Tgp chi Khoa hgc - Cong nghe Thuy sdn
s THONG BAO KHOA HOC
TINH DO CONG BE MAT CHO PHAN VUNG BE MAT TLT DO DITATREN PHAN MEM MATLAB
SURFACE CURVATURE COMPUTATION FOR FREE-FORM SURFACE PARTITIONING BASED ON MATLAB PROGRAM
Nguyen Van Tirdng' Ngay nh?n bai, 05/8/2014; Ngay phan bi?n th6ng qua, 11/8/2014; Ngay duyet dang 01/12/2014
T O M T A T
Cd the chia be mat ttr do thdnh cdc viing loi. lom. yen ngua khac nhau nha vdo dac diem dp cong bi mat tai cdc diim tren be mdt. Bdi bdo ndy trinh bdy vi^c tinh dg cong ciia bi mgi tyr do cho myic dich phdn vitng. Cac thong so do eong be mdt dugc sU dyng ldm dQ lieu cho qud tnnh phdn vitng vd xdc dinh bien cdc viing khi su dung cdc dgc tinh hlnh hoc cua bi mat vd ky thudi ma xich trong ITnh vifc xu ly dnh. Qud Irinh tinh todn dugc thuc hi?n nhd chuong trinh dugc viet bdng phdn mem Matlab. Dir lieu ddu vao cho chuang trinh Id phucmg trinh todn hge cua be mat tudo d dang tudng minh hogc bi mgt Bspline. Tpa dg cdc diem tren be mgt cSng nhu tren bien cua cae vitng trong dU lieu ddu ra duac sic dung cho viec mo kinh hda be mgt v&i cdc viing rieng biet trong mdi truang CAD (Computer Aided Design)
Tir khda. Be mat tu do. dg cong be mdt, phdn viing be mgt A B S T R A C T
A free-form surface can be partitioned into different convex, concave and saddle regions thanks to the characteristics of surface curvatures at points on ihe surface. This paper presents Ihe work of surface curvature computation for free-form surface partitioning. The surface curvatures are used as data for partitioning and defining the boundaries of regions on the surface when using the characteristics of surface geometry and the chain code technique in image processing field.
The computation process is performed by a Matlab program The input data of the program are mathematical equations of free-form surfaces in explicit form or Bspline surfaces. The coordinates of points on the surface and on Ihe region
boundaries in the output data are used for modelling the surface with separate regions in CAD environment.
Keywords- Free-form surface, surface curvatures, surface partitioning
I. DAT V A N D £ phap vec ta b l mgt de thdnh ldp mOt vec ta da chilu Be mgt ty do la mgt deu, tran, thudng dugc eho qud trinh chia vung b l mat ty do. Phuang phdp su dung trong cdc ngdnh thiet ke va che tgo khuon nhdm cum md (fuzzy clustering) va phuang phdp mau, thiet ke than d td, tdu thdy, mdy bay vd trong C-means md (fuzzy C-means) dd dugc ede tdc gid cdc tdc pham nghd thugt. Qua trinh gia cdng be mat sd dung d l chia be mat ty do thanh cac vung 111, t y do tren mdy CNC (Computer Numerical Control) Idm va yen ngya. Cac tinh chat hinh hpc ndi tren thudng t i n nhilu thdi gian do dudng kinh dao bi cung dugc Roman vd cs [7. 8] de xac djnh bien vd hgn che bdi ban kinh eong nhd nhat cCia b l mgt phan vCing b l mgt ty do.
c l n gia cdng. Mdt trong nhCrng phuang phap nang Beyvacdngsy [1]daxdpxibemgttydothanh eao nang suat gia cdng be mat t y do Id chia b l mgt cdc tam gidc. Cdc thong s i phdp vec ta vd dp eong thdnh cdc vung khde nhau vd mli vung cd the dugc b l mat dugc tinh de xae djnh hinh dgng cue bg vung gia edng bang cdc dao ed dudng kinh khde nhau. b l mgt tgi cdc dinh tam gidc. Tu dd, ede dinh ndy
Chen vd es [2] da tinh cdc tinh chat hinh dugc nhdm thanh cae vung cd do hinh khae nhau.
hpc cua b l mdt t y do nhu dp cong Gauss, d$ D l ndng cao hidu suit gia cdng bdng edch su eong trung binh, dp eong cyc dai vd eye tieu vd dung dao Idn nhat ed the, Li va Zhang [4] eung chia
'TS. NguySn Vdn Tudng; Khoa Co khi-Truing Ogi hpc Nha Trang
TRUONG DAI HOC NHA TRANG • 65
Tgp chi Khoa hoc - Cong nghe Thuy sdn So 4/2014
b l mgt ty do thanh cdc vung l6i. ldm vd ydn ngyanhd dp cong b l mgt Hp ggi cdc viling Idm vd ydn ngya Id nhung vung tdi han md d do ed t h i xay ra vide cat Igm. Cdc tdc gid da xdy dyng chuang trinh tinh todn chia vCing vd kilm tra cat Igm bing ngdn ngu C++.
Elber vd Cohen [3] da tien hdnh phdn tich d | eong cua b l mgt ty do de nghidn edu dgc tinh d l hinh be mgt. Hp da phat t n i n mgt phuang phap lai su dung ede todn tu ky higu vd todn tu s6 d l tfnh cdc dp cpng be mgt, td do xdc djnh bidn cdc dudng bien cua cae vung rieng bigt tren b l mgt. Tuy nhien, cac phuang trinh dudng bien Id nhung da thuc bdc eao, rit khd gidi.
Noi tom Igi, cho din nay, c6 nhilu cdng trinh nghien cdu phdn vung b l mat ty dc dya tren cdeh tilp can dung dp cong be mgt. Tuy nhidn da so cdc phuang phdp da dua ra la khd phuc tap, kho dp dgng. Tudng vd Pokomy [9] da dua ra mpt phuang phdp dan gidn nhung hieu qud d l phan vung be mat ty do. d ddy, be mat ty do dugc chia thdnh eac vCing 111, 16m vd yen ngya dua trdn dp eong t>e mgt. Dudng bidn cua cdc vung dugc xdc djnh nhd dp dung ky thudt ma xich diing trong xu ly dnh. Qud trinh tinh toan phan vCing va xdc djnh bien b l mgt dugc thyc hign nhd mpt chuang trinh Matlab dugc vilt cho b l mgt ty do d dang tudng minh hogc be mat Bspline.
Bai bdo nay tap taing gidl thtgu phuang phdp tinh toan do cong be mgt ty do cho muc dich phan vung vdi sy h i trg cua phln mim Matlab.
II. D 6 I TU'gNG NGHIDN CtPU VA PHUaNG P H A P NGHIEN CLTU
1. Thong so hinh hpc cua be mat t y do B I mgt ty do ed t h i dugc bilu dien theo:
-Dang an: f(x, y, z) = 0 (1) - Dgng tudng minh z = f(x, y) (2) - Dgng tham so:
S(u,v) = {S.(u,v), S/u,v), S^(u,v)} (3) Mgt s i thdng so hinh hpe chu y l u eua be mat ty do la:
a. Phdp vee ta tgi mdt dilm
Cho mOt be mgt ty do S(u, v) va mpt diem b i t k^ tren be mat. Tgi dilm ndy, Su vd Sv Id hai vde ta tilp tuyin theo hai phuang tham so u va Su vd Sv khong song song nhau, vd mOt vde ta vuong gde vdi ed hai vde ta nay Id vee ta dan vj, dugc xae djnh bdi d n g thde (4) va duac bilu diln nhu trdn hinh 1 [6],
S xS
Vee ta t bat ky vudng gdc vdi n^ dugc gpi Id vde ta tilp tuyin vdi S(u,v) tgi diem p. Mgt phing chua tat ed ede vde ta tiep tuyin vdi mgt S tgi dilm p dugc gpi Id mgt phang tilp tuyin tgi diem p, va dugc k:^ hi$u Id Tp(S) (hinh 1).
Hinh l.Phip victo'ainivjviit tai mgt diem
phing ti^p tuyen
b. Dgng todn phuang thu nhat, F1 Dgng todn phuang thd nhat cua mIt be mgi ^F do S bilu diln cac tinh chat khoi cda be mat, diroc xae dmh bdi [6]:
Fl = dS.dS = Edu^ + 2Fdudv + Gdv^ (5) trong dd
§1^ • F = — — • G - — — dudu ' du dv ' dv dv (6) Id eac h# s i cua dgng todn phuang thu nhat
e. Dgng toan phuang thu hai, F2:
Dgng todn phuang thu hai md td dp eong cua mdt b l mgt t y do, dugc xdc dinb bdi [35]:
F2= -dnS .dS = Ldu^ + 2Mdudv + Ndv^ (7) trong dd
' du" ' dudv ' av' (8) la cae hg so cua dgng todn phuang thi> hai.
d. GO eong Gauss (K) vd d l cong trung binh (H) Cho mIt b l mdt t y do S(u,v) va p la mpt diera bat ky tr§n nd. Gpi (Q) la mgt phang ehua phdp v6c ta cua mgt S tgi dilm p. Giao tuyin eCla Q Id S la mpt dudng cong cd dp eong nhat djnh (hinh 2). Khi mgt (Q) quay xung quanh phdp vec ta ndi trdn thi dO eong cOa dudng cong thay doi. Q-le dd chlnti minh rdng ton tgi ede hudng ma d dd dp cong oia dudng cong dgt dgt gia trj cue t i l u vd eye dgi [6].
Cdc dp cong d ede hudng ndy duge gpi Id cdc Si, eong chinh tdc vd cdc hydng dp cong chinh tai vudng gdc nhau
66 • TRUONG OAI HOC NHA TRANG
Tgp chi Khoa hoc - Cong nghe Thuy sdn So 4/2014
H m h 2. Dg cong cna be mat t u do Dp eong Gaussian (K), dO cong tmng binh (H) vd cac dO cong chinh t i e K ^ va K ^ cCia b l m§t S(u, v), tgi dilm p, dxiac tinh bang cae edng thue [6]:
LN-M-
= K,„.„K^
EG-F-
_\_fEN-2FM + GL\
(9)
^ ^ ™ ) (10) (11) (12) Dya vdo ede gia trj cua do cong Gauss, dp cong trung binh va ede dp cong ehinh tdc, cac dilm tren mpt be mat t y do ed the dugc chia thdnh sdu loai nhu sau [5]:
- Diem eliptic Idm: Neu K > 0 va H > 0.
- Diem eliptie l6i: Neu K > 0 va H < 0.
- Diem hyperbolic: Neu K <0.
- Diem parat)olic fom: Neu K = 0 va H > 0.
- Diem parabolic loi: Neu K = 0 va H < 0.
- Diem ron phang: Neu K = 0 va H = 0.
D l chia mgt be mat t y do thdnh cac vCing 111 ( k l ca vung phang), vung lom va vung yen ngya, hinh dang be mat cue ttd quanh mdt dilm cd t h i dugc chia thdnh ba logi vung khde nhau nhu sau [5]:
" K > 0 vd H £ 0' hinh dang be mat cue bp loi.
* K ^ 0 va H > 0: hinh dang be mat cue bp lom.
* K < 0 va H ' 0: hinh dgng be mat cue bo yen ngya.
2. Tinh do cong cilia b l mgt ty do
Trong nghidn euu ndy, thudt todn phdn vung b l m d t t y d o nhu sau:
(a) Tao tap hgp ludi dilm t>l mat {p} t u md hinh toan hpe cua t>e mat S vd luu t i t ca cac cfilm vao mdt ma trdn ehung.
(b) Tinh cdc thong sd K and H tgi m i l dilm p,^.
(e) Xet moi diem p^ thudc tdp {p}:
- Nlu K > 0 vd H £ 0: luu dilm vdo ma trdn cac dilm vOng loi, md hoa dilm luu thdnh so 1
Cdc ^ e m khdng cd tinh chat ndy dygc ma hda thdnh s6 0.
- Neu K a 0 vd H > 0: luu dilm vdo ma trgn cac dilm viing ldm, ma hda diem lyu thdnh so 2 Cdc dilm khong cd tinh chit ndy duge md hda thanh s6 0.
- N l u K < 0: luu dilm vao ma trdn eac dilm vung ydn ngya, ma hda dilm luu thanh so 3. Cdc diem khdng cd tinh chit nay dugc md hda thdnh soO.
C l u true ma trdn ede viing ndng bidt neu tren tuang ty nhy eau tnJc cOa ma trdn cua anh nhj phdn. Dilu ndy tao dieu kien de ddng eho vigc ap dgng ky thudt md xich trong xu ly dnh d l xdc dinh bien cdc viing da phdn. Cac diem bien se dugc dung de xdy dyng cdc dudng cong khdng gian ba ehilu diing eho viee chia be mgt t y do thanh cac vung khae nhau trong mdi tnrdng CAD.
D l thyc hien tinh todn theo thugt todn ndi tren va can cy cae phuang trinh (9) d i n (12), eac budc de tinh do cong cOa mot b l mdt ty do dugc thyc hien nhu sau:
- Tao tgp hgp cae diem tren b l mat ty do da eho.
- Tinh todn cdc phdp vee ta dan vj tai tat cd ede diem tren be mat.
- Tinh cac he so dgng toan phuang &iy nhit va thu hai.
- Tinh dp cong Gaussian, dp cong trung binh va cdc dp cong ehinh tie.
Trong nghien cuu ndy, vigc tinh todn phdn vung vd xdc djnh bidn cdc vung duac thye hien bdng mot chuang trinh Matlab Chuang trinh gom cae tgp tin M-funetion va M-seript de tgo mo hinh toan cua b l mgt, ti'nh d l cong b l mat, phdn viing vd xdc djnh bien ede viing. Ham tinh todn dp (X)ng ed npi dung ca ban nhy sau:
- Tinh dao hdm bac nhdt vd dgo ham bgc hai theo cdc bien u va v bdng each su dung hdm tieu chuan gradient.
[Xu,Xv] = gradlent(X);
|Yu,Yv] = gradient(Y);
[Zu.Zv] = gradient(Z);
[Xuu,Xuv] = gradient(Xu);
[Yuu,Yuv] = gradient(Yu);
[Zuu,Zuv]= gradient(Zu);
p(uv,Xw] = gradient(Xv);
[Yuv,Yw] = gradient(Yv);
[Zuv,Zw] = gradient(Zv);
X, Y vd Z d ddy Id nhung mdng 2 chilu cCia ede dilm trdn b l mdt. Nhung mdng ndy phdi duge ehuyen thanh cac vec ta de thyc hien cae phep tinh v l sau.
TRLfONG OAI HOC NHA TRANG • 67
Tgp chi Khoa hoc - Cong ngh? Thuy sdn SS 4/2014
- Tinh phdp vde ta tgi cdc dilm trdn be m^t:m = cross(Xu,Xv,2); q - Tinh cae hg s i dang todn phuang thd nhit:
E = dot(Xu,Xu,2); F - Tfnh cac hg so dang todn phuang thu hai:
L = dot(Xuu.n,2): M - Tinh dp cong Gauss:
K = (L.*N - M.''2)./(E.'G - R'^2);
- Tinh dd cong trung binh:
H = (E.-N + G."L - 2.*F.*M),/(2'(E.'G - Tinh dd cong chinh tac:
K „ „ = H + sqrt(H.'^2 - K);
111. K^T QUA NGHIDN CU'U VA T H A O LUAN Trong nghidn cuu nay, chuang trinh Matlab dygc viet eho tudng minh hogc b l mdt Bspline. Bdi bdo nay su dyng be mgt ty do d dang tudng minh d l minh hoa cho vigc tinh todn dp cong b l mdt.
Vi d{/: Cho b l mgt tu do dugc bilu diln bdi phuang trinh Z " , trong dd x vd y cd
= sqrt(dot(m,m,2)); n = m./[q q q];
= dot(Xu,Xv,2); G = dot(Xv,Xv,2);
dot(Xuv.n,2): N = dot(Xw,n,2);
F.^2));
^,„ = H - sqrt(H.''2 - K);
gia trj trong dogn [-1,3]. Gid s u ma trdn dilm lirdl can tgo trdn b l mgt c6 cd la 41 '41 theo hai phuang X vdy.
Trong nghidn cuu ndy, chuang trinh Matlab dugc chgy trdn mdy tinh xdeh tay (Intel Core 15, 1 ,aOGHz, RAM 4 GB) edi ddt hd dilu hdnh Windows 7. Hinh 3 vd hinh 4 trinh bay k i t qua tinh K, H, K^
va K tgi mdt so dilm ludi \r&n b l mdt.
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» TRl/ONG OAI HQC NHA TRANG
Tgp chi Khoa hpc - Cong nghe Thdy s So 4/2014 K i t qud tinh todn eho thiy b l mdt da cho cd
2 viing Idm, 1 viing i l l va 3 viing yen ngya. Chuang trinh cung cho k i t qud tga dp cac dilm bidn cua 6 vClng ndy. Tuy nhidn, doi vdi vi dg ndy, chi c l n su dgng cdc dilm bidn cua cac vung lom vd viing loi d l
xay dyng hai dudng cong khdng gian eho myc dieh chia vCing b l mgt trong moi tnj'dng CAD. Trdn hinh 5 Id md hinh b l mgt da cho vdi cdc dilm bien ciia cac vung lom vd viing III, dugc hiln thj trong mdi trudng Matlab.
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m n h S. MA b l a b \ik m§t \fn eke Cdc dilm bidn cOa vOng III dugc chda trong ma trgn PB24, gom 71 dilm (hinh 6a). Ma trgn PB13eell{1,1} trdn hinh 6b Id ma trgn chda 27 dilm bidn cOa vOng lom thu nhat vd ma trgn PB13eell{2,1}
tren hinh 6e chua 60 dilm bien cua vung ldm thd hai. Trong cdc ma trdn ndy, ede c^t 1, 2 vd 3 tuang ung vdi tga dp x, y va z ciJa cac diem.
Trong chuang trinh tinh todn ndy. cdc gid tri tpa d | cija ede dilm tren be mgt (d m$t ma trdn rieng) cung nhu trdn cdc bien cd t h i de ddng duge luu d dgng file Excel Dieu nay tgo thu§n Igi cho vide nhdp du lidu xdy dyng b l mdt da eho vd ede dudng cong bilu dien bien cua cac viing trdn b l
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TRUdNG DAI HOC NHA TRANG • 69
Tgp chi Khoa hoc - Cong nghe Thuy sdn So 4/2014
Chuang trinh Matlab trong nghidn cuu ndy mdiehi thyc hien tinh todn phdn tat ed eac viing hidn CO tren be mat dd cho thanh cdc viing ridng bidt.
Chuang trinh chua thyc hi#n vigc toi uu hda qud trinh phan viing nhy kit hgp eac viing cd didn tich qua nhd, ed bdn kinh cong nhd nhat khac bigt khing ddng k l so vdi bdn kfnh cong nho nhit cua cdc vung lien k l , thdnh mgt vting idn han nhdm ndng cao nang suat gia cdng. Hidn tgi, viee k i t hgp cdc viing nay e6 t h i dugc thyc hign thu eing trong mdi trudng CAD bing each khdng si> dgng bidn cdc vung CO di§n tieh nhd d l trong qud trinh chia mgt.
IV. K^T LUAN VA KI^N NGHj
Trong nghien ciru ndy, cae thdng so dO cong eua b l mgt dygc tinh toan d l phyc vg cho vigc
phdn viing be mat ty do thanh cdc viing i l l , 16m vd ydn ngya. Vige tinh todn dugc thyc hidn bdng chuang trinh Matlab DCr ligu dau vao trong nghidn cdu ndy Id phuang trinh todn hpc cua b l mat tv do dgng tudng minh hogc b l mat Bspline. Dd li$u d i u ra Id tdp hgp ede diem tren b l mgt vd cdc d i ^ trdn bien cdc viing cOa b l mgt ndy, de phyc vu cho vide xdy dyng b l mgt ty do vdi ede viing neng bi$t trong mdi trudng CAD. K i t qua eho thay chuang trinh dd tgo duge ede du lidu c l n thilt cho vl|c phSn vung b l mdt phdc tgp. Tuy nhidn, de nghidn dni nay hodn thidn han. c l n thyc hign t l i uu hda qui trinh phdn viing theo tieu chi thdi gian gia eong bS nhit trong trudng hgp da chgn dung cac dao kh^c nhau d l gia cdng cdc viing.
TAI LIEU THAM KHAO
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