Curvature of subdivision surfaces
|
|
- Mildred Freeman
- 6 years ago
- Views:
Transcription
1 Curvature of subdivision surfaces a differential geometric analysis and literature review Jörg Peters, jorg@cise.ufl.edu Georg Umlauf, georg.umlauf@gmx.de
2 Motivation Almost all subdivision algorithms in the current literature achieve tangent continuity but not curvature continuity. ( with infinite curvature!)
3 Motivation Almost all subdivision algorithms in the current literature achieve tangent continuity but not curvature continuity. ( with infinite curvature!) Why is it difficult to achieve curvature continuity at an extraordinary point (EOP)?
4 Motivation Almost all subdivision algorithms in the current literature achieve tangent continuity but not curvature continuity. ( with infinite curvature!) Why is it difficult to achieve curvature continuity at an extraordinary point (EOP)? The quantities to measure are Gaussian and mean curvature in a neighborhood of an EOP!
5 Motivation Almost all subdivision algorithms in the current literature achieve tangent continuity but not curvature continuity. ( with infinite curvature!) Why is it difficult to achieve curvature continuity at an extraordinary point (EOP)? The quantities to measure are Gaussian and mean curvature in a neighborhood of an EOP! Sample result: At EOP the determinant of the Jacobian of the subdominant eigenfunctions of a curvature continuous subdivision algorithm must have lower degree than the determinant of the Jacobian of the surface.
6 Motivation: Review Understand important lower bound results better: Sabin 91, ( bi-4) Reif 93,96, ( bi-6) Prauzsch,Reif 99, ( bi- ) (Lower bounds on parametrization, not surface) Understand constructions of curvature continuous piecewise polynomial subdivision algorithms Prautzsch 97, Prautzsch, Umlauf 98, Umlauf 99 (hybrid) Reif 98. Understand stiffness of such subdivision algorithms: infinite collection of polynomial pieces but generated by the same rule.
7 # # Talk Outline The (few) basics. (nomenclature) express curvatures of th spline ring converging towards the EOP # #1! " $ %'&)(* +&, -./0 2 -%'&)(* for scalar constants 3 and rational functions : implies necessary constraints Necessary and sufficient contraints: PDEs $, 2. Lower bounds Prautzsch s sufficient condition and construction. The key open problem! (well, sort of) preprint:
8 ?? Talk Outline The (few) basics. (nomenclature) express curvatures of 5 th spline ring converging towards the EOP 7?7 7? A'B)C*>+B D :.;/<=0> E 9-A'B)C*> for scalar constants : F < and rational functions :;< = : implies necessary constraints Necessary and sufficient contraints: E. Lower bounds Prautzsch s sufficient condition and construction. The key open problem! (well, sort of) preprint:
9 G G G G G G J R R Talk Outline The (few) basics. (nomenclature) express curvatures of H th spline ring converging towards the EOP RJ J R1J I J K LMNOP!Q"P S LT'U)V*Q+U W J K L-M.N/OP0Q X L-T'U)V*Q for scalar constants M Y O and rational functions MNO P : implies necessary constraints Necessary and sufficient contraints: PDEs S, X. Lower bounds Prautzsch s sufficient condition and construction. The key open problem! (well, sort of) preprint:
10 Z Z Z Z Z Z e e Talk Outline The (few) basics. (nomenclature) express curvatures of [ th spline ring converging towards the EOP ] e] ] e1] \ ] ^ _`abc!d"c f _g'h)i*d+h j ] ^ _-`.a/bc0d k _-g'h)i*d for scalar constants ` l b and rational functions `ab c : implies necessary constraints Necessary and sufficient contraints: PDEs f, k. Lower bounds Prautzsch s sufficient condition and construction. The key open problem! (well, sort of) preprint:
11 y Setting and definitions The talk focusses generic subdivision (GS): generalization of m n box-spline subdivision generating regular m o surfaces; affine invariant, symmetric, linear, local, stationary. However applies to non-generic cases [Reif 98 (habil), Zorin 98 (thesis)] and nonpolynomial cases. Surface rings are box-splines (with basis p q-r'sutwv ) xzy {w ~} s s)ƒ ˆ Š Œ s xzy q-r's)t*v Ž p qr.sut*v s
12 µ š š Setting and definitions œ œ œ ± Ÿ ³ œ ± ¹ º is square, stochastic subdivision matrix: diagonalizable with eigenvalues ž Ÿ ª «šµ where correspond to the st and ²-³ st block, (for ) to the nd z¹ and ²³ z¹ nd block and to the th block of the Fourier decomposition of. for all yields eigendecomposition ¹ ¹ ¹» ¹» ¹¼ ½ ¾
13 À Ù À À Setting and definitions Expanded in the eigenfunction wà Á*Â Ã*Ä Å Å Å Ä"Æ Ç È É Ê Ë Ì Í ÄÏÎ-Ð'Ä)Ñ*Ò ÓÌ Ô Î-Ð'Ä)Ñ*ÒÖÕ the surface ring zø is of the form zø ÎÐ'Ä)Ñ*Ò Ù Ú Ø À Ô Î-Ð'Ä)Ñ*ÒÖÕ À-Û À Ú Ø À wà Î-Ð'Ä)Ñ*Ò Û À Å
14 Ü Ý ã ã í í ä í à í à Gauss curvature Ü and the mean curvature Ý are Ü Þß'à"á*â ã ä Ý Þß'à"á*â ã ä Þ-ß'à)á*âæåçÞ-ß'àuáwâéè êëþ-ß'à)á*â)ì Þ-ß'à)á*â"î Þß.àuá*âëè ï Þß.à"áwâ ì Þß.à"áwâuî Þ-ß'à"á*âëè ðñêëþ-ß'à)á*â ï Þß.à"áwâò åçþ-ß'à)á*â ðóþ Þß.àuá*â"î Þ-ß'àuáwâ è ï Þ-ß'àôá*â ì â Þ-ß'à"á*â ã õzö õø ö à ï ã õzö õø ùà î ã õ ù õø ùà ã úzõø ö~ö à ê ã úzõø ö ù à å ã úzõø ù+ù à and ú ã Þûõzöüýõ ù âuþÿuõzöüýõ ù ÿ is the normal. Since õ is regular, í î è ï ì ã ÿuõzöüýõ ù ÿ ì is nonzero and Þ-õzö*à)õ ù à)õ ö ö â Þûõzö à"õ ù àôõ ù!ù â è Þûõzö àôõ ù àôõzö ù â"ì ÿuõzö ü õ ù ÿ à Þ-õ ö à)õ ù à)õ ö ö â Þ-õ ù õ ù â è ð Þûõ ö à)õ ù àôõ ö ù â Þûõ ö õ ù â ò Þûõ ö à"õ ù àôõ ù!ù â Þûõ ö õ ö â ð1ÿuõzö ü õ ù ÿ
15 Talk Outline The (few) basics. (nomenclature) express curvatures of th spline ring converging towards the EOP "!$#%&! ' ()*+, - ( "!$#% for. scalar constants and rational functions : implies necessary constraints Necessary and sufficient contraints: PDEs Lower bounds, -. Prautzsch s sufficient condition and construction. The key open problem! (well, sort of) preprint:
16 J T s A 6 J : : 9 9 = = K 0 J : 9 J T : = 9 0 = a A Gauss curvature and mean curvature at EOP Expand into eigenfunctions /%0 as in [Reif 93] V W X /9 2;: 9< / = 2*: 6 8 /CB2;: /FE2;: =?>@< A BD< EG< / H 2;: 6 K&L H&>@< I%J(A 1M2ON / 2PN 9 : 9< / 2PN = : 6 8 =&>@< A / B2ON : BD< / E2PN : EG< / 2ON H : H?>Q< ICJA 1 2 S 1 N 4 5 = 6 T S : 5 9U= J < I%J = 6 KL 1M2 L$13N;L1M2O2YKG4 5 = 6 V W X L : L : K ` 2O2 0 5 < ICJ = 6 6 K&R 0[Z B \]E^\_H 6 K&R where 0ba c ` fhg 0 c 0ba c 4 / 02 / aned / a2 / 0N L 4 T 9U= / 0fhg d T 9 0 / =fhg 4 V W X L < : L : K V W X = 0 / fhg 9 L L : L : ilujlk mon"lprqyl K&L
17 œ t u v wyx z{o { u }~? ƒ ~] ^~_ % }b yˆš } ŒŒŽ } Œ C O Žœ C {š { { is the Jacobi determinant of the subeigenfunctions ( characteristic map ). If x z { { is positive for almost all initial control nets ž Ÿ. Hence denominator ok. then the Gauss curvature at the EOP is infinite. [Catmull-Clark 78, Loop 87, Qu 90] If x z { 98] then the Gauss curvature at the EOP is zero. [Prautzsch & Umlauf If x t v u z { then the Gauss curvature at the EOP is bounded by the second factor of but is possibly non-unique [Sabin 91,Holt 96]. Note combination of tangent continuity and infinite curvature for x z {.
18 ³ º º º ¼ º É Ê º ² º º ¾ ¾ µ ¾ º ¼ º ¼ Ð É If then the limit for ª yields at the EOP ¼&¼Ž½ «± º? ]±^ _² µ Ž¹ ³»P» º ±»» a rational function in À and Á that must be constant! ÃÂ Ä Å Â É É Å Â É ÉÄ Å ³ Æ Ç È Æ Ç È arbitrary implies each summand has to be constant! É Eigenfunctions Ê PDE):»P» º ¼&¼Ž½ Ì»»P» º» ^ Ë ¼@Í ¼¼ ½  ¼¼ must satisfy the six partial differential equations (G-» ¼»P» Å ± ± b É ÏÎ const Ú É Î const for Ð for ÐÛ ÉŠÑ Ò ÓPÔ%ÉÕrÉÄÖØ YÉ7Ñ Ù Ô%É$ÕrÉÖ Summary A GS has for almost all initial nets non-zero Gauss curvature at the EOP if and only if and G-PDE holds. (9 additional partial differential equations for Ü )
19 ä ä à ù ä Þ ä General: GS is curvature continuous if Ý Þ ßà and the differential equations for á and â hold, because the principal curvatures converge like ë ìý Since ò óô ä í Þ ë ìõß à ß à ä î ä î ãrä å æ à Þ â ä ç for ï ð ñ. andãrä Ý ö ß ø ù åäæ à óô â à ä è é ä ê ë ìý à ä í ß à ä î ö ñ ú which implies [Reif Schröder 00] for û Þ ü : The principal curvatures of the limit surface of a GS are square integrable.
20 Talk Outline ý The (few) basics. (nomenclature) ý express curvatures of þ th spline ring converging towards the EOP ÿ for scalar constants and rational functions : implies necessary constraints Necessary and sufficient contraints: PDEs ý Lower bounds,. ý Prautzsch s sufficient condition and construction. ý The key open problem! (well, sort of) ý preprint:
21 & & R R & R R R R * 4 Lower bounds on the degree formal degree vs true degree! #" (= number of non-constant derivatives) Recall Gauss PDE $ % &'& $ ())+*,-$ % )#$ ( )/. $ ))#$ % ( &'& const%8(:9 for ; 9=< >?'@9AB9DCFEG9H< I ; 9 $ % $ )) % * JK$ % )ML 4 &'& const%n% 9 for ; Simple count with R 0!S" JTVU L parametrization. Left side of PDE,XJY,ZJ * [ L. total degree W,ZJ^, * [/. bi-degree W whereas right side of PDE formal total degree of is A JY, formal bi-degree of 1 23_4 is A *, L * [ L JY, total degree (resp. 0 \ R 0 \ R *, L * [ L * ]. A, bi-degree) of regular R 0 ` a b c Degree mismatch: (unless ) If the true degree equals the formal degree then GS is curvature continuous if and only if i.e. EOP is a flat point.,
22 t t i i t t g g e g t g t t t t t A GS with d e fg is curvature continuous only if the true degree of the Jacobian h i Options: (i) The true degree of j or j g is less than k. is less than its formal degree! ii) The leading terms in the Jacobian h i cancel. If not (ii) and not flat then kml n8e o!p#qsrj iut ozp#qsrj g total degree o!psqr left{v e }Zr^}~k l~ k and o!psqr^h i bi-degree o!psqr left{8 e }Zr^}~k l- k } and o!psqryh i Compare to find }~kml!e k :, k nve o!p#qrxwzy e ƒry}k l } e ƒry}k l. ): If not (ii) then GS is curvature continuous and not flat only if the true (bi-)degree of the surface is at least twice the true (bi-)degree of the subdominant eigenfunctions j and j g.
23 Comparison with earlier estimates m!ˆ is consistent with degree estimate of Reif 93, 96, Zorin 97: View surface as a function over the tangent plane parametrized by Š and!. Then non-flat implies non-tangential component at least quadratic in Š and, i.e. Œ ~. [Prautzsch, Reif 99] If the non-tangential component of the surface is at least of degree in Š and m then the surface representation has to be at least of degree Since Š and! have to have a minimal degree to form Ž rings, e.g. m ƒœ in the tensor-product case, a lower bound is Z (parametrization dependent reasoning about surfaces!)..
24 Or (i) the leading terms of 5 cancel total degree: šz #œs leftž8ÿ ' ª š! #œs Y 5 V«Q O X K V«F ± ² bi-degree: š! Sœ leftžvÿ³ ' s šz #œs Y _ V«O ~ /«± µ. Comparing with š! Sœ ^ 5 µ š! Sœ ^ 5. If the true degree of and is not less than then GS is curvature continuous and not flat only if the total degree š! #œ Y _ º¹» ½¼~, ( bi-degree š! Sœ ^ 5 º¹» ½¼~ ). That is possible! E.g. if bi- µ then šz #œs Y _ ¾ is needed as if š! #œ š! Sœ»
25 Ê Ê Talk Outline The (few) basics. (nomenclature) express curvatures of À th spline ring converging towards the EOP Á Â Ã Ä Å Æ ÇÈÉÈ Â ÊÂ Ë Ä ÌÍÎÉÍ Ï Â Ã ÄÅÆÇÈÉ Â ÊÂ Ð ÄÌÍÎÉ for scalar constants Å Ñ Ç and rational functions Å Æ Ç È : implies necessary constraints Necessary and sufficient contraints: PDEs Lower bounds Ë, Ð. Prautzsch s sufficient condition and construction. The key open problem! (well, sort of) preprint:
26 Curvature continuous subdivision constructions [Prautzsch & Umlauf 98]: induce flat spots to get low degree, small mask, curvature continuous subdivision algorithms. [Sabin 91, Holt 96]: adapt the leading eigenvalues to get non-zero bounded curvature. Otherwise need degree-reduced Jacobian. (Trivial) regular case of any Ò Ó box-spline: ÔÕ and Ô!Ó are linear. (Non-trivial) Projection of Prautzsch 97, Reif 98.
27 ó ó Ø Ø ó Ú Û Ú Û á á â â Ú å Û Prautzsch 98: Sufficient conditions ÖQ ÙØ _Û ÖÜDÝÞàß ÖÜ5Ö!Þàß 5Û Ö!ÞÝÞ ã ãoá ãdâ ä ã for æ Ø çãèxãoéqê Then (proof) ÖQ ë Ø ì~úf íöü_öü ë ß áî Ö ë'ë Ø ì~ú ªð Ö ëü Ý Þ ß ÖÜ ë Ö!Þàß ÖÜ_Ö!Þ ëgývß ì~â ïözþdö!þ ëã Ö Ü Ö ë'ë ñ Ü ß á ªð Ö ë'ë Ü Ö Þ ß ìòö ëü Ö ëþ ß Üô Ü5ÞòÛ á Ö Ü ß ì~â Ö Þ Ýã Þu õ ó Ü5Þ:Û ì~ú Ö Ü ß á Ö Þ Ý and ö ë'ë Ø ì ó Ü5Þ Û Û Ö ëü Ý Þ ß Ö ëü Ö ëþ ß Û Ö ëþ Ý Þ Ýê Ö Ü Ö ë'ë ñ Þ ß ì~â ªð Ö ëþ Ý Þ ß Ö Þ Ö ë'ë ñ Þ
28 ù û ù 0 ù ù ù ù ù / ù ù ( ù ø ù úüûîýùþ³ú ÿsú û ÿ û and ø "!#$% '& )+* þ ( )+* with constant (!) ø,.-#/,;/ 0 û 1 / ù û - 6 ù ø : 6 û for 8 9 for 8 ø : < ( )=* ø >?A@ ù3e for B ø CVø D for B ø CVø 5 5 for B ø9 C F
29 h X I X I N X K K X X I X qx X K M X M X M M K M Prautzsch s algorithm (Free-form splines) G HJI and HLK eigenvectors to the subdominant eigenvalue M of the Catmull-Clark algorithm. (Then N and N G IWV'KYX Set NQPSR TUN N[Z\R N and N^]SR T_N a I and a K are the control nets of N degree-doubled representation. have bi-degree O.) K=V`K and N with control nets a b, respectively, in a X'c X'deX f R O. G Subdivision matrix g R h i h j where kr lm a I X a K X a P a Z a ]on i kpr diagtrq K X K X K VsX h j kr Tth uvh Vxw I h u'y The only non-zero eigenvalues of g are corresponding to the eigenvectors m a I X y y y MLT{z VsX -fold M a ]. T#O -fold V
30 Š Š Talk Outline } The (few) basics. (nomenclature) } express curvatures of ~ th spline ring converging towards the EOP Uƒ ˆ + ' Šˆ UŒŽ ` s _ƒl 7 ˆ = Š _ŒŽ ` for ƒ ƒ scalar constants and rational functions : implies necessary constraints Necessary and sufficient contraints: PDEs } Lower bounds,. } Prautzsch s sufficient condition and construction. } The key open problem! (well, sort of) } preprint:
31 Big Question For what choices of eigenfunctions ˆ and of a GS is total degree š œ#ž Ÿ + ^š ˆœ_ bi-degree š œ#ž Ÿ ª ^š œ_ «?
32 Partial Answer µ^ Define the tensor-product mapping of the subeigenfunctions _ ±+²' ³s so that t ¹ µ^ bi-4 and #º _ ± ²W ³ Ẃ» µ^ _ ¼ ± ½S¾ ³ ¼ ³ ½ ±». À ³ quartics: knot insertion Á
33 Ê Partial Answer: Construction 1. Choose ˆà and Â^Ä of the Å7Æ corner patches initially to form Ç of true degree bi Choose the non-corner patches to be of true degree bi-3 and so that the ring of patches is È Ä. 3. Perturb the É -component of the common coefficient of the corner patch. Ê (no influence on next rings; ÃrÄÌË ÇÎÍSÏ ÉLÐÑÇ Ò Ï ÓÔ ). Then Õ^Ö ËtØ Ô»Ù Ú for the non-corner patches and for the corner patch Õ^Ö Ë Ë Â Ã Ð'Â Ä Ô'Ô»Ù Û ÜÌÝ Þ Ëtß Ð ß Ô+Ð Ë Ó Ð ÓÔsÐ Ë Û ÜÝ Þ Ë#ß Ð'Å[Ô Ï Ë#à Ð áâô+ð Ë ÅeÐ ß Ô Ï Ë á7ð à Ô ã;ð Ë Ó ÐoÓ;Ô ã Ù bi-úqä
34 Find the åqæ, å[ç, å^è (solve the PDEs for their coefficients). Any volunteers? Fits nicely with alternative answer: New é ê biquartic free-form surface splines (modification of my Oberwolfach construction 1998)
35 Conclusion express curvatures of ë th spline ring converging towards the EOP ì í î ïuðlñ7òˆó+ô'ó í õˆí ö ï_ Žø`ù ôsø ú í î ï_ðlñ7òˆó=ô í õ í û ïu Lø'ùQô ðlñ7ò ó : implies necessary constraints Necessary and sufficient contraints: PDEs Lower bounds Prautzsch s sufficient condition and construction. The key open problem preprint: It is worth looking for curvature continuous subdivision schemes whose regular rings are polynomial of degree less than 6!
Pointers & Arrays. CS2023 Winter 2004
Pointers & Arrays CS2023 Winter 2004 Outcomes: Pointers & Arrays C for Java Programmers, Chapter 8, section 8.12, and Chapter 10, section 10.2 Other textbooks on C on reserve After the conclusion of this
More informationPointers. CS2023 Winter 2004
Pointers CS2023 Winter 2004 Outcomes: Introduction to Pointers C for Java Programmers, Chapter 8, sections 8.1-8.8 Other textbooks on C on reserve After the conclusion of this section you should be able
More informationModules. CS2023 Winter 2004
Modules CS2023 Winter 2004 Outcomes: Modules C for Java Programmers, Chapter 7, sections 7.4.1-7.4.6 Code Complete, Chapter 6 After the conclusion of this section you should be able to Understand why modules
More informationAPPLESHARE PC UPDATE INTERNATIONAL SUPPORT IN APPLESHARE PC
APPLESHARE PC UPDATE INTERNATIONAL SUPPORT IN APPLESHARE PC This update to the AppleShare PC User's Guide discusses AppleShare PC support for the use of international character sets, paper sizes, and date
More informationCurved PN Triangles. Alex Vlachos Jörg Peters
1 Curved PN Triangles Alex Vlachos AVlachos@ati.com Jörg Peters jorg@cise.ufl.edu Outline 2 Motivation Constraints Surface Properties Performance Demo Quick Demo 3 Constraints 4 Software Developers Must
More informationTriangle Mesh Subdivision with Bounded Curvature and the Convex Hull Property
Triangle Mesh Subdivision with Bounded Curvature and the Convex Hull Property Charles Loop cloop@microsoft.com February 1, 2001 Technical Report MSR-TR-2001-24 The masks for Loop s triangle subdivision
More informationProbabilistic analysis of algorithms: What s it good for?
Probabilistic analysis of algorithms: What s it good for? Conrado Martínez Univ. Politècnica de Catalunya, Spain February 2008 The goal Given some algorithm taking inputs from some set Á, we would like
More informationConMan. A Web based Conference Manager for Asterisk. How I Managed to get Con'd into skipping my summer vacation by building this thing
ConMan A Web based Conference Manager for Asterisk -or- How I Managed to get Con'd into skipping my summer vacation by building this thing $90503&07 $:3.74889028,-47,94708 $90503&078:3.42 Sun Labs, slide
More information) $ G}] }O H~U. G yhpgxl. Cong
» Þ åî ïî á ë ïý þý ÿ þ ë ú ú F \ Œ Œ Ÿ Ÿ F D D D\ \ F F D F F F D D F D D D F D D D D FD D D D F D D FD F F F F F F F D D F D F F F D D D D F Ÿ Ÿ F D D Œ Ÿ D Ÿ Ÿ FŸ D c ³ ² í ë óô ò ð ¹ í ê ë Œ â ä ã
More informationNormals of subdivision surfaces and their control polyhedra
Computer Aided Geometric Design 24 (27 112 116 www.elsevier.com/locate/cagd Normals of subdivision surfaces and their control polyhedra I. Ginkel a,j.peters b,,g.umlauf a a University of Kaiserslautern,
More informationSubdivision Surfaces
Subdivision Surfaces 1 Geometric Modeling Sometimes need more than polygon meshes Smooth surfaces Traditional geometric modeling used NURBS Non uniform rational B-Spline Demo 2 Problems with NURBS A single
More informationCMPT 470 Based on lecture notes by Woshun Luk
* ) ( & 2XWOLQH &RPSRQHQ 2EMHF 0RGXOHV CMPT 470 ased on lecture notes by Woshun Luk What is a DLL? What is a COM object? Linking two COM objects Client-Server relationships between two COM objects COM
More informationASCII Code - The extended ASCII table
ASCII Code - The extended ASCII table ASCII, stands for American Standard Code for Information Interchange. It's a 7-bit character code where every single bit represents a unique character. On this webpage
More informationNormals of subdivision surfaces and their control polyhedra
Normals of subdivision surfaces and their control polyhedra I. Ginkel, a, J. Peters b, and G. Umlauf a, a University of Kaiserslautern, Germany b University of Florida, Gainesville, FL, USA Abstract For
More informationState of Connecticut Workers Compensation Commission
State of Connecticut Workers Compensation Commission Notice to Employees Workers Compensation Act Chapter 568 of the Connecticut General Statutes (the Workers Compensation Act) requires your employer,
More informationSubdivision Scheme Tuning Around Extraordinary Vertices
Subdivision Scheme Tuning Around Extraordinary Vertices Loïc Barthe Leif Kobbelt Computer Graphics Group, RWTH Aachen Ahornstrasse 55, 52074 Aachen, Germany Abstract In this paper we extend the standard
More informationSubdivision. Outline. Key Questions. Subdivision Surfaces. Advanced Computer Graphics (Spring 2013) Video: Geri s Game (outside link)
Advanced Computer Graphics (Spring 03) CS 83, Lecture 7: Subdivision Ravi Ramamoorthi http://inst.eecs.berkeley.edu/~cs83/sp3 Slides courtesy of Szymon Rusinkiewicz, James O Brien with material from Denis
More informationCassandra: Distributed Access Control Policies with Tunable Expressiveness
Cassandra: Distributed Access Control Policies with Tunable Expressiveness p. 1/12 Cassandra: Distributed Access Control Policies with Tunable Expressiveness Moritz Y. Becker and Peter Sewell Computer
More informationOOstaExcel.ir. J. Abbasi Syooki. HTML Number. Device Control 1 (oft. XON) Device Control 3 (oft. Negative Acknowledgement
OOstaExcel.ir J. Abbasi Syooki HTML Name HTML Number دهدهی ا کتال هگزاد سیمال باینری نشانه )کاراکتر( توضیح Null char Start of Heading Start of Text End of Text End of Transmission Enquiry Acknowledgment
More informationMAT 22B-001: Differential Equations
MAT 22B-001: Differential Equations Final Exam Solutions Note: There is a table of the Laplace transform in the last page Name: SSN: Total Score: Problem 1 (5 pts) Solve the following initial value problem
More informationLecture 5 C Programming Language
Lecture 5 C Programming Language Summary of Lecture 5 Pointers Pointers and Arrays Function arguments Dynamic memory allocation Pointers to functions 2D arrays Addresses and Pointers Every object in the
More informationSubdivision Curves and Surfaces
Subdivision Surfaces or How to Generate a Smooth Mesh?? Subdivision Curves and Surfaces Subdivision given polyline(2d)/mesh(3d) recursively modify & add vertices to achieve smooth curve/surface Each iteration
More informationThis file contains an excerpt from the character code tables and list of character names for The Unicode Standard, Version 3.0.
Range: This file contains an excerpt from the character code tables and list of character names for The Unicode Standard, Version.. isclaimer The shapes of the reference glyphs used in these code charts
More informationCartons (PCCs) Management
Final Report Project code: 2015 EE04 Post-Consumer Tetra Pak Cartons (PCCs) Management Prepared for Tetra Pak India Pvt. Ltd. Post Consumer Tetra Pak Cartons (PCCs) Management! " # $ " $ % & ' ( ) * +,
More information1. Oracle Mobile Agents? 2. client-agent-server client-server
1. Oracle Mobile Agents?!"#$ application software system%. &'( )'*+, -. */0 1 23 45 678 9:; >?, %@ +%. - 6A(mobility) : B? CDE@ F GH8!" * channel #I 1 = / 4%. ()'*, &', LAN) - * application
More informationSmooth Multi-Sided Blending of bi-2 Splines
Smooth Multi-Sided Blending of bi-2 Splines Kȩstutis Karčiauskas Jörg Peters Vilnius University University of Florida K. Karčiauskas, J. Peters (VU, UF) SMI14: Bi-3/4 Caps for bi-2 Splines 1 / 18 Quad
More informationTo provide state and district level PARCC assessment data for the administration of Grades 3-8 Math and English Language Arts.
200 West Baltimore Street Baltimore, MD 21201 410-767-0100 410-333-6442 TTY/TDD msde.maryland.gov TO: FROM: Members of the Maryland State Board of Education Jack R. Smith, Ph.D. DATE: December 8, 2015
More informationPersonal Conference Manager (PCM)
Chapter 3-Basic Operation Personal Conference Manager (PCM) Guidelines The Personal Conference Manager (PCM) interface enables the conference chairperson to control various conference features using his/her
More informationModels, Notation, Goals
Scope Ë ÕÙ Ò Ð Ò ÐÝ Ó ÝÒ Ñ ÑÓ Ð Ü Ô Ö Ñ Ö ² Ñ ¹Ú ÖÝ Ò Ú Ö Ð Ö ÒÙÑ Ö Ð ÔÓ Ö ÓÖ ÔÔÖÓÜ Ñ ÓÒ ß À ÓÖ Ð Ô Ö Ô Ú ß Ë ÑÙÐ ÓÒ Ñ Ó ß ËÑÓÓ Ò ² Ö Ò Ö Ò Ô Ö Ñ Ö ÑÔÐ ß Ã ÖÒ Ð Ñ Ó ÚÓÐÙ ÓÒ Ñ Ó ÓÑ Ò Ô Ö Ð Ð Ö Ò Ð ÓÖ Ñ
More informationUNIVERSITY OF CALGARY. Subdivision Surfaces. Advanced Geometric Modeling Faramarz Samavati
Subdivision Surfaces Surfaces Having arbitrary Topologies Tensor Product Surfaces Non Tensor Surfaces We can t find u-curves and v-curves in general surfaces General Subdivision Coarse mesh Subdivision
More informationRecursive Subdivision Surfaces for Geometric Modeling
Recursive Subdivision Surfaces for Geometric Modeling Weiyin Ma City University of Hong Kong, Dept. of Manufacturing Engineering & Engineering Management Ahmad Nasri American University of Beirut, Dept.
More information1 Swing 2006A 5 B? 18. Swing Sun Microsystems AWT. 3.1 JFrame JFrame GHI
' þ ³ š ³ œ ³ 2006 1 Swing! " # &%' ()+-,./0 1 2 45-6 &8% 9 : ; < = >@? 2006A 5 B? 18 C@D E F : G HJILK-M!NPO-Q R S-I!T R!U V-W X Y!Z[N GUI\ ] ^ O-Q R S _a` b-w!c dje!f g Swing Wh i Z j k l m n N VisualEditor
More informationA Parameter Study for Differential Evolution
A Parameter Study for Differential Evolution ROGER GÄMPERLE SIBYLLE D MÜLLER PETROS KOUMOUTSAKOS Institute of Computational Sciences Department of Computer Science Swiss Federal Institute of Technology
More informationThe course that gives CMU its Zip! Web Services Nov 26, Topics HTTP Serving static content Serving dynamic content
15-213 The course that gives CMU its Zip! Web Services Nov 26, 2002 Topics HTTP Serving static content Serving dynamic content Web History 1945: 1989: 1990: Vannevar Bush, As we may think, Atlantic Monthly,
More informationClaimSpotter: an Environment to Support Sensemaking with Knowledge Triples
ClaimSpotter: an Environment to Support Sensemaking with Knowledge Triples Bertrand Sereno, Simon Buckingham Shum & Enrico Motta Knowledge Media Institute The Open University Milton Keynes MK7 6AA, UK
More informationOn Smooth Bicubic Surfaces from Quad Meshes
On Smooth Bicubic Surfaces from Quad Meshes Jianhua Fan and Jörg Peters Dept CISE, University of Florida Abstract. Determining the least m such that one m m bi-cubic macropatch per quadrilateral offers
More informationRefinable C 1 spline elements for irregular quad layout
Refinable C 1 spline elements for irregular quad layout Thien Nguyen Jörg Peters University of Florida NSF CCF-0728797, NIH R01-LM011300 T. Nguyen, J. Peters (UF) GMP 2016 1 / 20 Outline 1 Refinable, smooth,
More informationG 2 Interpolation for Polar Surfaces
1 G 2 Interpolation for Polar Surfaces Jianzhong Wang 1, Fuhua Cheng 2,3 1 University of Kentucky, jwangf@uky.edu 2 University of Kentucky, cheng@cs.uky.edu 3 National Tsinhua University ABSTRACT In this
More informationGraphs (MTAT , 4 AP / 6 ECTS) Lectures: Fri 12-14, hall 405 Exercises: Mon 14-16, hall 315 või N 12-14, aud. 405
Graphs (MTAT.05.080, 4 AP / 6 ECTS) Lectures: Fri 12-14, hall 405 Exercises: Mon 14-16, hall 315 või N 12-14, aud. 405 homepage: http://www.ut.ee/~peeter_l/teaching/graafid08s (contains slides) For grade:
More informationUSB-ASC232. ASCII RS-232 Controlled USB Keyboard and Mouse Cable. User Manual
USB-ASC232 ASCII RS-232 Controlled USB Keyboard and Mouse Cable User Manual Thank you for purchasing the model USB-ASC232 Cable HAGSTROM ELECTRONICS, INC. is pleased that you have selected this product
More informationNon-Uniform Recursive Doo-Sabin Surfaces (NURDSes)
Non-Uniform Recursive Doo-Sabin Surfaces Zhangjin Huang 1 Guoping Wang 2 1 University of Science and Technology of China 2 Peking University, China SIAM Conference on Geometric and Physical Modeling Doo-Sabin
More informationO Type of array element
! " #! $ % % # & : ; a ontiguous sequene of variables. all of the sae type. Eah variable is identified by its index. Index values are integers. Index of first entry is. ' ( ) * + May /,. - ( & ( ( J K
More informationPolar Embedded Catmull-Clark Subdivision Surface
Polar Embedded Catmull-Clark Subdivision Surface Anonymous submission Abstract In this paper, a new subdivision scheme with Polar embedded Catmull-Clark mesh structure is presented. In this new subdivision
More informationERNST. Environment for Redaction of News Sub-Titles
ERNST Environment for Redaction of News Sub-Titles Introduction ERNST (Environment for Redaction of News Sub-Titles) is a software intended for preparation, airing and sequencing subtitles for news or
More informationPairs of Bi-Cubic Surface Constructions Supporting Polar Connectivity
Pairs of Bi-Cubic Surface Constructions Supporting Polar Connectivity Ashish Myles a, Kestutis Karčiauskas b Jörg Peters a a Department of CISE, University of Florida b Department of Mathematics and Informatics,
More informationGuided spline surfaces
Computer Aided Geometric Design 26 (2009) 105 116 www.elsevier.com/locate/cagd Guided spline surfaces K. Karčiauskas a, J. Peters b, a Department of Mathematics and Informatics, Naugarduko 24, 03225 Vilnius,
More informationSecond Year March 2017
Reg. No. :... Code No. 5052 Name :... Second Year March 2017 Time : 2 Hours Cool-off time : 15 Minutes Part III COMPUTER APPLICATION (Commerce) Maximum : 60 Scores General Instructions to Candidates :
More informationAppendix C. Numeric and Character Entity Reference
Appendix C Numeric and Character Entity Reference 2 How to Do Everything with HTML & XHTML As you design Web pages, there may be occasions when you want to insert characters that are not available on your
More informationCalligraphic Packing. Craig S. Kaplan. Computer Graphics Lab David R. Cheriton School of Computer Science University of Waterloo. GI'07 May 28, 2007
Calligraphic Packing Jie Xu Craig S. Kaplan Computer Graphics Lab David R. Cheriton School of Computer Science University of Waterloo GI'07 May 28, 2007 Outline 1 Background of NPR Packing Artistic Packing
More informationIII. CLAIMS ADMINISTRATION
III. CLAIMS ADMINISTRATION Insurance Providers: Liability Insurance: Greenwich Insurance Company American Specialty Claims Representative: Mark Thompson 142 N. Main Street, Roanoke, IN 46783 Phone: 260-672-8800
More informationBanner 8 Using International Characters
College of William and Mary Banner 8 Using International Characters A Reference and Training Guide Banner Support January 23, 2009 Table of Contents Windows XP Keyboard Setup 3 VISTA Keyboard Setup 7 Creating
More informationCommunication and processing of text in the Kildin Sámi, Komi, and Nenets, and Russian languages.
TYPE: 96 Character Graphic Character Set REGISTRATION NUMBER: 200 DATE OF REGISTRATION: 1998-05-01 ESCAPE SEQUENCE G0: -- G1: ESC 02/13 06/00 G2: ESC 02/14 06/00 G3: ESC 02/15 06/00 C0: -- C1: -- NAME:
More informationAdaptive techniques for spline collocation
Adaptive techniques for spline collocation Christina C. Christara and Kit Sun Ng Department of Computer Science University of Toronto Toronto, Ontario M5S 3G4, Canada ccc,ngkit @cs.utoronto.ca July 18,
More informationControl-Flow Graph and. Local Optimizations
Control-Flow Graph and - Part 2 Department of Computer Science and Automation Indian Institute of Science Bangalore 560 012 NPTEL Course on Principles of Compiler Design Outline of the Lecture What is
More informationElliptic vs. hyperelliptic, part 1. D. J. Bernstein
Elliptic vs. hyperelliptic, part 1 D. J. Bernstein Goal: Protect all Internet packets against forgery, eavesdropping. We aren t anywhere near the goal. Most Internet packets have little or no protection.
More information3D Modeling Parametric Curves & Surfaces. Shandong University Spring 2013
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2013 3D Object Representations Raw data Point cloud Range image Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Voxels
More informationApproximating Catmull-Clark Subdivision Surfaces with Bicubic Patches
Approximating Catmull-Clark Subdivision Surfaces with Bicubic Patches Charles Loop Microsoft Research Scott Schaefer Texas A&M University April 24, 2007 Technical Report MSR-TR-2007-44 Microsoft Research
More informationVersion /10/2015. Type specimen. Bw STRETCH
Version 1.00 08/10/2015 Bw STRETCH type specimen 2 Description Bw Stretch is a compressed grotesque designed by Alberto Romanos, suited for display but also body text purposes. It started in 2013 as a
More informationCS354 Computer Graphics Surface Representation III. Qixing Huang March 5th 2018
CS354 Computer Graphics Surface Representation III Qixing Huang March 5th 2018 Today s Topic Bspline curve operations (Brief) Knot Insertion/Deletion Subdivision (Focus) Subdivision curves Subdivision
More informationMachine Learning for Signal Processing Lecture 4: Optimization
Machine Learning for Signal Processing Lecture 4: Optimization 13 Sep 2015 Instructor: Bhiksha Raj (slides largely by Najim Dehak, JHU) 11-755/18-797 1 Index 1. The problem of optimization 2. Direct optimization
More informationUser Guide for Greek GGT-Fonts Revision date: 23 May, 2011
User Guide for Greek GGT-Fonts Revision date: 23 May, 2011 by Graham G Thomason Copyright Graham G Thomason, 2009. Permission is granted to copy or publish this document, provided this complete notice
More informationSubdivision Schemes for Variational Splines
Subdivision Schemes for Variational Splines Joe Warren Henrik Weimer Department of Computer Science, Rice University, Houston, TX 77251-1892 jwarren,henrik @rice.edu Abstract The original theory of splines
More informationConcurrent Execution
Concurrent Execution Overview: concepts and definitions modelling: parallel composition action interleaving algebraic laws shared actions composite processes process labelling, action relabeling and hiding
More information4: Parametric curves and surfaces
4: Parametric curves and surfaces Lecture 1: Euclidean, similarity, affine and projective transformations omogeneous coordinates and matrices Coordinate frames Perspective projection and its matri representation
More informationAdorn. Slab Serif Smooth R E G U LAR. v22622x
s u Adorn f Slab Serif Smooth R E G U LAR B OL D t 0 v22622x 9 user s guide PART OF THE ADORN POMANDER SMOOTH COLLECTION v O P E N T Y P E FAQ : For information on how to access the swashes and alternates,
More information3D Modeling Parametric Curves & Surfaces
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2012 3D Object Representations Raw data Point cloud Range image Polygon soup Solids Voxels BSP tree CSG Sweep Surfaces Mesh Subdivision
More informationCommunication and processing of text in the Chuvash, Erzya Mordvin, Komi, Hill Mari, Meadow Mari, Moksha Mordvin, Russian, and Udmurt languages.
TYPE: 96 Character Graphic Character Set REGISTRATION NUMBER: 201 DATE OF REGISTRATION: 1998-05-01 ESCAPE SEQUENCE G0: -- G1: ESC 02/13 06/01 G2: ESC 02/14 06/01 G3: ESC 02/15 06/01 C0: -- C1: -- NAME:
More informationAutomatic Verification of Finite State Concurrent Systems
Automatic Verification of Finite State Concurrent Systems Edmund M Clarke, Jr Computer Science Department Carnegie Mellon University Pittsburgh, PA 523 Temporal Logic Model Checking Specification Language:
More informationRSA (Rivest Shamir Adleman) public key cryptosystem: Key generation: Pick two large prime Ô Õ ¾ numbers È.
RSA (Rivest Shamir Adleman) public key cryptosystem: Key generation: Pick two large prime Ô Õ ¾ numbers È. Let Ò Ô Õ. Pick ¾ ½ ³ Òµ ½ so, that ³ Òµµ ½. Let ½ ÑÓ ³ Òµµ. Public key: Ò µ. Secret key Ò µ.
More informationComputer Graphics Curves and Surfaces. Matthias Teschner
Computer Graphics Curves and Surfaces Matthias Teschner Outline Introduction Polynomial curves Bézier curves Matrix notation Curve subdivision Differential curve properties Piecewise polynomial curves
More information09 - Designing Surfaces. CSCI-GA Computer Graphics - Fall 16 - Daniele Panozzo
9 - Designing Surfaces Triangular surfaces A surface can be discretized by a collection of points and triangles Each triangle is a subset of a plane Every point on the surface can be expressed as an affine
More informationAdorn. Serif. Smooth. v22622x
s u Adorn f Serif Smooth 9 0 t v22622x user s guide PART OF THE ADORN POMANDER SMOOTH COLLECTION v O P E N T Y P E FAQ : For information on how to access the swashes and alternates, visit LauraWorthingtonType.com/faqs
More information2D Spline Curves. CS 4620 Lecture 13
2D Spline Curves CS 4620 Lecture 13 2008 Steve Marschner 1 Motivation: smoothness In many applications we need smooth shapes [Boeing] that is, without discontinuities So far we can make things with corners
More informationApproximation by NURBS curves with free knots
Approximation by NURBS curves with free knots M Randrianarivony G Brunnett Technical University of Chemnitz, Faculty of Computer Science Computer Graphics and Visualization Straße der Nationen 6, 97 Chemnitz,
More informationEvaluation of Loop Subdivision Surfaces
Evaluation of Loop Subdivision Surfaces Jos Stam Alias wavefront, Inc. 8 Third Ave, 8th Floor, Seattle, WA 980, U.S.A. jstam@aw.sgi.com Abstract This paper describes a technique to evaluate Loop subdivision
More informationSubdivision overview
Subdivision overview CS4620 Lecture 16 2018 Steve Marschner 1 Introduction: corner cutting Piecewise linear curve too jagged for you? Lop off the corners! results in a curve with twice as many corners
More informationSheila. Regular Bold. User s Guide
Sheila Regular Bold User s Guide font faq HOW TO INSTALL YOUR FONT You will receive your files as a zipped folder. For instructions on how to unzip your folder, visit LauraWorthingtonType.com/faqs/. Your
More informationCurve Corner Cutting
Subdivision ision Techniqueses Spring 2010 1 Curve Corner Cutting Take two points on different edges of a polygon and join them with a line segment. Then, use this line segment to replace all vertices
More informationKnow it. Control points. B Spline surfaces. Implicit surfaces
Know it 15 B Spline Cur 14 13 12 11 Parametric curves Catmull clark subdivision Parametric surfaces Interpolating curves 10 9 8 7 6 5 4 3 2 Control points B Spline surfaces Implicit surfaces Bezier surfaces
More informationCurves and Surfaces. Shireen Elhabian and Aly A. Farag University of Louisville
Curves and Surfaces Shireen Elhabian and Aly A. Farag University of Louisville February 21 A smile is a curve that sets everything straight Phyllis Diller (American comedienne and actress, born 1917) Outline
More informationBUCKLEY. User s Guide
BUCKLEY User s Guide O P E N T Y P E FAQ : For information on how to access the swashes and alternates, visit LauraWorthingtonType.com/faqs All operating systems come equipped with a utility that make
More informationSmooth Patching of Refined Triangulations
Smooth Patching of Refined Triangulations Jörg Peters July, 200 Abstract This paper presents a simple algorithm for associating a smooth, low degree polynomial surface with triangulations whose extraordinary
More informationoptions (alternatives)
)! "!#$% #!&! '#($ * +,-./012-3/,-/ 452363/,7,89:636 ; ?@ABCDE=F@GGH>IJ@KLGMLBH=>JL>LGNCHC ; O@?HCH=> PQ=RG@EST=GKH>IDTBLI@C ; U@GLBH=> =V O W B= C=E@ =BX@Q O H C?HAGH>@C * 452363/,YZ8[3,\ 1,05. ],25.-83,-:
More informationSubdivision on Arbitrary Meshes: Algorithms and Theory
Subdivision on Arbitrary Meshes: Algorithms and Theory Denis Zorin New York University 719 Broadway, 12th floor, New York, USA E-mail: dzorin@mrl.nyu.edu Subdivision surfaces have become a standard geometric
More informationHoneyBee User s Guide
HoneyBee User s Guide font faq HOW TO INSTALL YOUR FONT You will receive your files as a zipped folder. For instructions on how to unzip your folder, visit LauraWorthingtonType.com/faqs/. Your font is
More informationA Mixed Fragmentation Algorithm for Distributed Object Oriented Databases 1
A Mixed Fragmentation Algorithm for Distributed Object Oriented Databases 1 Fernanda Baião Department of Computer Science - COPPE/UFRJ Abstract Federal University of Rio de Janeiro - Brazil baiao@cos.ufrj.br
More information1. Introduction. 2. Parametrization of General CCSSs. 3. One-Piece through Interpolation. 4. One-Piece through Boolean Operations
Subdivision Surface based One-Piece Representation Shuhua Lai Department of Computer Science, University of Kentucky Outline. Introduction. Parametrization of General CCSSs 3. One-Piece through Interpolation
More informationOracle Primavera P6 Enterprise Project Portfolio Management Performance and Sizing Guide. An Oracle White Paper December 2011
Oracle Primavera P6 Enterprise Project Portfolio Management Performance and Sizing Guide An Oracle White Paper December 2011 Disclaimer The following is intended to outline our general product direction.
More informationChap. 3. Chap. 3. Recall and Precision Alternative Measures. TREC Collection CACM and ISI Collections CFC (Cystic Fibrosis Collection)
b!"$#%&'(!) *,+.-0/1-0/2 3547698;:'=?@A8;BC
More informationSurfaces for CAGD. FSP Tutorial. FSP-Seminar, Graz, November
Surfaces for CAGD FSP Tutorial FSP-Seminar, Graz, November 2005 1 Tensor Product Surfaces Given: two curve schemes (Bézier curves or B splines): I: x(u) = m i=0 F i(u)b i, u [a, b], II: x(v) = n j=0 G
More informationModified Catmull-Clark Methods for Modelling, Reparameterization and Grid Generation
Modified Catmull-Clark Methods for Modelling, Reparameterization and Grid Generation Karl-Heinz Brakhage RWTH Aachen, 55 Aachen, Deutschland, Email: brakhage@igpm.rwth-aachen.de Abstract In this paper
More informationSolutions B B B B B. ( B/.B œ œ B/ ( /.B œ B/ / G
Solutions. (a) ( cosa& b. œ sina&b œ sina b sina& b œ Þ)$ & & & (b) Let? œ. Then.? œ., so.? % $ (. œ ( œ (?.? œ Œ? G œ G a % b?% $ ' a $ b (c) Let? œ and.@ œ /.. Then.? œ. and @ œ /, so ( /. œ (?.@ œ?@
More informationChemistry Hour Exam 2
Chemistry 838 - Hour Exam 2 Fall 2003 Department of Chemistry Michigan State University East Lansing, MI 48824 Name Student Number Question Points Score 1 15 2 15 3 15 4 15 5 15 6 15 7 15 8 15 9 15 Total
More information124 DISTO pro 4 / pro 4 a-1.0.0zh
0 30 40 50 DISTO PD-Z01 14 DISTO pro 4 / pro 4 a-1.0.0 DISTO pro 4 / pro 4 a-1.0.0 15 16 DISTO pro 4 / pro 4 a-1.0.0 DISTO pro 4 / pro 4 a-1.0.0 17 1 PD-Z03 3 7 4 5 6 10 9 8 18 DISTO pro 4 / pro 4 a-1.0.0
More informationTHE COMPUTER MODELLING OF GLUING FLAT IMAGES ALGORITHMS. Alekseí Yu. Chekunov. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69, 1 (2017), 12 22 March 2017 research paper originalni nauqni rad THE COMPUTER MODELLING OF GLUING FLAT IMAGES ALGORITHMS Alekseí Yu. Chekunov Abstract. In this
More informationAdorn. Serif. Smooth. v22622x. user s guide PART OF THE ADORN POMANDER SMOOTH COLLECTION
s u Adorn f Serif Smooth 9 0 t v22622x user s guide PART OF THE ADORN POMANDER SMOOTH COLLECTION v font faq HOW TO INSTALL YOUR FONT You will receive your files as a zipped folder. For instructions on
More informationTHE COMPUTER MODELLING OF GLUING FLAT IMAGES ALGORITHMS. Alekseí Yu. Chekunov. 1. Introduction
MATEMATIQKI VESNIK Corrected proof Available online 01.10.2016 originalni nauqni rad research paper THE COMPUTER MODELLING OF GLUING FLAT IMAGES ALGORITHMS Alekseí Yu. Chekunov Abstract. In this paper
More informationExact Evaluation Of Catmull-Clark Subdivision Surfaces At Arbitrary Parameter Values
Exact Evaluation Of Catmull-Clark Subdivision Surfaces At Arbitrary Parameter Values Jos Stam Alias wavefront Inc Abstract In this paper we disprove the belief widespread within the computer graphics community
More informationFinite curvature continuous polar patchworks
Finite curvature continuous polar patchworks Kȩstutis Karčiauskas 0, Jörg Peters 1 0 Vilnius University, 1 University of Florida Abstract. We present an algorithm for completing a C 2 surface of up to
More informationRSA (Rivest Shamir Adleman) public key cryptosystem: Key generation: Pick two large prime Ô Õ ¾ numbers È.
RSA (Rivest Shamir Adleman) public key cryptosystem: Key generation: Pick two large prime Ô Õ ¾ numbers È. Let Ò Ô Õ. Pick ¾ ½ ³ Òµ ½ so, that ³ Òµµ ½. Let ½ ÑÓ ³ Òµµ. Public key: Ò µ. Secret key Ò µ.
More information