MMO / mmouchpos

20

^_ X ^Z1, ... , wm} = {u1, ... , um}, v = {wm+1, ... , wp} = {v1, ... , vp}.

Lh^• kbkl_fm h[f_`_gv m \b]ey^• j•\ghkl_c fh`gZ ij_^klZ\blb lZdbf qbghf

ìïk1 (u, v) = 0, í................

ïîcm (u, v) = 0.

Ijb pvhfm kbkl_fZ h[f_`_gv m \b]ey^• j•\ghkl_c } aZfdgmlhx sh^h af•ggh€ X

G_oZc $ Ze]hjblf sh \bj•rm} kbkl_fm h[f_`_gv m \b]ey^• j•\ghkl_c sh^h af•ggh€ X ijb aZ^Zgbo \_ebqbgZo Y X $Y

L_i_j nmgdp•€ \ j•\gyggyo aZe_`Zlv l•evdb \•^ Y

f(w) = f(u,v) = f(A(v),v) = f′(v),

cj(w) = cj(u,v) = cj(A(v),v) = cj′(v); j = 1, ..., m;

ck(w) = ck(u,v) = ck(A(v),v) = ck′(v) ³ 0; k = m+1, ..., p.

Z aZ]ZevgZ aZ^ZqZ g_e•g•cgh]h ijh]jZfm\Zggy i_j_l\hjx}lvky \ aZ^Zqm g_e•g•cgh]h ijh]jZfm\Zggy [_a h[f_`_gv lbim j•\g•klv:

f′(v) → min,

ck′(v) ³ 0, k = m+1, ..., p;

u = A(v), w = u v.

QZkldh\bf \biZ^dhf aZ]Zevgh€ aZ^Zq• g_e•g•cgh€ hilbf•aZp•€ } aZ^ZqZ deZkbqgh€ hilbf•aZp•€ sh ij_^klZ\ey} kh[hx aZ]Zevgm aZ^Zqm g_e•g•cgh]h ijh]jZfm\Zggy m yd•c kbkl_fZ h[f_`_gv \•^kmlgy

21

f(w) → min.

DeZkbn•dZp•y f_lh^•\ g_e•g•cgh]h ijh]jZfm\Zggy

AZ^ZqZ g_e•g•cgh€ hilbf•aZp•€ ij_^klZ\ey} kh[hx aZ^Zqm ihrmdm _dklj_fmfm ^_ydh€ nmgdp•€ ?dklj_fmf ± p_ djZcg} agZq_ggy ± f•g•fmfm Z[h fZdkbfmfm nmgdp•€ Jhaj•agyxlv _dklj_fmfb ehdZevgbc ± _dklj_fmf m ^_yd•c ^h\Levgh fZe•c hdhebp• ^Zgh€ lhqdb • ]eh[Zevgbc ± _dklj_fmf m \k•c jha]eygml•c h[eZkl• agZq_gv af•ggbo

Ydsh ]eh[Zevgbc _dklj_fmf nmgdp•€ rmdZxlv gZ \kvhfm ijhklhj• g_aZe_`gbo af•ggbo lh lZdZ aZ^ZqZ hilbf•aZp•€ gZab\Z}lvky aZ^Zq_x gZ [_amfh\gbc _dklj_fmf Z f_lh^b €€ \bj•r_ggy –

f_lh^Zfb [_amfh\gh€ hilbf•aZp•€ Ydsh ` _dklj_fmf rmdZxlv ijb ^_ydbo h[f_`_ggyo gZ g_aZe_`g• af•gg• lh lZdZ aZ^ZqZ hilbf•aZp•€ gZab\Z}lvky aZ^Zq_x gZ mfh\gbc _dklj_fmf Z f_lh^b €€ \bj•r_ggy

– f_lh^Zfb hilbf•aZp•€ a h[f_`_ggyfb.

M aZe_`ghkl• \•^ d•evdhkl• _dklj_fmf•\ gZ •gl_j\Ze• ihrmdm p•evh\Z nmgdp•y fh`_ [mlb mg•fh^Zevghx h^bg _dklj_fmf Z[h ihe•fh^Zevghx ^_d•evdZ _dklj_fmf•\).

GZ ijZdlbp• [•evr•klv nmgdp•c ± ihe•fh^Zevg• lh[lh f•klylv ^_d•evdZ _dklj_fmf•\ Ihrmd ]eh[Zevgh]h _dklj_fmfm ihe•fh^Zevgh€ nmgdp•€ \djZc kdeZ^gZ aZ^ZqZ ijb \bj•r_gg• ydh€ \bgbdZ} ijh[e_fZ yjh\hkl•: a[•`g•klv f_lh^m hilbf•aZp•€

ih]•jrm}lvky ijb gZ[eb`_gg• ^h lhqgh]h j•r_ggy hilbf•aZp•cgh€ aZ^Zq• q_j_a l_ sh \•g [emdZ} ih ehdZevgbo f•g•fmfZo

I_j_\Z`gZ [•evr•klv f_lh^•\ qbk_evgh]h j•r_ggy aZ^Zq g_e•g•cgh]h ijh]jZfm\Zggy } •l_jZp•cgbfb M pbo f_lh^Zo

aZklhkh\m}lvky Zk lZ Zk+1 ± agZq_ggy gZ[hj•\ d_jh\Zgbo af•ggbo gZ

gZab\Zxlv f_lh^b sh ^ha\heyxlv hilbf•am\Zlb nmgdp•x aZ h^bg ijhoL^ [_a Ll_jZlb\gh]h ]hj_]m\Zggy agZq_gv d_jh\Zgbo afLggbo

22

AZ^Zq• deZkbqgh€ hilbf•aZp•€ ih d•evdhkl• af•ggbo m p•evh\•c nmgdp•€ ih^•eyxlvky gZ h^gh\bf•jg• h^gZ af•ggZ • [Z]Zlh\bf•jg•^_d•evdZ af•ggbo

Mk• f_lh^b j•r_ggy aZ^Zq g_e•g•cgh]h ijh]jZfm\Zggy aZ kihkh[hf \bdhjbklZggy ]jZ^•}glZ p•evh\h€ nmgdp•€ ih^•eyxlvky gZ ]jZ^•}glg• sh \bfZ]Zxlv h[qbke_ggy iho•^gbo • [_a]jZ^•}glg• sh aZf•klv iho•^gbo \bdhjbklh\mxlv agZq_ggy p•evh\h€ nmgdp•€ \ ]jmi• a ^_d•evdho lhqhd

>ey ijh]jZfgh€ j_Ze•aZp•€ ]jZ^•}glgbo f_lh^•\ g_h[o•^gh \f•lb jhajZoh\m\Zlb agZq_ggy iho•^gh€ nmgdp•€ aZ^Zgh€ lZ[ebqgh m ^_yd•c lhqp•

>ey hp•gdb agZq_ggy iho•^gh€ \bdhjbklh\mxlv f_lh^ d•gp_\bo j•agbpv g_oZc K ± djhd ^bn_j_gp•x\Zggy lh^• iho•^gZ Q]h ihjy^dm \bagZqZ}lvky aZ nhjfmehx

^_ n y d•gp_\Z j•agbpy Q]h ihjy^dm

P_gljZevgZ d•gp_\Z j•agbpy i_jrh]h ihjy^dm \bagZqZ}lvky aZ nhjfmehx

1y = yi+1 − yi−1 ,

2

^_ \s –

>ey \bagZq_ggy d•gp_\h€ j•agbp• Q]h ihjy^dm \bdhjbklh\mxlv •l_jZlb\gm nhjfmem

n y = ( n−1y) .

M aZe_`ghkl• \•^ j•\gy fZl_fZlbqgh]h h[‰jmglm\Zggy qbk_evg• f_lh^b g_e•g•cgh€ hilbf•aZp•€ ih^•eyxlv gZ fZl_fZlbqg• sh fZxlv kljh]_ fZl_fZlbqg_ h[‰jmglm\Zggy lZ _\jbklbqg• aZkgh\Zg• i_j_\Z`gh gZ ^hk\•^• lZ jhamf•ggyo jhajh[gbdZ pL f_lh^b g_ fZxlv kljh]h]h fZl_fZlbqgh]h h[‰jmglm\Zggy

23

Qbk_evg• f_lh^b hilbf•aZp•€ oZjZdl_jbamxlvky gZklmigbfb djbl_j•yfb

-r\b^d•klv a[•`ghkl• f_lh^m ± \bagZqZ} gZkd•evdb r\b^dh

af_grm}lvky a[•evrm}lvky agZq_ggy p•evh\h€ nmgdp•€ aZ h^bgbpx qZkm

-d•evd•klv •l_jZp•c f_lh^m ^ey agZoh^`_ggy _dklj_fmfm a aZ^Zghx lhqg•klx

-lhqg•klv f_lh^m sh ^hky]Z}lvky ijb aZ^Zg•c d•evdhkl• •l_jZp•c

-ljm^hf•kld•klv ijh]jZfgh€ j_Ze•aZp•€;

-\bfh]b f_lh^m ^h j_kmjk•\ ?HF ijb ijh]jZfg•c j_Ze•aZp•€

g_h[o•^gbc h[ky] iZf yl• c h[ky] h[qbke_gv

24

JHA>1E H>GH<BF1JG1 F?LH>B HILBF1A:P12

AZ^Zqm h^gh\bf•jgh€ hilbf•aZp•€ fh`gZ knhjfmex\Zlb lZdbf qbghf agZclb f•g•fmf nmgdp•€ I Z gZ \•^j•adm Z Î >D E@ ^_ Z ±

af•ggZ \•^ ydh€ aZe_`blv nmgdp•y I D lZ E ± ^_yd• aZ^Zg• dhgklZglb lZd• sh D E

Nmgdp•y I Z gZab\Z}lvky mg•fh^Zevghx gZ \•^j•adm >D E@ ydsh

\hgZ fhghlhggZ ih h[b^\• klhjhgb \•^ }^bgh€ gZ jha]eygml•c ^•eygp• hilbfZevgh€ lhqdb ZÎ [a,b].

Nmgdp•y I Z gZab\Z}lvky fhghlhgghx ydsh ^ey ^\ho ^h\•evgbo lhqhd Z1 lZ Z2 lZdbo sh Z1 £ w2 \bdhgm}lvky h^gZ a g_j•\ghkl_c

f(w1) £ f(w2 Z[h I Z1) ³ f(w2).

Nmgdp•y I Z gZab\Z}lvky [_ai_j_j\ghx \ ^_yd•c h[eZkl• ydsh \hgZ [_ai_j_j\gZ \ dh`g•c lhqp• p•}€ h[eZkl• lh[lh ydsh \ dh`g•c lhqp• p•}€ h[eZkl• nmgdp•y fZ} i_\g_ agZq_ggy ) • f_`Z j•\gZ ) m ijhlbe_`ghfm \biZ^dm nmgdp•y gZab\Z}lvky jhajb\ghx.

F_lh^b ijyfh]h ihrmdm

AZklhkm\Zggy f_lh^•\ ijyfh]h ihrmdm f_lh^•\ \bdexq_ggy •gl_j\Ze•\ gZdeZ^Z} }^bgm \bfh]m gZ ^hke•^`m\Zgm nmgdp•x \hgZ ih\bggZ [mlb mg•fh^Zevghx Hl`_ aZagZq_g• f_lh^b fh`gZ \bdhjbklh\m\Zlb ^ey ZgZe•am yd [_ai_j_j\gbo lZd • jhajb\gbo nmgdp•c Z lZdh` m \biZ^dZo dheb af•gg• ijbcfZxlv agZq_ggy a ^bkdj_lgh€ fgh`bgb

Eh]•qgZ kljmdlmjZ ihrmdm aZ ^hihfh]hx f_lh^•\ \bdexq_ggy •gl_j\Ze•\ aZkgh\ZgZ gZ ijhklhfm ihj•\gygg• agZq_gv nmgdp•€ \ ^\ho ijh[gbo lhqdZo Dj•f lh]h ijb lZdhfm ihj•\gygg• \ jhajZomghd ijbcfZ}lvky l•evdb \•^ghr_ggy ihjy^dm gZ fgh`bg• agZq_gv nmgdp•€

• g_ \jZoh\m}lvky \_ebqbgZ j•agbp• f•` agZq_ggyfb nmgdp•€

F_lh^ jhaih^•em •gl_j\Zem gZ\i•e

>Zgbc f_lh^ ^ha\hey} \bdexqblb \ lhqghkl• iheh\bgm •gl_j\Zem gZ dh`g•c •l_jZp•€ P_c f_lh^ lZdh` gZab\Zxlv ljvholhqdh\bf ihrmdhf gZ j•\gbo •gl_j\ZeZo hkd•evdb ch]h j_Ze•aZp•y aZkgh\ZgZ

25

gZ \b[hj• ljvho ijh[gbo lhqhd j•\ghf•jgh jhalZrh\Zgbo \ •gl_j\Ze• ihrmdm

Z IhdeZklb Zm D E / E D H[qbkeblb I Zm).

[ IhdeZklb Z1=a+L/4; w2 E / lhqdb Z1, w2, wm ± ih^•eyxlv •gl_j\Ze D E gZ j•\g• qZklbgb H[qbkeblb I Z1 lZ I Z2).

\ Ihj•\gylb I Z1 lZ I Zm).

Ydsh I Z1) < f(wm ± \bdexqblb •gl_j\Ze Zm E ihdeZklb E wm K_j_^gvhx lhqdhx gh\h]h •gl_j\Zem klZ} lhqdZ Z1 hl`_ g_h[o•^gh ihdeZklb Zm=w1 I_j_clb ^h djhdm ^

Ydsh I Z1) > f(wm i_j_clb ^h djhdm ] ] Ihj•\gylb I Z2 lZ I Zm) .

Ydsh I Z2) < f(wm \bdexqblb •gl_j\Ze D Zm), a=wm lhfm sh k_j_^gvhx lhqdhx gh\h]h •gl_j\Zem klZ} o2 ihdeZklb Zm=w2.

I_j_clb ^h djhdm ^ Ydsh I Z2) ³

a=w1, b=w2, wm ± g_ af•gx}lvky lh[lh ijh^h\`m} aZebrZlbky k_j_^gvhx lhqdhx gh\h]h •gl_j\Zem I_j_clb ^h djhdm ^

^ H[qbkeblb / E D ydsh \_ebqbgZ _/_ } fZehx aZd•gqblb ihrmd M ijhlbe_`ghfm \biZ^dm ih\_jgmlbky ^h djhdm Z

F_lh^ N•[hgZqq•

1kgm} •gl_j\Ze g_\bagZq_ghkl• Z1,w3 • \•^hf_ agZq_ggy nmgdp•€ f(w2 mk_j_^bg• pvh]h •gl_j\Zem g_h[o•^gh agZclb lZdm lhqdm Z4, sh[ h^_j`Zlb gZcf_grbc •gl_j\Ze g_\bagZq_ghkl•

IhdeZ^_fh Z2 - w1 / lZ Z3 – w 2 5 ijbqhfm / ! 5 • p• agZq_ggy [m^mlv n•dkh\Zg• ydsh \•^hf• Z1, w2, w3 Ydsh Z4 agZoh^blvky \ •gl_j\Ze• Z1,w2 lh

Z ydsh I Z4) < f(w2 lh gh\bf •gl_j\Zehf g_\bagZq_ghkl• [m^_

(w1,w2 ^h\`bghx Z2 - w1 = L ;

[ ydsh I Z4) > f(w2 lh gh\bf •gl_j\Zehf g_\bagZq_ghkl• [m^_ (w4,w3 ^h\`bghx Z3 – w 4 .

Hkd•evdb g_ \•^hfh ydZ a pbo kblmZp•c [m^_ fZlb f•kp_ \b[_j_fh w4 lZdbf qbghf sh[ f•g•f•am\Zlb gZc[•evrm a ^h\`bg Z3 – w 4 lZ w2 –w 1 P_ fh`eb\h ydsh ^h\`bgb Z3 – w 4 lZ Z2 –w 1 j•\g• lh[lh Z4

agZoh^blvky mk_j_^bg• •gl_j\Zem kbf_ljbqgh sh^h lhqdb Z2 sh \`_

26

e_`blv mk_j_^bg• •gl_j\Zem ;m^v yd_ •gr_ iheh`_ggy lhqdb Z4 fh`_ ijb\_klb ^h lh]h sh hljbfZgbc •gl_j\Ze [m^_ [•evr_ /

GZ Qfm h[qbke_gg• Qlm lhqdm \Zjlh ihf•klblb kbf_ljbqgh \•^ghkgh Qh€ lhqdb Iheh`_ggy p•}€ hklZggvh€ lhqdb \ ijbgpbi• aZe_`blv \•^ gZk >ey lh]h sh[ h^_j`Zlb gZc[•evr_ af_gr_ggy •gl_j\Zem gZ ^Zghfm _lZi• \Zjlh jha^•eblb gZ\i•e ihi_j_^g•c

•gl_j\Ze Lh^• lhqdZ Zn [m^_ a[•]Zlbky a lhqdhx Zn-1 A\bqZcgh lhqdb wn lZ Zn-1 \•^klhylv h^gZ \•^ h^gh€ gZ ^hklZlg•c \•^klZg• sh[ \bagZqblb \ yd•c iheh\bg• e•\•c Z[h ijZ\•c agZoh^blvky •gl_j\Ze g_\bagZq_ghkl• <hgb f•klylvky gZ \•^klZg• ih h[b^\• klhjhgb \•^

k_j_^bgb \•^j•adZ /n-1 ± \b[bjZ}lvky kZfhkl•cgh Z[h \b[bjZ}lvky j•\g•c f•g•fZevgh fh`eb\•c \•^klZg• f•` ^\hfZ lhqdZfb

1gl_j\Ze g_\bagZq_ghkl• [m^_ fZlb ^h\`bgm /n hl`_

Ln-1 = 2Ln ±

GZ ihi_j_^gvhfm _lZi• lhqdb Zn-1 lZ Zn-2 ih\bgg• [mlb

jhaf•s_gg• kbf_ljbqgh mk_j_^bg• •gl_j\Zem /n-2 gZ \•^klZg• /n-1 \•^ d•gp•\ pvh]h •gl_j\Zem Hl`_ /n-2 = Ln-1 + Ln

GZ i_j_^hklZggvhfm _lZi• Zn-2 aZebrZ}lvky yd \gmlj•rgy lhqdZ :gZeh]•qgh /n-3 = Ln-2 + Ln-1

M aZ]Zevghfm \biZ^dm /j-1 = Lj + Lj+1 ijb M Q LZdbf qbghf

Ln-1 = 2Ln ±

Ln-2 = Ln-1 + Ln = 3Ln ±

Ln-3 = Ln-2 + Ln-1 = 5Ln ±

Ln-4 = Ln-3 + Ln-2 = 8Ln ± • l ^

Ydsh \bagZqblb ihke•^h\g•klv qbk_e N•[hgZqq• gZklmigbf qbghf

F0 = 1, F1 = 1, Fk = Fk-1 + Fk-2 ^ey N «

lh^L

Lj-1 = Fj+1Ln - Fj-1 M « Q

27

Ydsh ihqZldh\bc •gl_j\Ze D E fZ} ^h\`bgm / E D lh^L

ihqZldh\hx ^h\`bghx g_olmxqb

Ydsh ihrmd ihqZlh lh ch]h g_kdeZ^gh ijh^h\`blb \bdhjbklh\mxqb hibkZg_ \bs_ ijZ\beh kbf_lj•€ Hl`_ g_h[o•^gh

agZclb iheh`_ggy i_jrh€ lhqdb sh f•klblvky gZ \•^klZg• /n \•^ h^gh]h a d•gp•\ ihqZldh\h]h •gl_j\Zem ijbqhfm g_\Z`eb\h \•^ ydh]h d•gpy hkd•evdb •grZ lhqdZ f•klblvky \•^ih\•^gh ^h ijZ\beZ

kbf_lj•€ gZ \•^klZg• /2 \•^ •grh]h d•gpy •gl_j\Zem

I•key lh]h yd agZc^_gh iheh`_ggy i_jrh€ lhqdb qbkeZ N•[hgZqq• [•evr_ g_ ihlj•[g• AgZq_ggy sh \bdhjbklh\m}lvky fh`_ \bagZqZlbky a ijZdlbqgbo jhamf•gv <hgh ih\bggh [mlb f_gr_ L1/Fn+1 m ijhlbe_`ghfm \biZ^dm [m^_ ^ZjfZ \bljZqZlbky qZk gZ h[qbke_ggy nmgdp•€

M ijhp_k• ihrmdm •gl_j\Zem Z1,w2 •a lhqdhx Z2 sh \`_ e_`blv

m pvhfm •gl_j\Ze• gZklmigZ lhqdZ Z4 aZ\`^b \b[bjZ}lvky lZdhx sh w3 – w 4 = w2 – w 1 Z[h Z4 – w 1 = w3 – w 2 lh[lh Z4 = w1 + w3 – w 2.

G_oZc I Z2) = f2 • I Z4) = f4 jha]eyg_fh qhlbjb \biZ^db ydsh Z Z4 < w2 , f4 < f2 ± gh\bc •gl_j\Ze Z1,w2 f•klblv lhqdm Z4; [ Z4 > w2 , f4 < f2 ± gh\bc •gl_j\Ze Z2,w3 f•klblv lhqdm Z4; \ Z4 < w2 , f4 > f2 ± gh\bc •gl_j\Ze Z4,w3 f•klblv lhqdm Z2; ] Z4 > w2 , f4 < f2 ± gh\bc •gl_j\Ze Z1,w4 f•klblv lhqdm Z2.

Ihrmd f_lh^hf ³ahehlh]h i_j_lbgm´

KljZl_]•y ^Zgh]h f_lh^m lZdZ ijh[g• lhqdb \•^klhylv \•^ ]jZgbqgbo lhqhd •gl_j\Zem gZ \•^klZg• Ijb lZdhfm kbf_ljbqghfm

28

jhalZrm\Zgg• lhqhd ^h\`bgZ •gl_j\Zem sh aZebrZ}lvky i•key \bdexq_ggy aZ\`^b ^hj•\gx} g_aZe_`gh \•^ lh]h yd_ a• agZq_gv nmgdp•€ \ ijh[gbo lhqdZo \by\ey}lvky f_grbf

Ijbimklbfh sh \bdexqZ}lvky ijZ\bc i•^•gl_j\Ze I•^•gl_j\Ze sh aZebrb\ky ^h\`bghx f•klblv h^gm ijh[gm lhqdm jhalZrh\Zgm gZ \•^ e•\h€ ]jZgbqgh€ lhqdb Sh[ kbf_lj•y ihrmdh\h]h ajZadZ

\•^klZg• j•\g•c ± •c qZklbg• ^h\`bgb •gl_j\Zem \•^ ijZ\h€ ]jZgbqgh€ lhqdb •gl_j\Zem

af_gr_gh]h •gl_j\Zem <bj•rmxqb d\Z^jZlg_ j•\gyggy h^_j`m}fh

τ = (−1 ± 5) / 2 ≈ 0,61803 .

F_lh^ j•\ghf•jgh]h ihrmdm

KihqZldm aZ^Z}lvky \•^j•ahd >D E@ lZ \b[bjZ}lvky qbkeh j•\gbo i•^•gl_j\Ze•\ Q gZ yd• jha[b\Z}lvky \•^j•ahd >D E@

Ihl•f agZoh^ylv ^h\`bgm i•^•gl_j\Zem K E D Q

H[qbkexxlv agZq_ggy nmgdp•€ I Z gZ fgh`bg• lhqhd Zi = a + kh, N Q • agZoh^ylv €€ f•g•fmf reyohf ijhklh]h i_j_[hjm agZq_gv nmgdp•€ I Zi m lhqdZo Zi G_oZc lhqdZ f•g•fmfm [m^_ Zm.

Ydsh mfh\Z mg•fh^Zevghkl• g_ \bdhgZgZ lh[lh ihlj•[gh agZclb

]eh[Zevgbc f•g•fmf lh^• Zm ijbcfZ}lvky yd gZ[eb`_g_ j•r_ggy aZ^Zq• Lhqg•klv j_amevlZlm sh h^_j`ZgZ aZe_`blv \•^ \b[hjm iZjZf_ljZ Q

Ydsh nmgdp•y I Z } mg•fh^Zevghx lh^• lhqg_ j•r_ggy

w [wm-1, wm+1].

29

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