r e c t V o l u m e R e n d e r i n g A l g o r i t h m s 76 D e c r e a s i n g C o m p l e x i t y 1 37 C o n c l u d i n g R e m a r k s 1 48 A c k n o w l e d g e m e n t s 1 6M e s h O p t i m i z a t i o n a n d M u l t i l e v e l F i n i t e E l e m e n t A p p r o x i -m a t i o n s 19Roberto Grosso, Thom as Ertl1 I n t r o d u c t i o n 1 92 M e s h R e d u c t i o n T e c h n i q u e s 2 03 L i n e a r A p p r o x i m a t i o n i n H u b e r t S p a c e s 2 14 A l g o r i t h m 2 55 R e s u l t s 276 C o nc lus io ns 28E f f i c i en t V i s u a l i z a t i o n o f D a t a o n S p a r s e G r i d s 3 1Norbert Heuer, Martin Rump f1 I n t r o d u c t i o n 3 12 B r i ef R e v i e w o f Fu nc t i on s on Sp a r s e G r id s 343 R e c u r s i v e S p a r s e G r i d I n t e r p o l a t i o n 3 64 P r o c e d u r a l D a t a A c c e ss 3 85 E s t i m a t i ng H igh e r O r de r Fu nc t i on O f fs et s 396 Im pr ov in g Ef f ic iency 42A M e t a S c h e m e fo r I t e r a t i v e R e f i n e m e n t o f M e s h e s 45Markus Kohler \1 I n t r o d u c t i o n 4 52 M e ta Sc he m e fo r Su bd iv i s i on 46

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VIII Table of Contents3 An alysis of th e Type Relation 484 Desc ription of Topology and th e Dou bling O pe rato r 505 The Averaging Op erator . i 536 Ob ject Linking ' 567 Con clusion 57A S c h e m e f o r E d g e - b a s e d A d a p t i v e T e t r a h e d r o n S u b d i v i s i o n 61Detlef Ruprecht, Heinrich Mller1 Int roduct ion 612 Triangle Subdivision 633 Tetra hed ron Subdivision 644 Discussion 67

I I G e o m e t r y a n d N u m e r i c sF i n i t e E l e m e n t A p p r o x i m a t i o n s a n d t h e D i r i c h le t P r o b l e m f orS u r fac e s o f P r e sc r i b e d M e a n C u r vatu r e 73Gerhard Dziuk, John E. Hutchinson1 i?-Harmo nic Maps 732 Disc rete _ff-Harmonic M aps 773 Proof of M ain Th eore m 794 Num erical Re sults 85E f f l c i e n t V o l u m e - G e n e r a t i o n D u r i n g t h e S i m u l a t i o n o f N C -Mi l l ing 89Georg Glaeser, Eduard Groller1 Introd uction 892 Swep t Volumes 923 Tool P at h Ge nerat ion 974 T he _T-Buffer R ep re se nt ati on of Surfaces 995 Conclusion and Fu ture W ork 1046 Ackn owledg emen ts 105C o n s t a n t M e a n C u r v a t u r e S u r f a c e s w i t h C y l i n d r i c a l E n d s 107Karsten Groe-Brauckm ann, Robert B. Kusner, John M. Sullivan1 Imm ersed Exam ples and Almost Em bedd edness 1082 Non existence Re sults for Cylindrical En ds 1083 Th e Necksize Pro blem 1104 Num erical Ex am ples 1115 Th e Fun dam ental Dom ains as Tru ncated Triunduloids 1126 Co njectures 114

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Table of Contents IXD i s c r e t e R o t a t i o n a l C M C S u r f a c e s a n d t h e E l l i p t i c B i l l i a r d 117Tim Ho ff mann1 Introd uctio n 1172 Discrete R ota tio na l Surfaces 1183 Unrolling Polygon s an d Discrete R ota tion al Surfaces 1194 Th e Sta nd ard Bil liard in an Ellipse and Hy perbo la 1195 Discrete R ota tion al CM C Surfaces 120Z o n o t o p e D y n a m i c s i n N u m e r i c a l Q u a l it y C o n t r o l 125Wolfgang Khn1 Discrete Dy nam ical System s 1252 Th e W rapp ing Effect 1273 Zo notop es, Intervals and th e Interv al Hll 1274 Zonotope Dynam ics 1305 The Cascade Red uction Algori thm 1306 Th e Performance of the Cascad e Red uction 1317 Exam ple: The Crem ona m ap 1318 Ex am ple: Lang ford's vector field 132S tr a i gh te s t G e o d e s i c s on P o l yh e d r a l S u r fac e s 135Konrad Polthier, Markus Schmies1 Introd uctio n 1352 Review of Geo desics on Sm oo th Surfaces 1373 C urv atu re of Po lyhe dral Surfaces 1384 Discrete Stra igh test Geodesics 1415 Discrete Geodesic C urv ature 1446 Parallel Tra nslatio n of Vectors 1467 Rung e K ut ta on Polyh edral Surfaces 1488 Co nclusion 150

I II G r a p h i c s A l g o r i t h m s a n d I m p l e m e n t a t i o n sS u p p or t o f Ex p l i c i t T i m e an d Ev e n t F l ow s i n th e O b je c t -O r i en t ed V i s u a l iz a t io n T o o lk it M A M / V R S 153Jrgen Dllner, Klaus Hinrichs1 A rchite ctura l Lim itation s of Visu alization Software 1532 Gra phic s Ob jects: Basic Visua lization En tities 1543 Behavior Gra ph s: Tim e and Event Flows 1574 Exa m ple: An An ima ted, Interact ive 3D Viewer 1615 Imp lem entat ion 1646 Conclusions and Fu tur e W ork 165

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X / - Table of Con tentsA S u rv ey o f Pa ra l l e l C o o rd in a t es 167Alfred Inselberg1 In th e Spirit of De scartes 1672 D ua lity in 2-D 1683 Lines, p-flats and Po lyto pe s in RN 1714 Rep resentat ion M apping 1765 Hype rsurfaces 177H i e r a r c h ic a l T e c h n i q u e s fo r G l o b a l I l l u m i n a t i o n C o m p u t a t i o n s R e c e n t T r e n d s a n d D e v e l o p m e n t s 181Philipp Slusallek, Marc Stamm inger, Hans-Pe ter Seidel1 Introd uct io n 1812 Fun dam entals 1823 Hierarchical Techniques 1844 Clus tering 1885 Refiners Based on Bo und ed T ran spo rt 1906 Con clusions 192T w o - D i m e n s i o n a l I m a g e R o t a t i o n 195Ivan Sterling, Thom as Sterling1 Introdu ct ion 1952 M athem atical Statem ent of the Prob lem 1963 N otation 1974 Th e P Arra y 1975 Th e T-Schem e 2006 T he Modified T-Sc hem e 2007 Period icity 2018 Lower Bo und s 2019 O ptim al Cases 20410 No n-optim al Cases 20511 Miscellaneous Com m ents and Quest ions 206A n Ob j ec t - Or ien t ed I n t era c t iv e S y s t em f o r S c ien t i f i c S imu la -t io n s : D e s i g n a n d A p p l i c a t i o n s 207A.C. Telea, C.W.AM. van Overveld1 Introdu ct ion 2072 Prev ious W ork 2083 Conc eptua l M odel and Design of the Simu lation System 2094 A Fini te Elem ents Ob ject-Oriented Library 2145 Stru ctu re of a Generic F E Sim ulation 2146 Exam ple of Modell ing a PD E : Th e Wave Eq uat ion 2167 Use of the Sim ulation System in Eng ineering Pro blem s 2188 Con clusion 218

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Table of Contents XI

I V G e o m e t r i e V i s u a l i z a t i o n T e c h n i q u e sA u d i t o r y M o r s e A n a l y s i s o f T r i a n g u l a t e d M a n i f o l d s 2 23Ulrike Axen, Herbert Edelsbrunner1 Audio as an Ex perim ental and An alytic Tool 2232 Wave Trav ersal 2243 Wave Traversal as a M orse Fu nction 2304 Co m pu tat ion of W aves, Cri t ical Poin ts and Sound 234C o m p u t i n g S p h e r e E v e r s i o n s 237George Francis, John M. Sullivan, Chris Hartman1 Introd uction 2372 Sym metrie Eversions Driven by W illmore Energy 2383 Visualizing the Dou ble Locus Surface of an Eversion 2454 Level Cu rve M etho ds for Ev erting Spheres 250M o r s e T h e o r y f o r I m p l i c i t S u r f a c e M o d e l i n g 257John C. Hart1 Introd uctio n 2572 Th e Pro blem of M odeling with Imp licit Surfaces 2573 Morse Th eory 2604 Ap plication to Imp licit Surfaces 2635 Con clusion 267S p e c i a l R e l a t i v i ty i n V i r tu a l R e a l i ty 269Rene T. Rau, Daniel Weiskopf, Hanns Ruder1 Introd uction 2692 Special Re lativistic Tran sform ation 2703 Special Re lativistic Re nde ring 2714 Vir tual Re ality for Re lativistic Fligh ts 2755 Des cription of th e System 2786 Conclusions and Fu rther W ork 278E x p l o r in g L ow D i m e n s i o n a l O b j e c t s in H i g h D i m e n s i o n a l S p a c e s 2 8 1Dennis Roseman1 Introd uction 2812 Some Term inology for High Dim ensional Viewing 2813 Four Sam ple Pro blem s 2824 Hew 2825 Gettin g from High Dim ensions Down to Four Dim ensions 2866 M ath em atics and Slicing 2877 Conclusions, Fu ture Developm ents 289

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X II T a b l e o f C o n t e n t s

V V e c t o r F i e l d s a n d F l o w V i s u a l i z a t i o n

F a s t L I C w i t h P i e c e w i s e P o l y n o m i a l F i l t e r K e r n e l s 29 5Han s-Christian Hege, Detlev Stalling1 I n t r o d u c t i o n 2 9 52 L i n e I n t e g r a l C o n v o l u t i o n 2 9 63 C o n v o l u t i o n T h e o r e m , F i l t e r K e r n e l s 3 0 04 A G e n e r a l F a s t L I C A l g o r i t h m 3 0 75 S t a t i s t i c a l A n a l y s i s of L I C I m a g e s 3 0 96 Re su l t s 3 1 2V i s u a l i z i n g P o i n c a r e M a p s t o g e t h e r w i t h t h e U n d e r l y i n g F l o w 3 1 5Helwig Lffelmann, Thomas Kucera, Eduard Grller1 I n t r o d u c t i o n 3 1 52 A b o u t P o i n c a r e M a p s 3 163 P r e v i o u s a n d R e l a t e d W o r k 3 1 74 V i s u a l i z i n g P o i n c a r e M a p s 3 1 85 E m b e d d i n g t h e V i s u a l i z a t i o n of P o i n c a r e M a p s w i t h i n t h e 3 D F l o w 3 2 16 A n i m a t i o n A s p e c t s 3 2 47 I m p l e m e n t a t i o n I s s u e s 3 2 48 C o n c l u s i o n s 3 2 59 A c k n o w l e d g e m e n t s 3 2 6A c c u r a c y i n 3 D P a r t i c l e T r a c i n g 3 2 9Adriano Lopes, Ken Brodlie1 I n t r o d u c t i o n 3 2 92 A c c u r a c y a n d t h e D a t a f l o w M o d e l 3 3 03 P a r t i c l e T r a c i n g 3 3 14 A c c u r a c y A s s e s s m e n t i n a R u n g e - K u t t a M e t h o d 3 325 R e su l t s 3 3 96 C o n c l u s i o n a n d f u t u r e w o r k 3 4 07 A c k n o w l e d g e m e n t s 3 4 0C l i f fo r d A l g e b r a i n V e c t o r F i e l d V i s u a l i z a t i o n 3 4 3Gerik Scheuermann, Hans Hagen, Heinz Krger1 I n t r o d u c t i o n 3 4 32 Cl if fo rd A lg eb ra 3443 Cl i fford A na lys is 3454 V e c t o r F i e l d V i s u a l i z a t i o n U s i n g C l if fo rd A l g e b r a 3 4 75 R e su l t s 3 5 06 A c k n o w l e d g e m e n t 3 5 0

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Table of Contents XIIIV i s u a l i z a t i o n o f C o m p l e x O D E S o l u t i o n s 353Laurent Testard1 Int roduct ion 3532 Extende d Pha se Po rt ra i t s 3563 Appl ica tions to CO DE s 3594 Conclusion 362A p p e n d i x : C o l o r P l a t e s 363