Synthetic generation of Monge Surfaces through folding and its applications to the design of architectural envelopes
Abstract
This work is based on the reinterpretation of the Monge Surface through the logic of growth by folding, in a context of hybridization between digital processes and digital fabrication. We propose the construction of a discrete Monge Surface from the movement of a set of flat surfaces superimposed and linked to each other. The generation law that we propose presents certain conditions that guarantee that the generated meshes have specific geometric properties, which will be relevant for the design and materialization of architectural envelopes. Finally, we present different case studies and their application in architecture.
Full text
012ÿ44 ÿ2567859659ÿ578757ÿ ÿ67ÿ 278 7895ÿ76ÿÿ69ÿ 89ÿ ÿ!"#$%&$'ÿ( )(* +,-./01 2ÿ,456789:4ÿ ;<=4+:>ÿ
ÿ11ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ 3ÿ45ÿ678ÿ896ÿ 3ÿÿ8ÿÿ678ÿ488ÿ865ÿÿ6ÿ7ÿ ÿ !!!" 9"#9$%$&ÿ 3ÿ'(ÿ'8)ÿ$%$&"ÿ 66*68ÿÿ'96ÿ7686*8ÿ+ÿ886 ÿÿ ( ,866ÿ68ÿ98ÿ6*5ÿ( ÿÿ *9)ÿ$$)ÿ%-%./'8)ÿÿÿ 08"ÿ12&32$ÿ4&.ÿ- %%ÿÿÿ !!!"8867686*8"!8845"ÿ#ÿ!!!"*"8ÿ ÿ 3ÿÿÿ6856)ÿ686ÿ9ÿ8)ÿ8ÿ!89ÿ45 ÿ678ÿ*67ÿ ÿ ,87ÿÿ989ÿ8ÿ45ÿ3ÿ486ÿ0"ÿ869,8:ÿ1ÿ;ÿ<"ÿ497)ÿ8848ÿ$%$ $ÿ ÿ 07ÿ*46ÿ7ÿ6ÿÿÿ678ÿ8ÿÿ678ÿÿ$%$&ÿÿ'96ÿ688 6ÿ 568ÿ888)ÿ*689ÿ45ÿ486ÿ0"ÿ869,8:ÿ7 ÿ9ÿ,9ÿ"ÿ084 =:ÿ7 )ÿ676ÿ!ÿ789ÿÿ(ÿ'8)ÿ$%$&ÿ>,848ÿ.2.4"ÿ ÿ 07ÿ*46ÿ7ÿ*988ÿ ÿ88ÿ8,8!ÿ8ÿÿ6ÿ6ÿ986"ÿ ÿ76ÿ88,89"ÿ07ÿ*46ÿ5ÿ6ÿ48ÿ8 9*89)ÿÿ!78ÿÿÿ6)ÿÿ 689ÿÿÿ868,ÿ568)ÿÿ66689ÿÿ5ÿÿÿ45ÿ5ÿ8)ÿ 886)ÿ 87)ÿ765)ÿ89ÿÿ678!8)ÿ!67*6ÿ678ÿÿ8ÿÿ !6ÿ ÿ678ÿ*478"ÿ078ÿ686ÿÿ67ÿ4?ÿÿ678ÿ8465ÿÿ678ÿ*67ÿ 9ÿ98ÿ6ÿ88 5ÿ886ÿ678ÿÿÿ678ÿ*478"ÿ07ÿ*46ÿ7ÿ *988ÿÿ88ÿ8,8!ÿ8ÿÿ 6ÿ6ÿ986"ÿ ÿ 896ÿ899ÿ488ÿ98ÿ@ÿ6ÿ '*8,ÿ88ÿ6ÿ.%2.)ÿ *99ÿ98ÿA68,98)ÿ(**5ÿ '>7ÿ3/-33.43B24$4ÿ
ÿ111ÿ 234566789ÿ7843ÿ ÿ ÿÿÿ !"#$%$ÿ&!"'%%ÿ$!ÿ(!)%#ÿ*+,)%$%#ÿ&!"'ÿ -ÿ./01ÿ2!3"4%ÿ5!)6ÿ7!#!%")6ÿ8# 99'9!ÿ:*ÿ ;<ÿ=ÿ>?<@A1BÿC*2D8(ÿC%")!,3%ÿ !"#9%9ÿ89!"%)3%,ÿ$!ÿ(%9%,'E%ÿ: +%ÿ FAGHÿ;ÿ=IJKFL1ÿ !"#$%$ÿ&39M)%ÿC3, %"%%ÿ(6,!D(3,3NO%ÿ ÿ 234566789ÿPQ4RS849ÿS97ÿT976US849ÿ ÿ .GÿW-1ÿ&"3)!!X4#Y,9!Yÿ5ÿZ!ÿ*'#9"%ÿ [ÿ\<I?ÿ('N8(*Xÿ5ÿZ!ÿ*'#9"%ÿ 56!#!ÿ+"3)!!$ 4#ÿ)39"O'9!$ÿ93ÿ96!ÿ$! !,3+N!9ÿ3Mÿ&"3)!!X4#ÿ]^_ÿ'$!"ÿ 96!ÿ)6%"9!"ÿ,%$ÿ3'9ÿÿ96!ÿ!`!)'9 !ÿN!!9 4ÿ3Mÿ96!ÿ:82"%X^ÿ ÿ a3S98bS849ÿ ÿ :82"%Xÿ]_]eÿDÿC3$49%,ÿ89!,,4!9ÿ:E#9!N#ÿ(3M!"!)!ÿ#ÿ63#9!$ÿOEÿ96!ÿ f:5855gÿhi7ÿCiX285*jÿ*7(k85g (57gÿlÿ2gfg58(:ÿmC*2nÿ%9ÿ !"#9%9ÿ89!"%)3%,ÿ$!ÿ(%9%,'E%ÿm8(nÿÿC%")!,3%ÿ:+%ÿ%$ÿ #'++3" 9!$ÿOEÿ96!ÿ:82"%Xÿg`!)'9 !ÿ89!"%93%,ÿ(3NN99!!ÿm(g8nÿ%$ÿ96!ÿ :82"%Xÿ*$ #3"Eÿ(3NN99!!ÿm(*n^ÿ ÿ ;< ÿ=ÿ>?<@A1BÿC*2D8(ÿC%")!,3%ÿ opqrsÿÿ ÿ FAGHÿ;ÿ=IJKFL1BÿC*2D8(ÿC%")!,3%ÿYÿ !"#$%$ÿ&3 9M)%ÿC3, %"% %ÿ ouvopqrsÿ ÿ w0Iÿxÿ;HyBÿC*2D8(ÿC%")!,3%ÿ z{|{sq}ÿouus~r|qru|ÿ ÿ ;I?ÿJG-GBÿC*2D8(ÿC%")!,3%ÿ |rru|q}ÿ{}qru|ÿ ÿ JÿIG1GIÿ00G<<BÿC*2D8(ÿC%")!,3%ÿ ',%ÿ*,N%"%ÿf%9%,%ÿ$!ÿ*$"%$!ÿ& %O,3ÿC%'!"3ÿ("#9O%,ÿC!)!"ÿ X%!,%ÿh"346!"ÿgMMN%ÿ2%3+3',3'ÿ&%O,3ÿj3"!3Dg"3%ÿ 3#9%9 %ÿ!,%)6"3+3',3'ÿ*,!`#ÿ$!ÿ'#9E%ÿ3"%#%ÿ f36!Nÿ&%)6!)3ÿ:%$"%ÿ:)6!ÿf%$ÿ5%"!ÿ*"!,ÿ%, !'!,%ÿ ("#9%ÿ)'#ÿ*4%$ÿZ%"%4ÿgN%$ÿ%"M%!"%N3#O%9ÿf%3Nÿ%')!$3ÿ ÿ 4943ÿÿÿÿÿÿ&,%9'Nÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ23,$ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿC"3!ÿ ÿ ÿ ÿ ÿ¡ÿ ¢£¤ÿ ¥¦§¦¨ ÿÿÿÿÿ
010234567589ÿ747584ÿÿ47ÿ297ÿ566ÿ84ÿ4ÿ85ÿ 89584ÿ5ÿ567ÿ784ÿÿ96857 95ÿ7477 ÿ !"# $%ÿ&'('$ÿ)$*+,-*"./( (, 00012ÿ 4ÿ347745ÿ1588587 6- (,ÿ7-'8 090:7 5;774ÿ567ÿ<4584ÿ4ÿ567ÿ=89757>ÿ?67ÿ=885ÿ84ÿ1968579 57 @ A-ÿ-ÿ-ÿ ($ 0BB?6 7ÿ<477497ÿÿ:88487ÿ=784Cÿ1856489ÿ=784ÿ4 ÿ=885ÿ 49584>ÿ1ÿ2557ÿÿ567ÿ15 @/ ÿ#D,$ÿ#$ (,,$ 0EF1 =ÿ=885ÿ5788587 @/ ÿ#D,$ÿ#$ (,,$%ÿ7/(ÿ&,$-%ÿ( -ÿG$%ÿ#/,$ÿ#--A 0FH 49584ÿIÿ24579587ÿ14ÿ<47745ÿ34844745 ÿJÿ K 34ÿ19685795ÿL7857 &- $ÿ)'!ÿM-%ÿNO,ÿP/*QÿRÿG(,S-(%ÿ($ÿTD-* 0UH1 89584ÿÿ158898ÿ245787497ÿ84ÿ567ÿ77745584ÿÿ9685 795ÿ 5 957ÿ8487ÿ53ÿ7589ÿ45674589ÿ49584 &,$ -'(ÿG+'8-*%ÿVW-,ÿMW$ 1X0?6 7ÿ<57ÿ7ÿÿY874ÿZÿ7ÿÿ<49798[4ÿ2449ÿ84ÿL 4 T -(ÿ$W (%ÿT( (ÿP $,$M-(%ÿG(M$-ÿ\-,-ÿ]$/-ÿ^(*($ ,( 100_7 47587ÿ247>ÿ2457584ÿ12ÿ845ÿ9685795ÿ4587 & '('$ÿT,S$ÿT (,,$%ÿ` 8-(-ÿT+'--ÿT$S,+%ÿGM/-,ÿ7,$ 11B?6 7ÿ285ÿ_44>ÿ154584ÿ885ÿ4ÿ885ÿ87ÿ;856 ÿ8ÿ 7 ]*/ "T8(-8ÿ`/ ÿ\$W%ÿ\- 8-ÿ7(W,- 10F:7 34ÿ567ÿ755673ÿÿ<45584ÿ=784 -/ÿW$ (8$ÿT $%ÿR-(-ÿa'$ÿ$ÿG $ 19H1 89584ÿÿ<1=ÿ<1ÿ84ÿ567ÿ774745ÿÿ58b8487ÿ78 4ÿ 47 5687ÿÿ567ÿ5796489ÿ4ÿ885ÿ58484ÿÿ845 8ÿ784 5 745 c$ ,dWWÿ,-e$ÿP-/-%ÿ)($ÿfM-g$ÿP/ $ h
ÿ ÿ 1 23456478ÿ636473ÿÿ36ÿ186ÿ455 ÿ 73ÿ3ÿ74ÿ78473ÿ4ÿ456ÿ673ÿÿ 8574684ÿ6366ÿ ÿ !"ÿ#$ÿ%&'&"ÿ(")*+,) -.''+ÿ/ÿ #0'1,'ÿ,ÿ2,1'++$ÿ345$ÿ/0'1,'ÿ,+ÿ!ÿ6&"$ÿ345ÿ ÿ ÿ &7'849.:,;ÿ<&'&":=")+ ,)9,7.:,.ÿ >?484@ÿA7'ÿB"Cÿ'ÿD,ÿ"ÿ7,ÿ,',4,'"ÿ"<ÿ7,ÿ"=,ÿ2.<&,ÿ7".=7ÿ7,ÿ +"='&ÿ "<ÿ="B7ÿDEÿ<"+'=$ÿ'ÿÿ&",Fÿ"<ÿ7ED'')'"ÿD,B,,ÿ'='+ÿ4"&,,ÿÿ '='+ÿ<D'&'" :ÿG,ÿ4"4",ÿ7,ÿ&".&'"ÿ"<ÿÿ'&,,ÿ"=,ÿ2.<&,ÿ<"8ÿ7,ÿ 8"1,8,ÿ"<ÿÿ,ÿ"<ÿ<+ÿ.<&,ÿ.4, '84",ÿÿ+'C,ÿ"ÿ,&7ÿ"7,:ÿA7,ÿ =,,'"ÿ+Bÿ7ÿB,ÿ4"4",ÿ4,,ÿ&,'ÿ&"''"ÿ7 ÿ=.,,ÿ7ÿ7,ÿ =,,,ÿ8,7,ÿ71,ÿ4,&'<'&ÿ=,"8,'&ÿ4"4,',$ÿB7'&7ÿB'++ÿD,ÿ,+,1ÿ<"ÿ7, ÿ ,'=ÿÿ8,'+')'"ÿ"<ÿ&7',&.+ÿ,1,+"4,:ÿ%'++E$ÿB,ÿ4,,ÿ'<<,,ÿ&,ÿ .',ÿ ÿ7,'ÿ44+'&'"ÿ'ÿ&7',&.,:ÿ H62IJÿ"=,ÿ2.<&,$ÿ!"+ÿ2.<&,$ÿK'&.+!-8,7,$ÿL,1,+"4D+,ÿ 2. <&,$ÿM',8'&ÿ",+ÿ NÿO34887P3ÿ Qÿ2.4,<'&',ÿ,ÿ"=,ÿR%'=.ÿSTÿ"ÿ.ÿ'4"ÿ,ÿ.4,<'&',ÿ8.Eÿ,4,&'+,ÿ &"ÿ',,,ÿ4+'&&' ",ÿ,ÿU.',&.:ÿ3ÿ<.,"ÿ,.'ÿ,ÿSVWXÿ YZ[ÿ]^_Y^[`ÿaZbcdÿefdÿg^_ÿ`dhibdÿjZkZlÿmnoÿqrstuvw xwtÿxruyoÿz{|ztÿ}{zoqÿ~oqÿ |{uo~tqÿq{|ÿ}o|t|}tqÿoÿr|oÿwqoÿqrstuvwxwtÿztqouu{~~o~tÿxro~rwtuoÿÿ ÿ ÿÿÿÿÿÿÿÿÿ {ywwt|}{ÿztÿr|oÿxruyoÿs~o|oÿouw}uouwoÿx{|q}o|}t ÿt|ÿv{uoÿÿ}oo{ÿxr{ÿ ÿÿÿÿÿÿÿÿR"=,ÿSVWXT:ÿÿ K "88,,ÿ,ÿ&&,')ÿ4"ÿ.ÿ&.1ÿ=,,')ÿ4+ÿU.,ÿ,&",ÿ ""="+8,,$ÿEÿ'ÿ"'$ÿ.ÿ& .1ÿ',&')ÿ&.+U.',$ÿ4+ÿ"ÿ+D,:ÿ K"8"ÿ4"4',,ÿ'84",$ÿ7EÿU.,ÿ,&ÿU.,ÿ+ÿ=, ,'&,ÿEÿ ',&'&,ÿ,ÿ+ÿ2.4,<'&',ÿ,ÿ"=,ÿ"ÿ+ÿQ,ÿ,ÿK.1.ÿ!'&'4+ÿ RQK!Tÿ,ÿ+ÿ. 4,<'&',:ÿ3"ÿ'84+'&ÿU.,ÿS: ÿA"ÿ+ÿ=,,'&,ÿ"ÿ&.1ÿ ¡
ÿ ÿ 123435ÿ678ÿ934ÿ432848ÿ3ÿ35ÿ235ÿ89985ÿÿÿ35ÿ235ÿ 89985ÿ54ÿ13328235ÿ84 8ÿ5ÿ3ÿ5834ÿ9735ÿ123435ÿÿ3238335ÿÿ 78ÿ743ÿ323ÿ8ÿÿ85ÿ33ÿ84ÿ57ÿ323ÿ1ÿ973 5ÿ13328235ÿ 94943ÿ139723848ÿ3ÿ23ÿ571898ÿÿ858ÿÿ235ÿ718985ÿ8ÿ 48ÿ34 ÿ58ÿ17884ÿ84ÿ9ÿ5718985ÿ8484335ÿ1ÿ9732678ÿ 94!74ÿ8ÿ9735ÿ13328235ÿ84ÿ82ÿ85139 ÿ"5ÿ3785ÿ235ÿ5#34ÿ9ÿ74ÿ 1ÿ8ÿ571898ÿ8ÿ3ÿ584ÿ82ÿ935ÿ8518932ÿ48ÿ4ÿ8$ 58ÿ5%4ÿ 848ÿ8483985ÿ1%$35ÿ ÿ ÿ &'()*+ÿ-ÿ71898ÿ8ÿ 48ÿ.7848/ÿ5ÿ3785ÿ01ÿ 35ÿ718985ÿ8ÿ 48ÿ234ÿ5ÿ857335ÿ858ÿ74ÿ174ÿ8ÿ53ÿ 3839ÿ94ÿ7239485ÿ678ÿ23 4ÿ827943ÿ858ÿ23ÿ183ÿ 117853ÿ8ÿ 48ÿ2353ÿ47855ÿ35ÿ45348ÿ638584ÿ0789ÿ462"72 ÿ :281ÿ009ÿ;34ÿ849435ÿ312939485ÿ9457935ÿ133ÿ82ÿ 935ÿ8ÿ89985ÿ123435ÿ82ÿ35ÿ392ÿ8ÿ33ÿ38393848ÿ<ÿ858ÿ =>?@ÿBCÿDE?CFG>H>CDÿDCÿICDÿJKÿBCL@M>LKB@ÿNO@EIB>LPÿQEFGKHCDRÿS 344ÿ8ÿ32ÿ 00T39ÿUÿK?KFCHCLÿ=KMV>WLÿFCH@P>BKDÿH@LÿCIÿ=WFM>L@ÿNXKFYCBÿQEFGKHCDRÿCLÿIKÿ 28373ÿ7 53ÿ4Z5231[ÿ\34ÿ07]9ÿ48ÿ58ÿ8984ÿ8575ÿ 38395ÿÿ825ÿ857973285ÿ8ÿ312939 %4ÿ84ÿ3678973ÿ4ÿ 83ÿ849435ÿ859355ÿ8575ÿ34ÿ8%95ÿ9ÿ13995ÿ8ÿ 718 985ÿ8ÿ 48ÿ848335ÿ94ÿ89985ÿ3238335ÿ85ÿ89ÿ94ÿ 9735ÿ4123435ÿÿ ÿ135ÿ8ÿ23ÿ 849%4ÿ5493ÿ117853ÿ1ÿ82ÿ3839ÿ3495ÿ 9135ÿ678ÿ23ÿ571898ÿ85322328ÿ5 8ÿ23ÿ678ÿ82ÿ1234ÿ8ÿ23ÿ PCLCFK=F>^ÿNFECBKÿD>LÿBCDI>^KFRÿPEKFBKÿELKÿFCIKH>_LÿMEUÿ>M?@F=KL=CÿH@LÿIKÿHEFYK ÿ 89`/ÿ58ÿ33ÿ8ÿ23ÿ71898ÿ23ÿ8ÿ923ÿ973ÿ 498ÿ23ÿ71898ÿ8ÿ 48ÿ858ÿ23ÿ 71898ÿ23ÿ8ÿ57ÿ89`ÿ 8$31234ÿ23ÿ83ÿ5493ÿ8ÿ 48ÿ3ÿ235ÿ3973285ÿ2838435ÿ 3935ÿ 3285ÿ1523ÿ23ÿ949819%4ÿ8ÿ9732678ÿ1ÿ8ÿ71898ÿ8ÿ 48ÿ 8518932848ÿ25ÿ 9355ÿ8ÿ89`ÿ4ÿ12343ÿ a885ÿ893ÿ678ÿ23ÿ71898ÿ23ÿ8ÿ743ÿ973ÿ85ÿ743ÿ8ÿ235ÿ85ÿ 5718985ÿ853223285ÿ3484932ÿ899348ÿÿ123ÿ359335ÿ32ÿ8ÿ bcc
ÿ ÿ 12ÿ342526ÿ78ÿ95942625628ÿÿ25241ÿ4ÿÿ252562ÿ12ÿÿ5ÿ 542ÿ241ÿ25ÿ ÿ6ÿ642ÿ2492ÿ42925ÿ42112ÿ 7ÿ2292ÿ59564ÿ4292562ÿ261ÿ 42ÿÿ! 2492ÿ"4ÿ 1212ÿ2ÿ25256ÿ4#9ÿ2561ÿ25ÿ$46%5&"648ÿ'5#2&(56ÿ )*)+ÿ7ÿ 42ÿ ÿ429,5ÿ25642ÿÿ!2492ÿ"4ÿ7ÿÿ!2492ÿ12ÿ$52ÿ25ÿ '5#2&(568ÿ$46%5&"64ÿ)*)- ÿÿ261ÿ2ÿ42256ÿ46ÿ 5ÿ592ÿ959426ÿ 42ÿ262ÿ.6ÿ62/6ÿ92564#512ÿ25ÿÿ 2562ÿ 2960ÿ &!2ÿ452ÿÿ25249,5ÿ56169ÿ12ÿ5ÿ!2492ÿ12ÿ$52ÿÿ464ÿ12ÿ 256ÿ1 2ÿ5ÿ95256ÿ12ÿ429462ÿ5ÿÿ211ÿ2ÿ16ÿ2ÿ25ÿ 42ÿ5ÿ!2492ÿ"4ÿ194261 ÿ25ÿ94ÿ5ÿ3ÿ12ÿ12ÿ21ÿ 25ÿ462964ÿ2ÿ5ÿ62ÿ2562ÿ2611ÿ45428ÿ4 1ÿ)*6ÿ %ÿ452ÿ5ÿ9ÿ7ÿ959426ÿ2ÿ42ÿ4ÿ194264ÿÿ 2492ÿ12ÿ$52ÿ25ÿ 5ÿÿ41ÿ4ÿ914#624ÿ5ÿ789:;<8ÿ 9594262562ÿ25ÿ=>?@ABC?78D:;<ÿ3ÿ949624%69ÿ12ÿ ÿ=>?@ABC?78D:;<ÿ 12564ÿ12ÿÿD:;<:;ÿF>G<ÿ:HC@GÿIJJ;:Gÿ12ÿ2595644ÿ25ÿ"6655ÿ26ÿ ÿ)**Kÿ 7ÿ#ÿ229%92562ÿ25ÿ"66558ÿL524ÿ)**M8ÿ1512ÿ2ÿ 5ÿÿ2562ÿ956496ÿ42296ÿ ÿ64ÿ4ÿ12ÿ1ÿ &!2ÿ452ÿ5ÿ62ÿ264964ÿ25ÿ2ÿ2ÿ7ÿÿ!2492ÿ12ÿ$52ÿ 194 261ÿ62ÿ62ÿ264964ÿ2ÿ959 2ÿ9ÿ5ÿ295ÿ12ÿ 22256ÿ21ÿ7ÿ2426ÿ25ÿ4 ÿ58ÿ97ÿ256ÿ2462ÿ 954ÿÿ4ÿ641255ÿ12ÿÿÿ6ÿ 6ÿÿ999, 5ÿ12ÿÿ 22256ÿ12ÿÿÿ12ÿ94ÿÿÿ56228ÿÿÿ2ÿ2ÿ25642ÿ412ÿÿ 95256ÿ5ÿ &N2 ÿÿ642ÿÿ12ÿ2492ÿ1244 2ÿ926 2ÿ12ÿ24ÿ 2562511ÿ9ÿ!2492ÿ"42 ÿ&2ÿ95148ÿ2ÿ95ÿ7ÿÿ2492ÿ 652592&ÿ 4142ÿ2ÿ9ÿ12ÿ95ÿ7ÿ12ÿ52912ÿ1 244 2ÿ!2ÿ 42256ÿ26ÿ.6ÿ2492ÿ9ÿ222ÿ4694ÿ12ÿÿ556ÿ 2492ÿ6 52592ÿ 2ÿÿÿ &ÿ95256ÿ12ÿ4ÿ46296,59ÿ2/26ÿ25ÿ2ÿ46%98ÿ%ÿ9ÿÿ 4 66ÿ4218ÿ2645ÿÿ5ÿ4261ÿ49ÿ12564ÿ5ÿ 2/49,5ÿ4,9ÿ95ÿ#ÿÿ7ÿ 9ÿ 42ÿ26ÿ 2492ÿ2ÿ2/5142ÿ25ÿ4,/ÿ 9952ÿ OÿQRSTUTVTWXYÿ Z[\]ÿ_`ab[\_c]d_e`ÿb[\fÿgh]`\b]i]ÿj]klÿbhÿdl`dbg\lÿibhÿmgb`[]n_b`\lÿcofp_dlÿ ]qnb`\]ilrsÿdqt][ÿj][b[ÿdl` dbg\q]hb[ÿtÿnb\lilhec_d][ÿb[\f`ÿbugqb[\][ÿb`ÿ $46%5&"648ÿv4&"2wÿ)*)-ÿ!2.5ÿ2ÿ98ÿÿ2ÿ5ÿ 2 5256ÿ4#9ÿ25618ÿ71ÿ25ÿ2ÿ52ÿ41ÿ12ÿÿ 9,5ÿ7ÿ95ÿÿ71ÿ12ÿÿ52442 56ÿ1628ÿ2ÿ9ÿ12ÿ 414ÿ 442562ÿ12ÿ21649ÿ922ÿÿ425ÿ12ÿÿ49,5ÿ 62#69ÿ xyz
ÿ ÿ ÿ 1234567874ÿ94ÿ4ÿ9494ÿ2ÿ1234567874ÿ5ÿ97854ÿ ÿ774ÿ94ÿ2ÿ825ÿ8479ÿ4ÿ2ÿ3 ÿ44ÿÿ2ÿ234567874ÿ 94554ÿ445ÿ2ÿ1234567874ÿ94ÿ4ÿÿ4979ÿ24ÿ4ÿ3ÿ24ÿ 8 744ÿÿÿ825ÿ4ÿ45ÿ54ÿ978ÿ1234567874ÿ5ÿ725ÿÿÿ44ÿ 8ÿÿ1234567874ÿ5 ÿ4ÿ48245ÿ9785479ÿ4ÿ85ÿ3ÿ35ÿÿ24ÿ 4ÿ3ÿ44ÿ4ÿ374ÿ54ÿÿ57ÿ94ÿÿ5 ÿ97854ÿ 234537 94ÿÿ4ÿ725ÿ! ÿ ÿ #$%&'(ÿ)*ÿ1234567874ÿ94ÿ4ÿ2ÿ+ÿ1234567874ÿ5ÿ97854ÿ5,ÿ244-ÿ .ÿ254ÿ/ 0 ÿ 1455ÿ3ÿ94ÿÿ1234567874ÿ5ÿ97854ÿ .ÿ1234567874ÿ5ÿ8ÿ822745ÿ94554ÿ32 494ÿ97854754ÿ 4974ÿ3ÿ44ÿ54ÿ2ÿ549ÿ8659ÿ2ÿ234567874ÿ 37 9578ÿ 178ÿ1234567874ÿ5ÿ97854ÿ4ÿ628744ÿ94554ÿ4ÿ4ÿ 3ÿ994ÿÿ97572873ÿ94ÿÿ 57ÿ94457ÿÿ2 ÿ523ÿ94ÿ234567874ÿ 94554ÿ3454484-ÿ47795ÿ3545ÿ8ÿ84 54ÿ4ÿ2ÿ 57845ÿ 487ÿ72ÿ94ÿÿ9ÿ725ÿ!ÿ ÿ #$%&'(ÿ6(*ÿ49ÿ549495ÿ94ÿÿ57ÿ94ÿÿ5ÿ#$%&'(ÿ67ÿ1455ÿ3ÿ 94ÿÿ1234567874 ÿ5ÿ97854ÿ+ÿ97572873ÿ94ÿ57ÿ244-ÿ.ÿ254ÿ/0 ÿ 89:
ÿ ÿ 123435623678598ÿ5ÿ898ÿ61ÿ51ÿ8592671ÿ85ÿ6ÿ15835ÿ8ÿ6ÿ82338ÿ 8ÿ1548ÿ6ÿ926ÿ8ÿ8 461ÿÿ283738591ÿ56926ÿ12ÿ8461ÿ89ÿ 288598ÿ96591ÿ85ÿ6ÿ5692686ÿ171ÿ85ÿ1ÿ2181ÿ8ÿ 623635ÿ34396ÿ6ÿ 8ÿ6ÿ436ÿ8ÿ8461ÿ356ÿ6ÿ926127635ÿ8ÿ6ÿ1276ÿ656ÿ85ÿ1926ÿ 12 76ÿ8636ÿ3828598ÿ7836598ÿ2183738591ÿ2678598ÿ48179231ÿ8ÿ 6821ÿ15ÿ891ÿ911ÿ1ÿ8ÿ88 ÿ158328ÿ12ÿ8461ÿ88ÿ 76982363628ÿ159293678598ÿ7836598ÿ76982368ÿ6735628ÿÿ9851146ÿ ! ÿ8ÿ1298ÿ651ÿ ÿ7181ÿ21891ÿ485826ÿ56ÿ82338ÿ8ÿ1548ÿ3289366ÿ85ÿ "#$%&'($)*+,-.ÿÿ 12ÿ81ÿ6ÿ766ÿ356ÿ88ÿ76982363628ÿ7836598ÿ 6239821ÿ651ÿ/01997655ÿ89ÿ6ÿ12234ÿ/5654ÿ 89ÿ6ÿ12674ÿ6ÿ6ÿ8ÿ8ÿ 288596ÿ213868ÿ8ÿ+,-.ÿ9#:.ÿ,;(%:ÿ<==-,:ÿ/01997655ÿ89ÿ6ÿ122>4ÿ 88 ÿ5ÿ591ÿ8ÿ396ÿ7121485931ÿ6ÿ133368ÿ15ÿ851278ÿ?6ÿ 1276ÿ2896598ÿ885825ÿ8ÿ6ÿ 82338ÿ0162ÿ88436ÿÿ8ÿ281298ÿ8ÿ 82338ÿ8ÿ8ÿ85216ÿ128ÿ86ÿ 012ÿ1921ÿ61ÿ@591ÿ6 ÿ6ÿ48582635ÿ8ÿ6ÿ766ÿ171ÿ8518598ÿ 623989536ÿ8ÿ21158ÿ5ÿ39876ÿ1592931ÿ8831 ÿ8ÿ1888ÿ6ÿ 6ÿ2136ÿ436ÿ485826936ÿ8ÿ6ÿ82338ÿ286315511ÿ15ÿ8ÿ8461ÿ128ÿ 6ÿ82338ÿ0162ÿ32896ÿ98ÿ39876ÿ8ÿ769823636ÿ15ÿ6ÿ28635ÿ8ÿ5ÿ 7865371ÿ8ÿ8878591ÿ651 ÿ629361ÿ85928ÿÿÿ15ÿ6636ÿ 28398598ÿ66ÿ8ÿ8567628ÿ85ÿ1276ÿ656ÿÿ63232ÿ35678 598ÿ6ÿ 1335ÿ8636ÿ8ÿ6ÿ766ÿ6ÿ6ÿ8ÿ328ÿ8ÿ11298ÿ 88ÿ5ÿ591ÿ8ÿ396ÿ1592931 ÿ658ÿ1ÿ2191931ÿ8A891ÿ 82365ÿ8ÿ56ÿ7376ÿ8ÿ8ÿ48582635ÿ15ÿ56ÿ213868ÿ48179236 ÿ ÿ5ÿ39876ÿ1592931ÿ17B5ÿC6ÿ8ÿ2876262ÿ8ÿ2885965ÿ 629362368ÿ393596ÿ28891ÿ6ÿ6ÿ623 635ÿÿ8ÿ71596@8ÿ8ÿ8825ÿ 82ÿ656366ÿ8ÿ1276ÿ1278512366ÿ8ÿ378785966ÿ85ÿ8ÿ921ÿ / C3628 6ÿ62935D06912ÿ126E4ÿ/F158DG35936ÿ6295D06912ÿ126H4ÿ ÿ ÿ IJKJLJMNOPQÿ ÿ S26582ÿÿTÿF26885ÿUÿ/12634ÿ1548ÿ268ÿ65ÿ6562ÿ48183ÿ1369315ÿ V<&$W('ÿ<=ÿX,<+,:$YZÿ6 2H/64ÿ[ÿ C36286ÿÿTÿ\6261ÿ]Fÿ/126E4ÿ^18ÿ 17139315ÿ35ÿ\2C398928_ÿ6936ÿ 02182938 ÿ65ÿ698236ÿ!8Aÿ!89`ÿUÿ6>ÿ71ab7aHÿ C99_cc13124c62622>c2222[D26[D2117D[ÿ C36286ÿÿTÿ62935D069 12ÿ\ÿ/126E4ÿ0856738591ÿF231ÿÿ38d1ÿ 16126931ÿ F8178926ÿ8622168ÿ85ÿ17133158ÿ846 6ÿ\23989536ÿefeÿ 154281ÿ6ÿ13868ÿf821D\7823656ÿ8ÿF236ÿ34396ÿ126EÿSC82ÿ8345ÿ 02188354ÿg1ÿ1ÿ126Eÿ/ÿ>21D>2>4ÿC99_ccA13124c626267c821D 34263126ED61262>ÿ F3DhÿÿTÿ !1738ÿ\Ufÿ/12124ÿ\393156ÿ48178923ÿ35893469315ÿ1ÿ628ÿ 1548ÿ268ÿU1256ÿ1ÿC3 ÿ 1582858ÿ8238ÿ673>/64ÿ6122aÿ iji
ÿ ÿ 123456748936ÿÿÿ382ÿÿÿ237ÿ 7ÿ173723ÿ ! 723ÿ3"ÿ##6 23 ÿ3ÿ $7 7ÿ%7&ÿ%7'ÿ(ÿÿ $#)**"22*++,*--,.-.ÿ 12345674-936ÿÿÿ 3-2ÿÿ+.ÿ/23026ÿ-ÿ!70762#167ÿ#ÿ3"ÿ "6ÿ1 7"ÿ6$'7$ÿ $7 7ÿ2 77"3ÿ2ÿ$7ÿ2,$ÿ7/!7ÿ3"ÿ 2"ÿ31!ÿ/23773 7ÿ##ÿ454-4.ÿ $#)**"22*+45* 23 7 "7+.+454ÿ 3467"2ÿ73ÿÿ+..7ÿ89:;9<=>?ÿA9BC=DE<DF?ÿBGE9=D:=ÿHÿ?EIDC?A?Jÿÿ7"ÿ 332ÿ K3##7ÿ(ÿ% L76ÿKÿ1ÿÿ#7 LÿMÿ+7ÿND:;D;9<DCÿO9B9?=CPÿQ:=ÿR=CPD<9C<G=9ÿ?SAÿ NGDIADS TÿU:SB<=GC<D:Sÿ#37ÿ/$Vÿ$#)**"22*++,*.,5-2-2+.-72,-ÿ K02$#L2ÿ%ÿ30320ÿW %ÿ+4ÿXSCHCI:E9AD?ÿ:Qÿ?S?IH<DC?IÿBG=Q?C9Bÿ/$V)ÿ #37ÿ$#)**"22*++,*.,5-2-2+.-++,, 2-,ÿ YÿZ[ÿ2V33ÿ\[ÿ]6637ÿ([ÿZ3ÿZY[ÿ]3ÿ]ÿ7ÿ172V7 ÿV2"7663ÿ '$ÿ 23 6ÿV7$7ÿ 3"ÿ"70762#167ÿ 7ÿ/ÿM3ÿ1#$ÿ42)ÿ75+^75.ÿ 3-2ÿÿÿY_#74-374ÿÿ ÿ!70762#167ÿ`67"ÿ 7ÿ2Vÿÿ /a63" 6ÿ\76&ÿ3"ÿM$7ÿ##6 23ÿÿ $7 6ÿ 7 ÿ2 77"3ÿ3ÿ $7V ÿÿ ÿ026ÿ27ÿ#37ÿ/$Vÿ##ÿ+,-+ÿ $#)**"22*++,*.,5-2- 2-75-b5ÿ 3-2ÿÿÿ12345674-936ÿÿÿcdefdghÿjklmnoeÿpkqdgÿrsgtdunvÿwlÿ ##2 $ÿ2VÿV737"ÿ#$ ÿ$3L3ÿx1ÿx_3ÿ15 ÿ67 _3 ÿ .4ÿ,^5,ÿ$#)**"22 *+..4*7+.,.ÿ 3-2ÿÿÿW-7yÿ`ÿÿ172V7ÿ"7 #0ÿaÿ#73V732ÿ 5 2ÿV73"2ÿx1xÿ`70ÿ!7ÿx_3ÿ15 ÿ73ÿ6ÿx" _3ÿÿ^ +ÿ$#)**"22*+. .4*77,ÿ 237ÿ1ÿ+5,ÿREEIDC?<D:SÿA9ÿIz?S?IHB9ÿ{ÿI?ÿ8|:;|<=D9ÿ} $767)ÿÿ 2$6ÿ1ÿ ÿ% $66ÿ]ÿ+4ÿND:;D;9<DCBÿQ:=ÿR=CPD<9C<G=9ÿ~ÿ9BDTSÿ#37ÿ /$Vÿ$#)**"22*++,*.,5 -2-2+.-+.+-+ÿ 2V33ÿ\ÿ#76ÿÿ\27ÿÿÿK63ÿÿ,ÿR=CPD<9C<G=?IÿT9:;9<=Hÿ}7367 aÿ 337ÿ7ÿ 2V33ÿ\ÿYÿZÿ]6637ÿ(ÿ}2173L2ÿÿÿ]3ÿ]ÿ,1ÿ172V7aÿ2ÿV6- 6 a7ÿ772Vÿ 7ÿ2ÿ $7 7ÿ33ÿ/ÿ311`\ÿ,ÿ%7'ÿZ2Lÿ $#)**"22*+++4*+,455 +,745ÿ 2V33ÿ\ÿ $37ÿÿ}2ÿÿ $V7"$27ÿ\ÿ]3ÿ]ÿ}6"3ÿ%ÿÿ ]6637ÿ(ÿ 5ÿ772Vÿ 7ÿ2Vÿ367ÿ 07"ÿ#376ÿ/ÿM3ÿ 1#$ÿ,ÿ$#)**"22*+++4*+277+ +277,4ÿ 2V33ÿ\ÿ]6637ÿ(ÿ5ÿM$7ÿ2 6ÿ72V7aÿ2ÿ 6ÿ3"ÿ 23 6ÿV7$7ÿ "0ÿ/2V#ÿ $ÿ.ÿ.^75ÿ$#)**"22*++,*+-,-.4-ÿ M3ÿ/ÿK63ÿÿ}2ÿÿ(ÿ]6637ÿ(ÿ ÿ2V33ÿ\ÿ+7ÿ36aÿ3"ÿ"73ÿ 2ÿ 07"ÿ##2ÿ 7ÿ33ÿ"7373ÿ[ÿ1V42ÿ ÿK2$67ÿÿ 737ÿÿÿ6aÿÿx"ÿ"03 7ÿ3ÿ $7 6ÿ172V7aÿ##ÿ5^2ÿ W!ÿ\2 $ $6076ÿxM\ÿ $ÿ