Abstract: In the paper, a new constructive approach to solving geometric ... Newton-Raphson iteration are used to solve these ... new form that is easy to draw. ... circles, circular arcs, and line segments form the ... the constraints: Point(p, x, y), AngleValue(l, a), .... row and the j .... usually does not present an interior angle (for.
$*(20(75,&&21675$,1762/9(5:,7+'(&20326$%/( &21675$,176(7 'DYLG3RGJRUHOHF%RUXWäDOLN
)DFXOW\RI(OHFWULFDO(QJLQHHULQJDQG&RPSXWHU6FLHQFH 8QLYHUVLW\RI0DULERU6PHWDQRYD 6,0DULERU 6ORYHQLD HPDLOV GDYLGSRGJRUHOHF#XQLPEVL]DOLN#XQLPEVL
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
.H\ZRUGV &$'FRQVWUDLQWEDVHGGHVLJQJHRPHWULFFRQVWUDLQWVJHRPHWULFPRGHOOLQJ
,1752'8&7,21
*U|EQHU EDVLV PHWKRG >$MZD @ ILUVW $IWHU WKLV WKH WUDQVIRUPHG V\VWHP LV VROYHG QXPHULFDOO\ ,Q WKH D
SUHSURFHVVLQJ
LV
$JHRPHWULFFRQVWUDLQWLVDUHODWLRQDPRQJJHRPHWULF
FRQVWUXFWLYH
REMHFWV WKDW VKRXOG EH VDWLVILHG >)UHHP @ 7KH
SHUIRUPHGWRWUDQVIRUPWKHFRQVWUDLQWSUREOHPLQWRD
DSSURDFK
JHRPHWU\ FDQ EH GHVFULEHG E\ GHILQLQJ UHODWLRQV
QHZIRUPWKDWLVHDV\WRGUDZ,IDJUDSKLVHPSOR\HG
FRQVWUDLQWV DPRQJ SDUWLFXODU JHRPHWULF HOHPHQWV
IRUWKLVWDVNVRFDOOHGF\FOHVKDYHWREHEURNHQ7KH
$XWRPDWLF
EDVLF
VROYLQJ
RI
JHRPHWULF
FRQVWUDLQWV
UHSUHVHQWV DQ LQWHUIDFH EHWZHHQ D GHFODUDWLYH DQG D
LGHD
LV
WR
VSOLW
WKH
FRQILJXUDWLRQ
LQWR
FRPSRQHQWVVXFKWKDWHDFKFRPSRQHQW KDV QR F\FOHV
SURFHGXUDO GHVFULSWLRQ RI WKH JHRPHWU\ 7KH XVHU
DQG DOO WKH FRPSRQHQWV FDQ EH PHUJHG WRJHWKHU LQ
RQO\VSHFLILHVREMHFW¶VVKDSHDQGVL]HLQDGHFODUDWLYH
VRPH ZD\ >*DR D@ (IILFLHQW PHWKRG RI EUHDNLQJ
ZD\ DQG WKH V\VWHP WDNHV FDUH RI PDNLQJ WKH
WKH F\FOHV ZDV SUHVHQWHG E\ )XGRV DQG +RIIPDQ
GUDZLQJ LQ DFFRUGDQFH WR WKH VSHFLILFDWLRQ 7KH XVHU
>)XGRV @ IRU H[DPSOH %HVLGHV JUDSKV WHFKQLTXHV
VSHFLILHVZKDWWRGUDZQRWKRZWRGUDZLW>6XQGH@
RI DUWLILFLDO LQWHOOLJHQFH DV VHDUFKLQJ DQG UXOHEDVHG V\VWHPV FDQ EH HPSOR\HG >*DR D@ ,Q WKLV VKRUW
JHRPHWULF
SDSHU ZH FDQQRW PHQWLRQ DOO WKH DOJRULWKPV WKDW ZH
GHVLJQ ZHUH SUHVHQWHG LQ OLWHUDWXUH 7KH\ FDQ EH
9DULRXV
PHWKRGV
KDYH VWXGLHG EXW ZLWK DOO RI WKHP VRPH RI WKH
FODVVLILHG LQWR WKUHH PDLQ JURXSV WKH QXPHULFDO
IROORZLQJGUDZEDFNVZHUHQRWLFHG
DSSURDFK
WKH
RI
FRQVWUDLQWEDVHG
V\PEROLF
DSSURDFK
DQG
WKH
FRQVWUXFWLYH DSSURDFK >*DR D@ ,Q WKH QXPHULFDO DSSURDFK FRQVWUDLQWV DUH WUDQVODWHG LQWR DOJHEUDLF HTXDWLRQV
DQG
YDULRXV
QXPHULFDO
WHFKQLTXHV
DV
1HZWRQ5DSKVRQ LWHUDWLRQ DUH XVHG WR VROYH WKHVH HTXDWLRQV 7KH V\PEROLF DSSURDFK LV VLPLODU WR WKH QXPHULFDO DSSURDFK EXW WKH DOJHEUDLF HTXDWLRQV DUH WUDQVIRUPHG LQ WKH V\PEROLF IRUP E\ HPSOR\LQJ JHQHUDO
V\PEROLF
PHWKRGV
DV
:X5LWW¶V
FKDUDFWHULVWLF PHWKRG >.DSXU *DR E@ RU WKH
- 7KH VHW RI JHRPHWULF HOHPHQWV LV QRW ULFK HQRXJK IRU
YDULRXV
JHRPHWULF
WDVNV
7KH
DOJRULWKPV
XVXDOO\ HPSOR\ DQG FRQVWUDLQ SRLQWV DQG LQ WKH EHVWFDVHOLQHVRUOLQHVHJPHQWVDQGFLUFOHV - 7KH FRQVWUDLQW VHW VKRXOG GHVFULEH JHRPHWU\ LQ D QDWXUDOZD\DQGHQDEOHGHVLJQHUVWRXVHWKHLURZQ GHVLJQVW\OHV - ,Q VRPH FDVHV D V\VWHP GRHV QRW ILQG D VROXWLRQ DOWKRXJKLWH[LVWV
- 7KH PDMRULW\ RI V\VWHPV DUH QRW DEOH WR KDQGOH HYHQWXDOPXOWLSOHVROXWLRQV - 7KH VROXWLRQ VKRXOG QRW GHSHQG RQ RUGHU RI DGGLQJ DQG VROYLQJ FRQVWUDLQWV 6XFK D FRQVWUDLQW VROYHULVFDOOHGYDULDWLRQDO>%RXPD @
2XU QHZ DOJRULWKP SUHVHQWHG LQ WKLV SDSHU WULHV WR ILW DOO WKHVH FULWHULD %HVLGHV WKLV ZH LQWHQG WR SURYLGH FRQVLVWHQF\ RI DOO OHYHOV RI JHRPHWULF GDWD SUHVHQWDWLRQ7KHDOJRULWKPLVDUHSUHVHQWDWLYHRIWKH FRQVWUXFWLYH
DSSURDFK
:H
VKDOO
VHH
LQ
WKH
FRQWLQXDWLRQ WKDW WKH PRVW RI WKH ZRUN LV GRQH LQ DQ H[WHQVLYH SUHSURFHVVLQJ SKDVH 7KH SUHSURFHVVLQJ LVWKHUHIRUHWKHPDLQVXEMHFWRIWKLVSDSHU7KH³UHDO´ FRQVWUDLQWVROYHUZKLFKSRVLWLRQVJHRPHWULFHOHPHQWV WRUHTXLUHGPXWXDODQGRUDEVROXWHSRVLWLRQVDFWXDOO\ RSHUDWHV LQ D IHZ VLPSOH VWHSV DW WKH HQG RI WKH SURFHVV ,Q WKH ODVW SDUW RI WKH SDSHU ERWK WKH SUH SURFHVVLQJ DQG WKH FRQVWUDLQW VROYLQJ FDQ EH WUDFHG WKURXJKDSUDFWLFDOH[DPSOH
ZKHUH D VLQJOH FOXVWHU LV REWDLQHG 7KHUHIRUH WKH VWHSV RI FUHDWLQJ ,, DQG PHUJLQJ WKH FOXVWHUV ,, DUHQRWGHVFULEHGLQWKHSDSHU
9LVLEOHDQGDX[LOLDU\JHRPHWU\
:LWK %)R)' DOUHDG\ D WZROHYHO RUJDQLVDWLRQ RI JHRPHWULF REMHFWV ZDV LQWURGXFHG DQG LW LV DGRSWHG LQ RXU QHZ V\VWHP DV ZHOO %p]LHU FXUYHV HOOLSVHV FLUFOHV FLUFXODU DUFV DQG OLQH VHJPHQWV IRUP WKH YLVLEOH JHRPHWU\ DQG WKH\ DUH PDSSHG RQWR FRQWURO SRLQWV DQG OLQHV IRUPLQJ WKH DX[LOLDU\ JHRPHWU\ (DFKHQWLW\RIWKHYLVLEOHJHRPHWU\KDVLWVHTXLYDOHQW RU PRUH RI WKHP LQ WKH DX[LOLDU\ SDUW DQG WKH RSSRVLWHLVQRWQHFHVVDU\7KHWZROHYHORUJDQLVDWLRQ LVLPSRUWDQWEHFDXVHRIWKHIROORZLQJIDFWV
- ,W VXIILFHV WR HPSOR\ FRQVWUDLQWV WKDW RSHUDWH RQ HOHPHQWV RI DX[LOLDU\ JHRPHWU\ RQO\ :KHQ WKH DX[LOLDU\JHRPHWU\ LV FRQVWUDLQHG WKH YLVLEOH SDUW LVFRQVWUDLQHGWRR - /HVV FRPSOH[ FRQVWUDLQWV FDQ EH HPSOR\HG DQG
7+(0(7+2'
7KH
DOJRULWKP
RSHUDWHV
LPSOHPHQWDWLRQLVDOVRVLPSOLILHGLQWKLVZD\
LQ
'
VSDFH
,W
ZDV
GHYHORSHG DV D VXFFHVVRU RI WZR FRQVWUDLQWEDVHG
- 7KH PDMRULW\ RI WKH DX[LOLDU\ JHRPHWU\ DQG DOO VHOIXQGHUVWDQGDEOH
FRQVWUDLQWV
DUH
FUHDWHG
DXWRPDWLFDOO\E\WKHV\VWHP>äDOLN @
GUDZLQJ V\VWHPV LPSOHPHQWHG E\ RXU JURXS LQ WKH
- 1HZ W\SHV RI HOHPHQWV RI WKH YLVLEOH JHRPHWU\
ERWK ZHUH HPSOR\LQJ D ORFDO SURSDJDWLRQ DV WKH
WUDQVIRUPDWLRQ LQWR WKH DX[LOLDU\ JHRPHWU\ KDV WR
SDVW ± )/(;, DQG %)R)' VHH >äDOLN @ 7KH\ EDVLF WKH ROGHVW DQG WKH PRVW QDWXUDO JUDSKEDVHG
PHWKRG 7KH ORFDO SURSDJDWLRQ PRVWO\ IDLOV ZKHQ
FDQ EH GHILQHG LQ DQ HDV\ ZD\ 2QO\ WKH FRUUHFW
EHSURYLGHG
F\FOHVDUHSUHVHQWLQWKHFRQILJXUDWLRQDQGWKHUHIRUH UHTXLUHV WKH SUHSURFHVVLQJ 1DPHO\ LQ D F\FOH D JURXS RI YDULDEOHV FDQQRW EH GHWHUPLQHG EHFDXVH HDFK RI WKHP UHTXLUHV WKDW VRPH RWKHU YDULDEOH IURP WKH JURXS LV GHWHUPLQHG ILUVW >%RXPD @ 7KHUHIRUH WKH PDLQ JRDO RI RXU QHZ DOJRULWKP LV WR ³FOHDQ´ D JUDSK
IURP
F\FOHV
EHIRUH
HPSOR\LQJ
WKH
ORFDO
SURSDJDWLRQ 7R VDWLVI\ DOO GLVWLQFW JRDOV PHQWLRQHG LQ WKH LQWURGXFWLRQ WKH DOJRULWKP ZDV GHVLJQHG LQ VHYHUDOSKDVHV
,
35(352&(66,1* ,
PDSSLQJ WKH YLVLEOH JHRPHWU\ RQWR WKH DX[LOLDU\JHRPHWU\
,
GHFRPSRVLWLRQRIFRPSOH[FRQVWUDLQWV
,
DGGLQJ UHGXQGDQW FRQVWUDLQWV RI GLVWDQFHV
$%p]LHUFXUYHDQGLWVDX[LOLDU\JHRPHWU\ )LJXUH
DQGDQJOHV ,, &21675$,1762/9,1* ,, ,,
FUHDWLRQRIFOXVWHUV VROYLQJFRQVWUDLQWVLQSDUWLFXODUFOXVWHUVE\ ORFDOSURSDJDWLRQ
,,
PHUJLQJWKHFOXVWHUV
,,
DEVROXWHSRVLWLRQLQJ
,,,35(3$5,1*7+(5(68/76 ,,, HVWDEOLVKLQJ WKH YLVLEOH JHRPHWU\ IURP WKH LQYLVLEOHJHRPHWU\
,Q WKLV SDSHU ZH GHDO RQO\ ZLWK FRQVWUDLQW SUREOHPV
,Q)LJXUHD%p]LHUFXUYHDQGLWVFRQWUROSRO\OLQHLV SUHVHQWHG 7KH DX[LOLDU\ JHRPHWU\ FRQVLVWV RI IRXU SRLQWVS ±S FRQVWUDLQWV
DQGWKUHHOLQHVO ±O %HVLGHVWKLVVL[
HVWDEOLVKLQJ
WKH
WRSRORJ\
DUH
DGGHG
DXWRPDWLFDOO\ 2Q S O 2Q S O 2Q S O
2Q S O 2Q S O DQG2Q S O
7KHFRQVWUDLQWVHW
$IWHULQWURGXFLQJWKHDX[LOLDU\JHRPHWU\RQO\SRLQWV DQG OLQHV KDYH WR EH FRQVWUDLQHG ,Q FRQVHTXHQFH RI
WKLV WKH QXPEHU RI QHFHVVDU\ FRQVWUDLQW W\SHV KDV
,W LV XVXDOO\ GLIILFXOW IRU KXPDQ GHVLJQHUV WR VSHFLI\
DOVR
H[DFWO\ JHRPHWULF FRQVWUDLQWV QHHGHG WR GHILQH DQ
EHHQ
FRQVLGHUDEO\
UHGXFHG
:H
KDYH
LPSOHPHQWHG FRQVWUDLQW W\SHV DQG WKH\ FDQ EH
REMHFW XQDPELJXRXVO\ $ ZHOOFRQVWUDLQHG SUREOHP
FODVVLILHGLQWRIRXUJURXSV
KDV D ILQLWH QXPEHU RI VROXWLRQV ,I DQ LQILQLWLYH
7RSRORJLFDO FRQVWUDLQWV HVWDEOLVK WRSRORJ\ RI WKH
FRQVWUDLQHGDQGLIWKHUHDUHQRWVROXWLRQVLWLVRYHU
QXPEHU RI VROXWLRQV H[LVW D SUREOHP LV XQGHU
VFHQH 7KH FRQVWUDLQW 2Q S O WKDW UHTXLUHV WKDW
FRQVWUDLQHG 7R GHILQH WKH WRSRORJ\ VKDSH DQG
WKH SRLQW S OLHV RQ WKH OLQH O LV WKH RQO\
GLPHQVLRQV
UHSUHVHQWDWLYHRIWKLVJURXS
FKDUDFWHULVWLF SRLQWV Q ± LQGHSHQGHQW FRQVWUDLQWV
'LPHQVLRQDO FRQVWUDLQWV GHWHUPLQH GLVWDQFHV DQG
RI
D
SODQDU
VFHQH
FRQVLVWLQJ
RI
Q
DUHQHFHVVDU\>/DWKD @
DQJOHV EHWZHHQ SDLUV RI JHRPHWULF HOHPHQWV 7KLV
JURXS LQFOXGH FRQVWUDLQWV 'LVWDQFH S S G
$QJOH O O a 3DUDOOHO O O 3HUSHQGLFXODU O
&21675$,17'(&20326,7,21
O 5HO3RV S S [ \ &RLQFLGHQFH S S
&RLQFLGHQFH O O DQG'LVWDQFH O O G 3RVLWLRQLQJ SRVLWLRQV
FRQVWUDLQWV
RI
JHRPHWULF
GHWHUPLQH HOHPHQWV
,Q WKH VHFRQG VWHS RI SUHSURFHVVLQJ WKH FRQVWUDLQW
DEVROXWH LH
SRLQW
VHW &6 LV WUDQVIRUPHG LQWR D IRUP FRQVLVWLQJ RQO\ RI DQ\ QXPEHU RI WRSRORJLFDO SUHGLFDWHV 2Q S O GLPHQVLRQDO
WKH FRQVWUDLQWV 3RLQW S [ \ $QJOH9DOXH O a
$QJOH O O a QR PRUH WKDQ RQH SRVLWLRQLQJ
SUHGLFDWHV
'LVWDQFH S
FRRUGLQDWHV DQG OLQH VORSHV 7KLV JURXS FRQWDLQV
S
DQG
G
SUHGLFDWH3RLQW S[\ DQG DOVR D VLQJOH RU QRQH
+/LQH O DQG9/LQH O 6WUXFWXUDO FRQVWUDLQWV GHILQH UHODWLRQV EHWZHHQ GLPHQVLRQV ,Q WKLV JURXS ZH FDQ ILQG WKH
SRVLWLRQLQJ
SUHGLFDWH
$QJOH9DOXH O
a
7KHVH
SUHGLFDWHV DUH GLUHFWO\ FRSLHG IURP WKH &6 LQWR WKH
FRQVWUDLQWV 6\PPHWULF S S S 6\PPHWULF O
WUDQVIRUPHG FRQVWUDLQW VHW $OO RWKHU SUHGLFDWHV DUH
O O DQG GLIIHUHQW FRQVWUDLQWV ZKLFK HVWDEOLVK
GHILQHGDVDFRQMXQFWLRQRIWKHSUHGLFDWHVPHQWLRQHG
DERYH7KHWUDQVIRUPHGFRQVWUDLQWVHWZLOO EH QDPHG
VXPVDQGUDWLRVRIGLVWDQFHVRUDQJOHV
WKHPLQLPDOFRQVWUDLQWVHW&6 (DFK
FRQVWUDLQW
LV
GHVFULEHG
E\
LWV
SUHGLFDWH
$QJOH9DOXH IRU H[DPSOH WKH OLVW RI FRPSULVHG
&RQVWUDLQWV IURP WKH
VHW
&6
LQWKHFRQWLQXDWLRQ
ZLOO
EH
FDOOHG
H[SOLFLWHO\ GHILQHG FRQVWUDLQWV MXVW WR GLVWLQJXLVK
SRLQWV WKH OLVW RI FRPSULVHG OLQHV DQG WKH OLVW RI
EHWZHHQ WKHP DQG WKH UHGXQGDQW FRQVWUDLQWV LQVHUWHG
QXPHULFDO SDUDPHWHUV 7KH OHQJWK RI D SDUWLFXODU OLVW
LQWKHQH[WVWHS DQGFRQVWUDLQWVIURPWKHVHW&6DUH
GHSHQGVRQWKHFRQVWUDLQWSUHGLFDWH6RPHRIWKHOLVWV
RULJLQDORUXVHUGHILQHGFRQVWUDLQWV
FDQEHHPSW\EXWQRWDOOWKUHHRIWKHP 6RPH RULJLQDO FRQVWUDLQWV FDQQRW EH SUHVHQWHG E\
HPSOR\LQJ WKH SUHGLFDWHV IURP WKH VHW &6
$FRQVWUDLQWSUREOHP
RQ
H[LVWLQJ HOHPHQWV RI WKH DX[LOLDU\ JHRPHWU\ RQO\ :KLOH RXU FRQVWUDLQW VROYHU RSHUDWHV ZLWK DX[LOLDU\
7KH\ UHTXLUH DGGLWLRQDO JHRPHWULF HOHPHQWV ZKLFK
SRLQWV DQG OLQHV RQO\ D FRQVWUDLQW SUREOHP LV
SOD\ WKH VDPH UROH DV WKH HOHPHQWV RI WKH DX[LOLDU\
GHILQHGDV&3
3/&6 ZKHUH
JHRPHWU\ EXW D XVHU QHHG QRW EH DZDUH RI WKHLU
H[LVWHQFH7KH\SUHVHQWVRFDOOHGLQYLVLEOHJHRPHWU\
- 3 LV WKH VHW RI SRLQWV S «S $ SRLQW S
7KH\ FDQQRW EH PRGLILHG VHSDUDWHO\ EXW RQO\ E\
«P LV SUHVHQWHG E\ LWV DEVROXWH FRRUGLQDWHV
FRQVWUDLQWV WKDW KDYH FUHDWHG WKHP ,Q WKLV ZD\ WKH
L
[ \
FRQVLVWHQF\RIWKHFRQVWUDLQWSUREOHPLVSURYLGHG
- /LVWKHVHWRIDX[LOLDU\OLQHVO «O )RUHDFKOLQH
O L
«Q LWV VORSH a > p DQG WKH DEVROXWH
FRRUGLQDWHV O[ O\ RI D SRLQW O\LQJ RQ WKH OLQH DUHJLYHQ7KHVORSHFRUUHVSRQGVWRDKRUL]RQWDO OLQH DQG DQJOHV DUH PHDVXUHG LQ PDWKHPDWLFDO SRVLWLYH FRXQWHUFORFNZLVH GLUHFWLRQ
- &6 LV WKH VHW RI FRQVWUDLQWV F «F GHILQLQJ UHODWLRQVEHWZHHQHOHPHQWVRI3DQG/
7KHPHWKRGWDNHVDFRQVWUDLQWSUREOHP DV LQSXW DQG WULHV WR FDOFXODWH YDOXHV RI DWWULEXWHV RI SRLQWV DQG OLQHV IURP WKH VHWV 3 DQG / ,QLWLDO YDOXHV DUH REWDLQHG IURP WKH XVHU¶V VNHWFK $IWHU VROYLQJ WKH FRQVWUDLQWV WKH VROXWLRQ RI WKH FRQVWUDLQW SUREOHP LV
7KHGLPHQVLRQDOSUHGLFDWH&RLQFLGHQFH O O )LJXUH
DOVR UHWXUQHG LQ WKH VHWV 3 DQG / 7KLV VROXWLRQ LV WKHQHPSOR\HGWRHVWDEOLVKWKHYLVLEOHJHRPHWU\
/HW XV GHVFULEH GHFRPSRVLWLRQ RI WKH SUHGLFDWH
&RLQFLGHQFH O O ,W UHTXLUHV WKDW WKH OLQHV O DQG O FRLQFLGH$GGLWLRQDOSRLQWS
IROORZLQJ FRQVWUDLQWV DUH LQVHUWHG LQWR WKH VHW &6 LQVWHDG 2Q S
RI
WKH
RULJLQDO
SUHGLFDWH
LVDGGHGILUVWDQGWKH
2Q S
O
O $QJOH O O 2I FRXUVH WKH QXPEHU
RI SRLQWV P KDV WR EH LQFUHPHQWHG E\ RQH DIWHU WKLV
WKHVORSHRIWKHOLQHO
WKURXJKSDQGS
7UDQVIRUP$QJOH9DOXH O Q
Q
DOSKD
HQG HQG HQG
VXEVWLWXWLRQ7KHVLWXDWLRQLVVKRZQLQ)LJXUH
/HW XV KDYH WKUHH SRLQWV S S S DQG WKUHH 7UDQVIRUPDWLRQ
LQWR
WKH
PLQLPDO
FRQVWUDLQW
VHW
SUHGLFDWHV
3RLQW S
[
\
3RLQW S
[
\
3RLQW S [ \ 7KHVLWXDWLRQLVKDQGOHGDVVKRZQLQ
UHVXOWVLQWKHIROORZLQJDGYDQWDJHV
)LJXUH (PSOR\LQJ RQO\ ILYH FRQVWUDLQW W\SHV VLPSOLILHV GHVLJQDQGLPSOHPHQWDWLRQRIWKHV\VWHP %RWK
WKH
SRVLWLRQLQJ
SUHGLFDWHV
3RLQW
DQG
$QJOH9DOXH FDQ EH LJQRUHG GXULQJ WKH VROYLQJ SURFHVV DQG SHUIRUPHG DW WKH HQG E\ VLPSOH JHRPHWULFWUDQVIRUPDWLWLRQVRIWKHZKROHVFHQH &RQVWUDLQW GHSHQGHQFLHV DQG EHKDYLRXU RI WKH FRQVWUDLQWVROYHUDUHPXFKPRUHSUHGLFWDEOH $GGLQJ QHZ FRQVWUDLQW W\SHV LQWR WKH RULJLQDO FRQVWUDLQW VHW LV DOVR LPSRUWDQWO\ VLPSOLILHG 1R PDWWHU KRZ FRPSOH[ D QHZ FRQVWUDLQW LV LW GRHV QRW UHTXLUH ZULWLQJ DQG WHVWLQJ PHWKRGV IRU LWV VROYLQJ EHFDXVH FRUUHFW WUDQVIRUPDWLRQ DOUHDG\ DVVXUHVWKHGHVLUHGEHKDYLRXU
7UDQVIRUPDWLRQRIWKUHHSUHGLFDWHV3RLQW )LJXUH
/HW XV FORVH WKLV VHFWLRQ E\ DQ H[DPSOH SUHVHQWLQJ WKH PHFKDQLVP WKDW DVVXUHV WKDW
RQO\
D
VLQJOH
SUHGLFDWH 3RLQW DQG D VLQJOH SUHGLFDWH $QJOH9DOXH FDQ EH SUHVHQW LQ WKH VHW &
7KLV PHFKDQLVP LV
LPSOHPHQWHGE\WKHIROORZLQJWZRIXQFWLRQV
3RLQW S [ \ LVGLUHFWO\FRSLHGLQWR&6
O
O
S S LVDGGHGDQGLWVVORSHDQGWKHGLVWDQFH
^3RLQW S [ \ 2Q S O
$QJOH9DOXH O
2Q S
a 'LVWDQFH S S G `
S S LV DGGHG LQWR WKH
6LPLODUO\ WKH 3RLQW S [ \ FDQQRW EH DGGHG
LQWR &6
HQG
7KH OLQH O
LQYLVLEOH JHRPHWU\ DQG LWV VORSH DQJOH DQG WKH
$GG3UHGLFDWH $QJOH O ODOSKD±DOSKD
&6
WKHQEHJLQ
$GG3UHGLFDWH $QJOH9DOXH ODOSKD
HOVH
EHWZHHQ ERWK SRLQWV DUH FDOFXODWHG :H REWDLQ
EHJLQ
QXPBRIBIL[HGBOLQHV
DOUHDG\ FRQWDLQV D SUHGLFDWH 3RLQW 7KH OLQH
IXQFWLRQ7UDQVIRUP$QJOH9DOXH ODOSKD
LI QXPBRIBIL[HGBOLQHV
3RLQW S [ \ FDQQRW EH DGGHG EHFDXVH &6
GLVWDQFH
DOSKD LVWKHVORSHRIWKHDOUHDG\IL[HGOLQHO
EHWZHHQ
WKH
SRLQWV
S
DQG
S
DUH
FDOFXODWHG %XW ZH FDQQRW XVH DQRWKHU SUHGLFDWH
HQG
$QJOH9DOXH ZKLOH RQH LV DOUHDG\ SUHVHQW LQ WKH &6
IXQFWLRQ7UDQVIRUP3RLQW S[\ EHJLQ
2Q S O
$GG3UHGLFDWH 3RLQW S[\ QXPBRIBIL[HGBSRLQWV
,QVWHDGRIWKLVZHDGGWKHSUHGLFDWH$QJOH
:HREWDLQ&6
WKHQEHJLQ
LI QXPBRIBIL[HGBSRLQWV
GHILQLQJWKHDQJOHEHWZHHQWKHOLQHV O
$QJOH9DOXH O
HQG
a
a 'LVWDQFH S S
2Q S O $QJOH O 'LVWDQFH S S G ` G 2Q S O
DQG O
^3RLQW S [ \ 2Q S O
O
HOVHEHJLQ
ZHDOUHDG\KDYHDIL[HGSRLQWS [ \
(
)
) (
GLVWDQFH_SS _
$'',1* 5('81'$17 &21675$,176 2)',67$1&(6$1'$1*/(6
$GG3UHGLFDWH 'LVWDQFH S SG :H KDYH GLVFXVVHG VHYHUDO DGYDQWDJHV RI SUHYLRXV
LI G ! WKHQEHJLQ DGGOLQHO
LQWRWKHLQYLVLEOHJHRPHWU\
$GG3UHGLFDWH 2Q SO
$GG3UHGLFDWH 2Q S O DOSKD
$UFWJ
VWHSV RI SUHSURFHVVLQJ EXW WKH PDLQ SUREOHP VWLOO UHPDLQV XQVROYHG F\FOHV RI FRQVWUDLQWV KDYH QRW EHHQ UHPRYHG \HW 7KH LGHD RI RXU DOJRULWKP LV WR DGG UHGXQGDQW FRQVWUDLQWV RI GLVWDQFHV DQG DQJOHV DQGWKHQWRFKRRVHVXFKDFRPELQDWLRQRIFRQVWUDLQWV
ZKLFK GRHV QRW FRQWDLQ F\FOHV 7KH ODVW VWHS RI WKH SUHSURFHVVLQJLQWURGXFHVWZRPDWULFHV
DQGWKHLUOHQJWKVDQGabgIRUWKHDQJOHVDQGWKHLU YDOXHV 2QO\ WKUHH RI WKHVH VL[ GLPHQVLRQV DUH LQGHSHQGHQW DQG WKH\ DUH HQRXJK
WR
GHILQH
D
7KH PDWUL[ RI GLVWDQFHV 0 VWRUHV WKH GLVWDQFHV
WULDQJOH 2WKHU WKUHH GLPHQVLRQV DUH FDOFXODWHG E\
EHWZHHQ DOO SDLUV RI SRLQW IURP 3 7KH HOHPHQW LQ
XVLQJWKHIROORZLQJZHOONQRZQUHODWLRQV
WKHL G
URZDQGWKHM
FROXPQRIWKHPDWUL[LVPDUNHG
DQG SUHVHQWV WKH GLVWDQFH EHWZHHQ WKH SRLQWV S
D WKHFRVLQHVWDWHPHQW
DQG S 7KH VL]H RI WKH PDWUL[ LV P P ZKHUH P LV
D
E F ±EFFRVa
WKH WRWDO QXPEHU RI SRLQWV RI WKH DX[LOLDU\ DQG
E
D F ±DFFRVb
F
D E ±DEFRVg
LQYLVLEOH JHRPHWU\ :KLOH WKH GLVWDQFH LV XQVLJQHG WKHPDWUL[ LV V\PPHWULF DQG WKHUHIRUH LW VXIILFHV WR XVH RQO\ WKH HOHPHQWV EHORZ WKH PDLQ GLDJRQDO %HVLGH WKH QXPHULFDO YDOXH RI WKH GLVWDQFH HDFK HOHPHQW G
VWRUHV LWV SULRULW\ DQG D SRLQWHU WR WKH
OLQH SDVVLQJ WKURXJK WKH SRLQWV S DQG S 7KH SULRULW\ GHQRWHV D ZD\ KRZ WKH YDOXH ZDV REWDLQHG ,WFRQWDLQVRQHRIWKHIROORZLQJYDOXHV
WKHDSSUR[LPDWHYDOXHIURPWKHXVHUV¶VVNHWFK ±KLJK SULRULW\ WKH YDOXH ZDV FDOFXODWHG E\ XVLQJ GHWHUPLQHG
YDOXHV
DQG
FDQQRW
EH
FKDQJHGDQ\PRUH ±WKH KLJKHVW SULRULW\ WKH YDOXH SUHVHQWV WKH QXPHULFDO DWWULEXWH RI DQ H[SOLFLWHO\ GHILQHG SUHGLFDWH'LVWDQFH
VWRUHG LQ WKH PDWUL[ RI DQJOHV 0 7KH HOHPHQW LQ
D
URZDQGWKHM
FROXPQRIWKHPDWUL[LVPDUNHG
DQGSUHVHQWVWKHDQJOHEHWZHHQWKHOLQHVO DQGO
7KH VL]H RI WKH PDWUL[ LV Q Q ZKHUH Q LV WKH WRWDO QXPEHU RI OLQHV RI WKH DX[LOLDU\ DQG WKH LQYLVLEOH JHRPHWU\ $QJOHV DUH RULHQWHG DQG WKHUHIRUH WKH PDWUL[ LV QRW V\PPHWULF %XW GLDJRQDOO\ V\PPHWULF HOHPHQWV DUH VWLOO VWURQJO\ FRUUHODWHG D
p ± D
(OHPHQWVRIWKHPDWUL[0 DOVRFRQWDLQSULRULWLHV
%DVLF RSHUDWLRQV IRU ILOOLQJ WKH PDWULFHV DUH VROYLQJ DQG
GHWHUPLQLQJ
WKH
VXPV
DQG
WKH
GLIIHUHQFHV RI DQJOHV VKDULQJ FRPPRQ HGJHV 7KH PDWUL[LVIXOOLILWFRQWDLQVRQO\WKHHOHPHQWVZLWKWKH KLJK RU WKH KLJKHVW SULRULW\ 7KH HOHPHQWV ZLWK WKH KLJK SULRULW\
UHSUHVHQW
QXPHULFDO
SDUDPHWHUV
RI
UHGXQGDQW GLPHQVLRQDO FRQVWUDLQWV 7KH UHGXQGDQW FRQVWUDLQWV SOD\ DQ LPSRUWDQW UROH EHFDXVH WKH\ FDQ EH HPSOR\HG WR GHWHUPLQH RWKHU YDOXHV LQ ERWK PDWULFHV DQG WKH\ FDQ EH FKRVHQ IRU FRQVWUDLQW VROYLQJ LQVWHDG RI H[SOLFLWO\ GHILQHG FRQVWUDLQWV ,Q WKLVZD\F\FOHVFDQEHDYRLGHG
E WKHVLQHVWDWHPHQW
D VLQ a
=
E VLQ b
=
F VLQ g
F WKHVXPRILQWHULRUDQJOHVLVp abg
p
FRRUGLQDWHV DUH QRW LPSRUWDQW \HW D WULDQJOH FDQ EH GHILQHGLQILYHGLIIHUHQWZD\V
DVLGHDQGERWKQHLJKERXULQJDQJOHVDUHNQRZQ D VLGH WKH DQJOH DJDLQVW LW DQG RQH RI WKH QHLJKERXULQJDQJOHVDUHNQRZQ WZRVLGHVDQGWKHDQJOHEHWZHHQWKHPDUHNQRZQ DOOWKUHHVLGHVDUHNQRZQ
NQRZQ 1RWH WKDW WKLV FDVH FDQ SURGXFH WZR GLIIHUHQWVROXWLRQV
7KH VWHS RI VROYLQJ WKH WULDQJOHV WHVWV DOO SRVVLEOH WULDQJOHVGHILQHGE\DUELWUDU\WKUHHSRLQWVDQGVROYHV WKRVH WULDQJOHV ZKHUH DW OHDVW WKUHH GLPHQVLRQV DUH NQRZQ$GLVWDQFHRUDQDQJOHLVNQRZQLILWVSULRULW\ LVKLJKRU WKH KLJKHVW ,I PRUH WKDQ WKUHH GLPHQVLRQV DUH NQRZQ WKHQ WKH\ KDYH WR ILW VRPH UHTXLUHG UHODWLRQ IRU H[DPSOH WKH FRVLQH VWDWHPHQW DOUHDG\ RWKHUZLVH WKH FRQVWUDLQW SUREOHP LV UHFRJQLVHG DV
7KH LGHD LV WR ILOO WKH PDWULFHV DV PXFK DV SRVVLEOH
WULDQJOHV
WZR VLGHV DQG WKH DQJOH DJDLQVW RQH RI WKHP DUH
6LPLODUO\ DOO WKH DQJOHV EHWZHHQ SDLUV RI OLQHV DUH
WKHL
,Q WKLV SKDVH RI WKH DOJRULWKP ZKHQ WKH DEVROXWH
±WKH ORZHVW SULRULW\ WKH YDOXH ZDV LQLWLDOLVHG E\
DOUHDG\
6ROYLQJWKHWULDQJOHV
7KHVKDSHDQGWKHVL]HRIDWULDQJOHDUHGHVFULEHGE\ VL[ GLPHQVLRQV WKUHH VLGHV D E DQG F DQG WKUHH LQWHULRU DQJOHV a b DQG g 7KH DQJOH a OLHV DJDLQVW WKHVLGHDbDJDLQVWWKHVLGHEDQGgDJDLQVWWKHVLGH F :H VKDOO XVH WKH VDPH ODEHOV D E F IRU WKH VLGHV
RYHUFRQVWUDLQHG
7KLV
VWHS
DOVR
GHWHFWV
FRQILJXUDWLRQV WKDW DUH FRQWUDGLFWRU\ WR VRPH ZHOO NQRZQPDWKHPDWLFDOWKHRUHPV7KLVSUREOHPUHIOHFWV LQRQHRIWKHIROORZLQJHUURUV
$VLGHRIDWULDQJOHLVORQJHUWKDQWKHVXPRIRWKHU WZRVLGHV ,Q WKH VLQH RU FRVLQH VWDWHPHQW WKH DWWULEXWH RI D WULJRQRPHWULFDO IXQFWLRQLVQRWLQWKHUDQJH>@
6LGHV RI WULDQJOHV DUH GLUHFWO\ VWRUHG LQWR WKH PDWUL[ RI GLVWDQFHV 0 EXW DQJOHV UHTXLUH VRPH DGGLWLRQDO FRQVLGHUDWLRQEHIRUHEHLQJ ZULWWHQ LQWR WKH PDWUL[ RI DQJOHV 0 RU UHDG IURP LW 1DPHO\ WKH DQJOH EHWZHHQ WKH FDUULHU OLQHV RI WZR VLGHV RI D WULDQJOH XVXDOO\ GRHV QRW SUHVHQW DQ LQWHULRU DQJOH IRU H[DPSOHa RIWKHWULDQJOH,WFDQDOVRSUHVHQWRQHRI WKHIROORZLQJWKUHHDQJOHV
- WKHH[WHULRUDQJOHRIDWULDQJOH p±a - WKH GLIIHUHQFH EHWZHHQ WKH IXOO DQJOH DQG WKH
WULHVWRVROYHWKHPDOO6LPLODUO\DOOFRPELQDWLRQVRI
LQWHULRUDQJOH p±a - WKH GLIIHUHQFH EHWZHHQ WKH IXOO DQJOH DQG WKH H[WHULRUDQJOH pa
:KHQHYHU
DQ
DQJOH
GLIIHUHQFHV RI DQJOHV VKDULQJ FRPPRQ HGJHV 2WKHU VWHSVRIWKHDOJRULWKPDUHPXFKIDVWHU
IURP
WKH
PDWUL[
0
LV
WUDQVIRUPHG WR DQ LQWHUQDO DQJOH RI D WULDQJOH RU WKH LQYHUVH WUDQVIRUPDWLRQ LV GRQH DOO IRXU DQJOHV DUH FRPSDUHG ZLWK WKH FRUUHVSRQGLQJ DQJOH IURP WKH XVHU¶V VNHWFK DQG WKH FORVHVW RQH LV VHOHFWHG 2I FRXUVHWKLVLVFRUUHFWDQGUHDVRQDEOHRQO\LIWKHXVHU GHVLJQVVWULFWO\HQRXJK
WKUHH OLQHV DUH WHVWHG WR FDOFXODWH WKH VXPV DQG
7R
VSHHG
XS
LQWHUDFWLYLW\
WKH
VRPH
SURFHVV SDUWV
RI
DQG WKH
LPSURYH
WKH
DOJRULWKP
DUH
GHVLJQHG LQFUHPHQWDOO\ $Q LQFUHPHQWDO FRQVWUDLQW VROYHU LV DEOH WR WDNH DGYDQWDJHV RI SUHYLRXV VWHSV DQGGRHVQRWFDOFXODWHDOOWKHYDOXHVDIWHUHDFKXVHU¶V LQWHUDFWLRQ >)UHHP @ )RU H[DPSOH WKH PDMRULW\ RI YDOXHV IURP WKH PDWULFHV 0 DQG 0 GHWHUPLQHG
'HWHUPLQLQJWKHVXPVDQGWKHGLIIHUHQFHVRI DQJOHVVKDULQJFRPPRQHGJHV
DIWHU D SDUWLFXODU GHVLJQHU¶V LQWHUDFWLRQ QHHG QRW EH FDOFXODWHGDJDLQDIWHUWKHQH[WRSHUDWLRQ
8VXDOO\ D ELJ DPRXQW RI GLVWDQFHV DQG DQJOHV LV FDOFXODWHGE\VROYLQJWKHWULDQJOHVEXWWKLVRSHUDWLRQ LV
QRW
DOZD\V
VXIILFLHQW
WR
FRPSOHWHO\
ILOO
WKH
PDWULFHV0 DQG0 )RUWKLVUHDVRQLWLVDVVLVWHGE\ D VLPSOH RSHUDWLRQ RI GHWHUPLQLQJ WKH VXPV DQG WKH GLIIHUHQFHV RI DQJOHV VKDULQJ FRPPRQ HGJHV 7KLV WULYLDORSHUDWLRQLVEDVHGRQWKHIROORZLQJUXOH
DQJOH EHWZHHQ OLQHV O DQG O LV b WKHQ WKH DQJOH
EHWZHHQWKHOLQHVO DQGO LVab
D
D
2IFRXUVHLW
FDQ EH DOVR HPSOR\HG WR FDOFXODWH WKH GLIIHUHQFH RI WZRNQRZQDQJOHVVKDULQJDFRPPRQHGJH%HVLGHWR WKHWULYLDOVLWXDWLRQZKHQDOOWKUHHOLQHVLQWHUVHFWLQD FRPPRQ
SRLQW
FRUUHVSRQGLQJ SURYLGHV
WKH
UXOH
DQJOHV
WUDQVLWLYLW\
WR
RI
KDQGOHV VRPH
DOWHUQDWH
NQRZQ
SDUDOOHOLVP
RU
DQG
DQJOHV ILWV
WKH
UHODWLRQWKDWDQH[WHUQDODQJOHRIDWULDQJOHLVHTXDOWR WKHVXPRIERWKRSSRVLWHLQWHUQDODQJOHV
SDVVHG DQG DIWHU WKDW WKH VXPV DQG GLIIHUHQFHV RI DQJOHV DUH FDOFXODWHG GXULQJ SDVVLQJ WKH PDWUL[ 0 &DOFXODWHG GLPHQVLRQV DUH HPSOR\HG WR VROYH RWKHU WULDQJOHVDQGFDOFXODWHWKHVXPVRUWKHGLIIHUHQFHVRI RI
DQJOHV
6RPH
/2&$/
$IWHU WKH SUHSURFHVVLQJ WKH DOJRULWKP VSOLWV WKH VHWV RI OLQHV / DQG SRLQWV 3 LQWR WKH FOXVWHUV VROYDEOH ZLWK ORFDO SURSDJDWLRQ (DFK FOXVWHU FRQWDLQV OLQHV
WKHP ,Q WKLV SDSHU ZH RQO\ GLVFXVV FRQILJXUDWLRQV ZKHUH D VLQJOH FOXVWHU LV REWDLQHG :KLOH WKH DQJOHV EHWZHHQDOOSDLUVRIOLQHVLQDFOXVWHUDUHNQRZQWKLV
WULDQJOH
2Q WKH RWKHU KDQG WKH GLVWDQFHV EHWZHHQ SDLUV RI SRLQWV
QHHG
QRW
EH
GHWHUPLQHG
5HJDUGLQJ
WKH
SULRULWLHV RI GLVWDQFHV D FOXVWHU FRQVLVWV RI RQH RU PRUHVXEVHWVRISRLQWV:HQDPHWKHPWKHFRQQHFWHG VXEVHWV RI SRLQWV :LWK ZHOOFRQVWUDLQHG SUREOHPV WKH PDWUL[ 0 LV DOVR IXOO DQG WKH FOXVWHU FRQWDLQV D VLQJOHFRQQHFWHGVXEVHW
7KHVXEVHWRIWKHFRQVWUDLQWVHW&6
EHORQJLQJWRD
SDUWLFXODU FOXVWHU FRQWDLQV WKH SUHGLFDWHV 2Q $QJOH DQG 'LVWDQFH RQO\ 7KHVH FRQVWUDLQWV DUH VROYHG LQ
2UGHURIVROYLQJWKHWULDQJOHV
SDLUV
%*DR E@ *DR;±6 &KRX6±& 6ROYLQJ JHRPHWULF FRQVWUDLQW V\VWHPV ,, $ V\PEROLF DSSURDFK
FOXVWHUV :KLOH WKH SUREOHP FRQVLVWV RI VROYLQJ WKH
DQG
V\VWHP RI HTXDWLRQV SUHVHQWLQJ GLVWDQFHV RQO\ ZH
&RPSXWHU$LGHG
DUHVWXGLQJGLIIHUHQWJHRPHWULFUHOD[DWLRQPHWKRGV
SS±
GHFLVLRQ
RI
5FFRQVWUXFWLELOLW\
'HVLJQ
9RO
1R
>.DSXU@ .DSXU' 0XQG\-/ :X¶V 0HWKRG DQG ,WV
$SSOLFDWLRQ
.DSXU'
$&.12:/('*(0(176
WR
3HUVSHFWLYH
0XQG\-/
HGV
9LHZLQJ *HRPHWULF
5HDVRQLQJ(OVHYLHU6FLHQFH 7KLV UHVHDUFK KDV EHHQ VXSSRUWHG E\ WKH 0LQLVWU\ RI
>/DWKD @ /DWKDP56 0LGGOHWLFK$( &RQQHFWLYLW\
6FLHQFHDQG7HFKQRORJ\RI5HSXEOLFRI6ORYHQLDDQG
DQDO\VLV D WRRO IRU SURFHVVLQJ JHRPHWULF
E\
FRQVWUDLQWV &RPSXWHU$LGHG 'HVLJQ 9RO
WKH
%ULWLVK
&RXQFLO
6ORYHQLD
LQVLGH
WKH
9DOYD]RU$/,6SURMHFW $/,6
1RSS ± >6XQGH@6XQGH* $ &$' 6\VWHP ZLWK'HFODUDWLYH 6SHFLILFDWLRQ ZRUNVKRS
5()(5(1&(6
RQ
RI
6KDSH
,QWHOOLJHQW
(XURJUDSKLFV &$'
6\VWHPV
1RRUGZLMNHUKRXW7KH1HGHUODQGV >$MZD @ $MZD,$ /LX= :DQJ36 *U|EQHU %DVHV $OJRULWKP ,&0 7HFKQLFDO 5HSRUWV 6HULHV
äDOLN%
*XLG1
&ODSZRUWK\*
&RQVWUDLQWEDVHG 2EMHFW 0RGHOOLQJ -RXUQDO RI (QJLQHHULQJ 'HVLJQ 9RO 1R SS ±
>%RXPD @ %RXPD: )XGRV, +RIIPDQQ& &DL- 3DLJH5
>äDOLN @
*HRPHWULF
&RPSXWHU$LGHG SS
&RQVWUDLQW
'HVLJQ
9RO
6ROYHU 1R
