%!PS-Adobe-2.0 %%Creator: dvips 5.54 Copyright 1986, 1994 Radical Eye Software %%Title: seg99n.dvi %%CreationDate: Thu Mar 22 16:56:37 2001 %%Pages: 4 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: C:\EMTEX\DVIPS16.EXE -O0.0mm,-8.8mm seg99n %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 2001.03.22:1656 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies 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b(1755{1758,)c(So)q(c.)j(Explor.)h (Geoph)o(ysicists)q(,)d(T)m(ulsa,)e(1999.)p eop %%Page: 2 2 2 1 bop 670 -75 a Fd(In)o(terp)q(olation)12 b(within)i(ra)o(y)h(cells) 13 606 y @beginspecial 18 @llx 143 @lly 577 @urx 750 @ury 1404 @rwi @setspecial %%BeginDocument: cell1.ps /Win35Dict 290 dict def Win35Dict begin/bd{bind def}bind def/in{72 mul}bd/ed{exch def}bd/ld{load def}bd/tr/translate ld/gs/gsave ld/gr /grestore ld/M/moveto ld/L/lineto ld/rmt/rmoveto ld/rlt/rlineto ld /rct/rcurveto ld/st/stroke ld/n/newpath ld/sm/setmatrix ld/cm/currentmatrix ld/cp/closepath ld/ARC/arcn ld/TR{65536 div}bd/lj/setlinejoin ld/lc /setlinecap ld/ml/setmiterlimit ld/sl/setlinewidth ld/scignore false def/sc{scignore{pop pop pop}{0 index 2 index eq 2 index 4 index eq and{pop pop 255 div setgray}{3{255 div 3 1 roll}repeat setrgbcolor}ifelse}ifelse}bd /FC{bR bG bB sc}bd/fC{/bB ed/bG ed/bR ed}bd/HC{hR hG hB sc}bd/hC{ /hB ed/hG ed/hR ed}bd/PC{pR pG pB sc}bd/pC{/pB ed/pG ed/pR ed}bd/sM matrix def/PenW 1 def/iPen 5 def/mxF matrix def/mxE matrix def/mxUE matrix def/mxUF matrix def/fBE false def/iDevRes 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt def/fPP false def/SS{fPP{ /SV save def}{gs}ifelse}bd/RS{fPP{SV restore}{gr}ifelse}bd/EJ{gsave showpage grestore}bd/#C{userdict begin/#copies ed end}bd/FEbuf 2 string def/FEglyph(G )def/FE{1 exch{dup 16 FEbuf cvrs FEglyph exch 1 exch putinterval 1 index exch FEglyph cvn put}for}bd/SM{/iRes ed/cyP ed /cxPg ed/cyM ed/cxM ed 72 100 div dup scale dup 0 ne{90 eq{cyM exch 0 eq{cxM exch tr -90 rotate -1 1 scale}{cxM cxPg add exch tr +90 rotate}ifelse}{cyP cyM sub exch 0 ne{cxM exch tr -90 rotate}{cxM cxPg add exch tr -90 rotate 1 -1 scale}ifelse}ifelse}{pop cyP cyM sub exch 0 ne{cxM cxPg add exch tr 180 rotate}{cxM exch tr 1 -1 scale}ifelse}ifelse 100 iRes div dup scale 0 0 transform .25 add round .25 sub exch .25 add round .25 sub exch itransform translate}bd/SJ{1 index 0 eq{pop pop/fBE false def}{1 index/Break ed div/dxBreak ed/fBE true def}ifelse}bd/ANSIVec[ 16#0/grave 16#1/acute 16#2/circumflex 16#3/tilde 16#4/macron 16#5/breve 16#6/dotaccent 16#7/dieresis 16#8/ring 16#9/cedilla 16#A/hungarumlaut 16#B/ogonek 16#C/caron 16#D/dotlessi 16#27/quotesingle 16#60/grave 16#7C/bar 16#82/quotesinglbase 16#83/florin 16#84/quotedblbase 16#85 /ellipsis 16#86/dagger 16#87/daggerdbl 16#89/perthousand 16#8A/Scaron 16#8B/guilsinglleft 16#8C/OE 16#91/quoteleft 16#92/quoteright 16#93 /quotedblleft 16#94/quotedblright 16#95/bullet 16#96/endash 16#97 /emdash 16#99/trademark 16#9A/scaron 16#9B/guilsinglright 16#9C/oe 16#9F/Ydieresis 16#A0/space 16#A4/currency 16#A6/brokenbar 16#A7/section 16#A8/dieresis 16#A9/copyright 16#AA/ordfeminine 16#AB/guillemotleft 16#AC/logicalnot 16#AD/hyphen 16#AE/registered 16#AF/macron 16#B0/degree 16#B1/plusminus 16#B2/twosuperior 16#B3/threesuperior 16#B4/acute 16#B5 /mu 16#B6/paragraph 16#B7/periodcentered 16#B8/cedilla 16#B9/onesuperior 16#BA/ordmasculine 16#BB/guillemotright 16#BC/onequarter 16#BD/onehalf 16#BE/threequarters 16#BF/questiondown 16#C0/Agrave 16#C1/Aacute 16#C2 /Acircumflex 16#C3/Atilde 16#C4/Adieresis 16#C5/Aring 16#C6/AE 16#C7 /Ccedilla 16#C8/Egrave 16#C9/Eacute 16#CA/Ecircumflex 16#CB/Edieresis 16#CC/Igrave 16#CD/Iacute 16#CE/Icircumflex 16#CF/Idieresis 16#D0/Eth 16#D1/Ntilde 16#D2/Ograve 16#D3/Oacute 16#D4/Ocircumflex 16#D5/Otilde 16#D6/Odieresis 16#D7/multiply 16#D8/Oslash 16#D9/Ugrave 16#DA/Uacute 16#DB/Ucircumflex 16#DC/Udieresis 16#DD/Yacute 16#DE/Thorn 16#DF/germandbls 16#E0/agrave 16#E1/aacute 16#E2/acircumflex 16#E3/atilde 16#E4/adieresis 16#E5/aring 16#E6/ae 16#E7/ccedilla 16#E8/egrave 16#E9/eacute 16#EA /ecircumflex 16#EB/edieresis 16#EC/igrave 16#ED/iacute 16#EE/icircumflex 16#EF/idieresis 16#F0/eth 16#F1/ntilde 16#F2/ograve 16#F3/oacute 16#F4 /ocircumflex 16#F5/otilde 16#F6/odieresis 16#F7/divide 16#F8/oslash 16#F9/ugrave 16#FA/uacute 16#FB/ucircumflex 16#FC/udieresis 16#FD/yacute 16#FE/thorn 16#FF/ydieresis ] def/reencdict 12 dict 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Fk(is)h(used)h(to)f(obtain)h(the)f(v)n (alues)h Fh(\034)837 1375 y Fc(A)865 1379 y Fb(k)899 1391 y Fk(of)f(an)g(in)o(terp)q(olated)i(quan)o(tit)o(y)f Fh(\034)j Fk(at)c(p)q(oin)o(ts)h Fl(A)1600 1398 y Fg(k)1635 1391 y Fk(from)e(kno)o(wn)h(v)n(alues)0 1438 y Fh(\034)21 1422 y Fc(B)48 1426 y Fb(k)81 1438 y Fk(and)f Fh(\034)176 1422 y Fc(C)203 1426 y Fb(k)237 1438 y Fk(at)g(the)g(v)o(ertices)h(of)e (the)i(ra)o(y)f(cell.)18 b(Linear)c(in)o(terp)q(olation)706 1494 y Fh(\034)727 1476 y Fc(X)768 1494 y Fk(=)d Fh(w)837 1500 y Fi(1)855 1494 y Fh(\034)876 1476 y Fc(A)904 1480 y Fa(1)931 1494 y Fk(+)d Fh(w)997 1500 y Fi(2)1016 1494 y Fh(\034)1037 1476 y Fc(A)1065 1480 y Fa(2)1092 1494 y Fk(+)g Fh(w)1158 1500 y Fi(3)1176 1494 y Fh(\034)1197 1476 y Fc(A)1225 1480 y Fa(3)1901 1494 y Fk(\(5\))0 1538 y(with)13 b(resp)q(ect)g(to)f(\()p Fh(w)307 1544 y Fi(1)325 1538 y Fh(;)6 b(w)370 1544 y Fi(2)388 1538 y Fk(\))12 b(is)h(then)g(used)g(to)g(obtain)h(the)e(v)n(alues)i(at)e(p)q(oin)o(t)i Fl(X)p Fk(.)j(This)c(in)o(terp)q(olation)i(\014ts)e(the)g(functional)h (v)n(alues)g(at)0 1576 y(the)d(six)h(v)o(ertices)g(of)f(the)g(ra)o(y)h (cell.)17 b(Since)c(the)e(v)n(alues)h(at)f(eac)o(h)h(cell)g(face)f(are) h(dep)q(enden)o(t)g(on)g(the)f(v)n(alues)h(at)f(the)h(v)o(ertices)g(of) f(the)g(face)0 1613 y(only)m(,)h(the)g(in)o(terp)q(olated)h(v)n(alues)g (are)e(con)o(tin)o(uous)j(across)d(the)h(cell)g(faces.)17 b(This)11 b(in)o(terp)q(olatio)q(n)j(seems)d(con)o(v)o(enien)o(t)i(in)f (in)o(terp)q(olatin)q(g)0 1650 y(all)h(the)f(quan)o(tities)i(resulting) h(from)c(ra)o(y)h(tracing,)h(except)g(for)e(the)h(tra)o(v)o(el)h(time)g (whic)o(h)f(should)i(b)q(e)e(in)o(terp)q(olated)j(more)d(accurately)m (,)0 1688 y(see)h(\014gures)h(2a,b,c,d.)0 1759 y Fl(3.2)30 b(Bicubic)12 b(in)o(terp)q(olation)0 1830 y Fk(Assume)h(that)g(partial) i(deriv)n(ativ)o(es)899 1892 y Fh(p)918 1874 y Fg(i)943 1892 y Fk(=)995 1868 y Fh(@)r(\034)p 988 1883 59 2 v 988 1918 a(@)r(x)1032 1907 y Fg(i)1901 1892 y Fk(\(6\))0 1951 y(are)e(kno)o(wn)h(in)h(addition)g(to)f(the)f(functional)j(v)n (alues)f(of)e Fh(\034)k Fk(at)c(six)i(v)o(ertices)f Fl(B)1148 1957 y Fi(1)1167 1951 y Fk(,)f Fl(B)1222 1957 y Fi(2)1241 1951 y Fk(,)g Fl(B)1296 1957 y Fi(3)1315 1951 y Fk(,)g Fl(C)1371 1957 y Fi(1)1389 1951 y Fk(,)g Fl(C)1445 1957 y Fi(2)1464 1951 y Fk(,)g Fl(C)1520 1957 y Fi(3)1552 1951 y Fk(of)g(the)g(ra)o(y)h(cell.)19 b(W)m(e)14 b(ma)o(y)0 1988 y(then)f(wish)h(to)f(use)g(the)g(bicubic)i(in)o(terp)q(olation)h (with)e(resp)q(ect)f(to)g Fh(w)1008 1995 y Fg(B)1049 1988 y Fk(and)g(\()p Fh(w)1166 1994 y Fi(1)1184 1988 y Fh(;)7 b(w)1230 1994 y Fi(2)1248 1988 y Fk(\))12 b(to)h(\014t)g(b)q (oth)h(the)f(functional)i(v)n(alues)f(and)g(the)0 2025 y(\014rst)f(partial)i(deriv)n(ativ)o(es)g(at)e(the)g(six)h(v)o (ertices.)k(T)m(o)13 b(\014t)g(the)g(functional)i(v)n(alues,)f(w)o(e)f (de\014ne)h(cubic)g(function)815 2080 y Fh(a)p Fk(\()p Fh(w)q Fk(\))c(=)h Fh(w)974 2062 y Fi(2)992 2080 y Fk(\(3)d Ff(\000)h Fk(2)p Fh(w)q Fk(\))765 b(\(7\))0 2132 y(with)17 b(functional)j(v)n(alues)e Fh(a)p Fk(\(0\))g(=)f(0,)g Fh(a)p Fk(\(1\))h(=)f(1)g(and)h(\014rst)f(deriv)n(ativ)o(es)i Fh(a)1122 2117 y Fm(0)1134 2132 y Fk(\(0\))e(=)g(0,)h Fh(a)1315 2117 y Fm(0)1327 2132 y Fk(\(1\))f(=)g(0)g(at)g(the)g(endp)q (oin)o(ts)i(of)e(in)o(terv)n(al)0 2170 y Fh(<)11 b Fk(0)p Fh(;)6 b Fk(1)11 b Fh(>)p Fk(.)16 b(T)m(o)d(\014t)g(the)g(deriv)n(ativ) o(es,)i(w)o(e)d(de\014ne)i(cubic)h(functions)665 2262 y Fh(b)p Fk(\()p Fh(w)724 2269 y Fg(B)752 2262 y Fk(\))c(=)f(\()p Fh(w)861 2269 y Fg(B)889 2262 y Fk(\))904 2244 y Fi(2)951 2211 y(3)929 2223 y Fe(X)932 2312 y Fg(i)p Fi(=1)996 2262 y Fh(p)1015 2244 y Fg(i)1015 2270 y Fc(B)1044 2262 y Fk([)p Fh(A)1084 2244 y Fg(i)1097 2262 y Fk(\()p Fh(w)1140 2269 y Fg(B)1168 2262 y Fk(\))e Ff(\000)h Fh(B)1261 2244 y Fg(i)1275 2262 y Fk(])615 b(\(8\))0 2365 y(with)15 b(functional)j(v)n(alues)e Fh(b)p Fk(\(0\))e(=)g(0,)h Fh(b)p Fk(\(1\))f(=)g(0)h(and)g(deriv)n(ativ)o(es)j Fh(b)999 2349 y Fm(0)1011 2365 y Fk(\(0\))c(=)f(0,)i Fh(b)1178 2349 y Fm(0)1190 2365 y Fk(\(1\))f(=)1297 2335 y Fe(P)1341 2345 y Fi(3)1341 2379 y Fg(i)p Fi(=1)1403 2365 y Fh(p)1422 2349 y Fg(i)1422 2376 y Fc(B)1451 2365 y Fk([)p Fh(B)1493 2349 y Fg(i)1516 2365 y Ff(\000)c Fh(C)1586 2349 y Fg(i)1600 2365 y Fk(])k(at)h(the)g(endp)q(oin)o(ts)i(of)0 2409 y(in)o(terv)n(al)e Fh(<)10 b Fk(0)p Fh(;)d Fk(1)j Fh(>)p Fk(.)17 b(Here)706 2446 y Fl(A)p Fk(\()p Fh(w)782 2453 y Fg(B)810 2446 y Fk(\))11 b(=)f Fh(w)904 2453 y Fg(B)933 2446 y Fl(B)f Fk(+)f(\(1)h Ff(\000)f Fh(w)1120 2453 y Fg(B)1148 2446 y Fk(\))p Fl(C)38 b Fh(;)657 b Fk(\(9\))p eop %%Page: 3 3 3 2 bop 670 -75 a Fd(In)o(terp)q(olation)12 b(within)i(ra)o(y)h(cells)0 720 y @beginspecial 71 @llx 431 @lly 496 @urx 771 @ury 2246 @rwi @setspecial %%BeginDocument: homltt.ps %----------------------------------------------------------- % Default UNDEFINED procedure: /UNDEFINED { 0.95 setgray} bind def % Default VALUEtoCOLOR procedure: /VCIRC 0.200000 def % Range of grid values: 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FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1110100EFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFF PAPERSIZE 2 div dup translate ROTATE neg rotate PAPERSIZE -2 div dup translate showpage %%EndDocument @endspecial 0 1709 a(Figure)11 b(2c:)k(Di\013erences)c(of)f(the)f (bilinearl)q(y)k(in)o(terp)q(olated)f(tra)o(v)o(el)0 1747 y(times)j(from)e(the)i(exact)f(solution.)22 b(The)14 b(maxim)o(um)i(di\013erence)0 1784 y(is)e(ab)q(out)f(0.308)h(s.)1014 1709 y(Figure)21 b(2d:)30 b(Di\013erences)22 b(of)d(the)h(bicubicall)q (y)i(in)o(terp)q(olated)1014 1747 y(tra)o(v)o(el)13 b(times)f(from)g (the)g(exact)h(solution.)18 b(The)12 b(maxim)o(um)h(dif-)1014 1784 y(ference)g(is)h(ab)q(out)g(0.017)f(s.)0 1854 y(see)g(\(2\).)0 1926 y(Analogously)j(for)d(functions)641 2011 y Fh(b)p Fk(\()p Fh(w)700 2018 y Fg(C)728 2011 y Fk(\))e(=)f(\()p Fh(w)837 2018 y Fg(C)865 2011 y Fk(\))880 1993 y Fi(2)926 1960 y(3)905 1972 y Fe(X)908 2061 y Fg(i)p Fi(=1)971 2011 y Fh(p)990 1993 y Fg(i)990 2019 y Fc(C)1019 2011 y Fk([)p Fh(A)1059 1993 y Fg(i)1073 2011 y Fk(\()p Fh(w)1116 2018 y Fg(C)1143 2011 y Fk(\))f Ff(\000)f Fh(C)1235 1993 y Fg(i)1249 2011 y Fk(])38 b Fh(:)573 b Fk(\(10\))0 2133 y(Cubic)14 b(in)o(terp)q(olations)p 0 2154 1950 2 v 0 2310 2 156 v 248 2244 a Fh(\034)269 2226 y Fc(A)297 2230 y Fb(k)329 2244 y Fk(=)c Fh(a)p Fk(\()p 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Fk(=)10 b Fh(a)p Fk(\()p Fh(w)767 2427 y Fg(B)795 2420 y Fk(\))p Fh(p)829 2402 y Fg(i)829 2428 y Fc(B)856 2432 y Fb(k)885 2420 y Fk(+)e Fh(a)p Fk(\()p Fh(w)986 2427 y Fg(C)1014 2420 y Fk(\))p Fh(p)1048 2402 y Fg(i)1048 2428 y Fc(C)1075 2432 y Fb(k)1134 2420 y Fh(;)44 b(i)11 b Fk(=)g(1)p Fh(;)6 b Fk(2)p Fh(;)g Fk(3)p Fh(;)516 b Fk(\(12\))p 1948 2444 V 0 2446 1950 2 v eop %%Page: 4 4 4 3 bop 670 -75 a Fd(In)o(terp)q(olation)12 b(within)i(ra)o(y)h(cells)0 7 y Fk(along)f(the)f(edges)g(of)g(the)g(ra)o(y)g(cell,)h(parametrized)g (b)o(y)f Fh(w)838 14 y Fg(B)866 7 y Fk(,)g(are)f(used)i(to)f(obtain)h (functional)h(v)n(alues)f Fh(\034)1522 -9 y Fc(A)1550 -5 y Fb(k)1583 7 y Fk(at)f(p)q(oin)o(ts)h Fl(A)1778 14 y Fg(k)1811 7 y Fk(together)0 53 y(with)f(their)h(\014rst)g(deriv)n (ativ)o(es)h Fh(p)469 38 y Fg(i)469 64 y Fc(A)497 68 y Fb(k)517 53 y Fk(.)i(The)c(deriv)n(ativ)o(es)i(will)g(b)q(e)e(used)h (for)f(the)g(in)o(terp)q(olation)j(within)f(triangle)f Fl(A)1674 59 y Fi(1)1693 53 y Fl(A)1726 59 y Fi(2)1745 53 y Fl(A)1778 59 y Fi(3)1797 53 y Fk(.)0 132 y(Cubic)g(in)o(terp)q (olation)i(within)f(triangle)g Fl(A)633 138 y Fi(1)652 132 y Fl(A)685 138 y Fi(2)704 132 y Fl(A)737 138 y Fi(3)p 0 150 1950 2 v 0 307 2 157 v 443 240 a Fh(\034)464 222 y Fc(X)505 240 y Fk(=)568 189 y Fi(3)546 201 y Fe(X)546 291 y Fg(k)q Fi(=1)612 240 y Fh(a)p Fk(\()p Fh(w)675 247 y Fg(k)696 240 y Fk(\))p Fh(\034)732 222 y Fc(A)760 226 y Fb(k)789 240 y Fk(+)849 189 y Fi(3)827 201 y Fe(X)827 291 y Fg(k)q Fi(=1)887 240 y Fk(\()p Fh(w)930 247 y Fg(k)950 240 y Fk(\))965 222 y Fi(2)1012 189 y(3)990 201 y Fe(X)993 290 y Fg(i)p Fi(=1)1057 240 y Fh(p)1076 222 y Fg(i)1076 248 y Fc(A)1104 252 y Fb(k)1124 240 y Fk([)p Fh(X)1170 222 y Fg(i)1192 240 y Ff(\000)8 b Fl(A)1263 222 y Fg(i)1263 249 y(k)1284 240 y Fk(])g(+)h Fh(w)1370 246 y Fi(1)1388 240 y Fh(w)1416 246 y Fi(2)1434 240 y Fh(w)1462 246 y Fi(3)1480 240 y Fh(\030)375 b Fk(\(13\))p 1948 307 V 0 309 1950 2 v 0 351 a(with)12 b(resp)q(ect)g(to)f(\()p Fh(w)304 357 y Fi(1)322 351 y Fh(;)6 b(w)367 357 y Fi(2)385 351 y Fk(\))12 b(is)g(then)f(used)i(to)e(obtain)i(the)e(v)n(alues)i(at) e(p)q(oin)o(t)i Fl(X)p Fk(.)k(The)11 b(three)h(functional)h(v)n(alues)g (and)f(six)g(\014rst)g(partial)0 388 y(deriv)n(ativ)o(es)k(at)f(the)f (v)o(ertices)h(of)f(planar)i(triangle)g Fl(A)789 394 y Fi(1)808 388 y Fl(A)841 394 y Fi(2)860 388 y Fl(A)893 394 y Fi(3)926 388 y Fk(are)e(\014tted)h(b)o(y)f(the)g(cubic)i (function)g(of)d Fh(w)1542 394 y Fi(1)1575 388 y Fk(and)h Fh(w)1678 394 y Fi(2)1697 388 y Fk(.)20 b(Since)15 b(a)g(cubic)0 425 y(function)e(of)e(t)o(w)o(o)g(v)n(ariables)j(has)e(10)f(co)q (e\016cien)o(ts,)i(there)f(is)g(a)f(free)g(co)q(e\016cien)o(t)i Fh(\030)g Fk(in)f(the)g(last)g(equation.)18 b(Co)q(e\016cien)o(t)12 b Fh(\030)h Fk(corresp)q(onds)0 463 y(to)g(the)f(basis)i(function)g Fh(w)384 469 y Fi(1)402 463 y Fh(w)430 469 y Fi(2)449 463 y Fh(w)477 469 y Fi(3)507 463 y Fk(whic)o(h)g(has)f(b)q(oth)g(the)g 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