Curvature of subdivision surfaces

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!