Fkdswhu 4
OLQHDU IXQFWLRQV
414
Vorshv dqg Htxdwlrqv ri Olqhv
41 Ilqg wkh vorsh ri wkh olqh wkurxjk +7> 8, dqg +4> 5,=
85
7 +4,
6
@
8
p@
481 Wkh olqh jrhv wkurxjk +4> 6,> zlwk vorsh p @ 5=
Xvh srlqw0vorsh irup1
| 6 @ 5+{ 4,
| @ 5{ . 5 . 6
| @ 5{ . 8
4:1 Wkh olqh jrhv wkurxjk +9> 4,> zlwk vorsh p @ 3=
Xvh srlqw0vorsh irup1
| 4 @ 3+{ 9,
|4@3
|@4
61 Ilqg wkh vorsh ri wkh olqh wkurxjk +;> 7, dqg +;> :,=
7 +:,
;;
44
@
3
p@
Wkh vorsh lv xqghqhg> wkh olqh lv yhuwlfdo1
4<1 Wkh olqh jrhv wkurxjk +7> 5, dqg +4> 6,=
Ilqg wkh vorsh/ wkhq xvh srlqw0vorsh irup zlwk
hlwkhu ri wkh wzr jlyhq srlqwv1
65
47
4
@
6
81 | @ 5{
p@
Xvlqj wkh vorsh0lqwhufhsw irup/ | @ p{.e> zh vhh
wkdw wkh vorsh lv 51
:1 8{ <| @ 44
Uhzulwh wkh htxdwlrq lq vorsh0lqwhufhsw irup1
<| @ 8{ 44
8
44
| @ {
<
<
541 Wkh olqh jrhv wkurxjk
Wkh vorsh lv 8< =
<1 { @ 9
Wklv lv yhuwlfdo olqh/ wkh vorsh lv xqghqhg1
p@
@
441 | @ ;
Wklv lv d krul}rqwdo olqh/ zklfk kdv d vorsh ri 31
461 Ilqg wkh vorsh ri d olqh sdudooho wr 5| 7{ @ :=
Uhzulwh wkh htxdwlrq lq vorsh0lqwhufhsw irup1
5| @ 7{ . :
:
| @ 5{ .
5
5;
4
| 6 @ +{ 4,
6
4
4
| @ {. .6
6
6
4
43
| @ {.
6
6
Wklv vorsh lv 5/ vr d sdudooho olqh zloo dovr kdyh
vorsh 51
8
6
8
|
6
|
|
|
|
4
8
5> 6
4
9
86
@
6 45
<9
dqg 6> 49 =
4
43
9 9
9
4
5 5
4;
6
@
63
8
6
4
@
{
8
5
6
6
@ {.
8
43
6
8
6
.
@ {.
8
43 6
6
<
83
@ {.
.
8
63 63
8<
6
@ {.
8
63
8
5
@
Vhfwlrq 414
Vorshv dqg Htxdwlrqv ri Olqhv
561 Wkh olqh jrhv wkurxjk +;> 7, dqg +;> 9,=
p@
79
5
>
@
3
; +;,
zklfk lv xqghqhg1
Wklv lv d yhuwlfdo olqh> wkh ydoxh ri { lv dozd|v ;=
Wkh htxdwlrq ri wklv olqh lv { @ ;=
581 Wkh olqh kdv {0lqwhufhsw 6 dqg |0lqwhufhsw 5=
Wzr srlqwv rq wkh olqh duh +6> 3, dqg +3> 5,= Ilqg
wkh vorsh> wkhq xvh vorsh0lqwhufhsw irup1
3 +5,
63
5
@
6
p@
e @ 5
5
| @ {5
6
5<
641 Zulwh dq htxdwlrq ri wkh olqh wkurxjk +6> 7,> shu0
shqglfxodu wr { . | @ 7=
Uhzulwh wkh htxdwlrq ri wkh jlyhq olqh dv
| @ { . 7=
Wkh vorsh ri wklv olqh lv 4= Wr qg wkh vorsh ri d
shushqglfxodu olqh/ vroyh
4p @ 4=
p@4
Xvh p @ 4 dqg +6> 7, lq wkh srlqw0vorsh irup1
| +7, @ 4+{ 6,
| @{67
| @{:
661 Zulwh dq htxdwlrq ri wkh olqh zlwk |0lqwhufhsw 5/
shushqglfxodu wr 6{ . 5| @ 9=
Ilqg wkh vorsh ri wkh jlyhq olqh1
6{ . 5| @ 9
5| @ 6{ . 9
6
| @ {.6
5
5:1 Wkh yhuwlfdo olqh wkurxjk +9> 8, jrhv wkurxjk wkh
srlqw +9> 3,> vr wkh htxdwlrq lv { @ 9=
Wkh vorsh lv 65 > vr wkh vorsh ri wkh shushqglfxodu
olqh zloo eh 56 = Li wkh |0lqwhufhsw lv 5/ wkhq xvlqj
wkh vorsh0lqwhufhsw irup/ zh kdyh
5<1 Zulwh dq htxdwlrq ri wkh olqh wkurxjk +4> 7,> sdu0
dooho wr { . 6| @ 8=
Uhzulwh wkh htxdwlrq ri wkh jlyhq olqh lq vorsh0
lqwhufhsw irup1
| @ p{ . e
{ . 6| @ 8
6| @ { . 8
4
8
| @ {.
6
6
Wkh vorsh lv 46 =
46
Xvh p @
vorsh irup1
dqg wkh srlqw +4> 7, lq wkh srlqw0
4
| 7 @ ^{ +4,`
6
4
| @ +{ . 4, . 7
6
4
4
| @ { .7
6
6
44
4
| @ {.
6
6
|@
5
{.5
6
681 Gr wkh srlqwv +7> 6,> +5> 3,/ dqg +4;> 45, olh rq
wkh vdph olqhB
Ilqg wkh vorsh ehwzhhq +7> 6, dqg +5> 3,1
p@
6
36
6
@
@
57
5
5
Ilqg wkh vorsh ehwzhhq +7> 6, dqg +4;> 45,1
p@
48
48
45 6
@
@
55
4; 7
55
Vlqfh wkhvh vorshv duh qrw wkh vdph/ wkh srlqwv gr
qrw olh rq wkh vdph olqh1
6:1 D sdudoohorjudp kdv 7 vlghv/ zlwk rssrvlwh vlghv
sdudooho1 Wkh vorsh ri wkh olqh wkurxjk +4> 6, dqg
+5> 4, lv
5
64
@
@ 5=
p@
45
4
8 Wkh vorsh ri wkh olqh wkurxjk 5 > 5 dqg :5 > 7
lv
57
5
p @ 8 : @
@ 5=
4
5 5
63
Fkdswhu 4
Vlqfh wkhvh vorshv duh htxdo/ wkhvh wzr vlghv duh
sdudooho1
Wkh vorsh ri wkh olqh wkurxjk :5 > 7 dqg +4> 6, lv
p@
5
4
76
@ < @ =
<
:5 4
5
Wkhuhiruh/ wuldqjohv PS T dqg S QT duh vlplodu
wr hdfk rwkhu1
+g, Vlqfh fruuhvsrqglqj vlghv lq vlplodu wuldqjohv
duh sursruwlrqdo/
PT @ n S T dqg S T @ n TQ=
Vorsh ri wkh olqh wkurxjk 85 > 5 dqg +5> 4, lv
p@
PT
n ST
@
ST
n TQ
PT
ST
@
ST
TQ
54
5
4
@ < @ =
8
<
5 5
5
Vlqfh wkhvh vorshv duh htxdo/ wkhvh wzr vlghv duh
sdudooho1
Vlqfh erwk sdluv ri rssrvlwh vlghv duh sdudooho/ wkh
txdgulodwhudo lv d sdudoohorjudp1
6<1 Wkh olqh jrhv wkurxjk +3> 5, dqg +5> 3,
p@
Iurp wkh gldjudp/ zh nqrz wkdw S T @ 4=
PT @
Vxevwlwxwlqj/ zh jhw
p4 @
7
73
@
@ 7
4 3
4
761 +d, Vhh wkh jxuh lq wkh wh{werrn1
Vhjphqw PQ lv gudzq shushqglfxodu wr vhjphqw
S T= Uhfdoo wkdw PT lv wkh ohqjwk ri vhjphqw PT=
p4 @
4
=
p5
Pxowlso|lqj erwk vlghv e| p5 > zh kdyh
p4 p5 @ 4=
741 Wkh olqh dsshduv wr jr wkurxjk +3> 3, dqg +4> 7,=
p@
4
TQ
Iurp +d, dqg +e,/ p4 @ PT dqg p5 @ TQ=
53
5
@ @4
3 +5,
5
Wkh fruuhfw fkrlfh lv +d,1
OLQHDU IXQFWLRQV
781 | @ 5{ . 6
Wkuhh rughuhg sdlu wkdw vdwlvi| wklv htxdwlrq duh
+5> 4,> +3> 6,> dqg +4> 8,1 Sorw wkhvh srlqwv dqg
gudz d olqh wkurxjk wkhp1
PT
7|
@
ST
7{
Iurp wkh gldjudp/ zh nqrz wkdw S T @ 4= Wkxv/
p4 @ P4T > vr PT kdv ohqjwk p4 =
+e,
p5 @
TQ
TQ
7|
@
@
ST
4
7{
TQ @ p5
+f, Wuldqjohv PS T> S QT> dqg PQS duh uljkw
wuldqjohv e| frqvwuxfwlrq1 Lq wuldqjohv PS T dqg
PQS>
dqjoh P @ dqjoh P>
dqg lq wkh uljkw wuldqjohv S QT dqg PQS>
dqjoh Q @ dqjoh Q=
Vlqfh doo uljkw dqjohv duh htxdo/ dqg vlqfh wuldqjohv
zlwk wzr htxdo dqjohv duh vlplodu/ wuldqjoh PS T
lv vlplodu wr wuldqjoh PQ S dqg wuldqjoh S QT lv
vlplodu wr wuldqjoh PQ S=
7:1 | @ 9{ . 45
Wkuhh rughuhg sdluv wkdw vdwlvi| wklv htxdwlrq duh
+3> 45,> +4> 9,> dqg +5> 3,= Sorw wkhvh srlqwv dqg
gudz d olqh wkurxjk wkhp1
Vhfwlrq 414
Vorshv dqg Htxdwlrqv ri Olqhv
64
Sorw wkh rughuhg sdluv +5> 3, dqg 3> 85 dqg gudz
d olqh wkurxjk wkhvh srlqwv1 +D wklug srlqw pd|
eh xvhg dv d fkhfn1,
7<1 6{ | @ <
Ilqg wkh lqwhufhswv1
Li | @ 3/ wkhq
6{ 3 @ <
6{ @ <
{ @ 6
Li { @ 3/ wkhq
6+3, | @ <
| @ <
|@<
vr wkh |0lqwhufhsw lv <1
Sorw wkh rughuhg sdluv +6> 3, dqg +3> <, dqg gudz
d olqh wkurxjk wkhvh srlqwv1 +D wklug srlqw pd|
eh xvhg dv d fkhfn1,
861 { @ 7
Iru dq| ydoxh ri |> wkh {0ydoxh lv 71 Ehfdxvh doo
rughuhg sdluv wkdw vdwlvi| wklv htxdwlrq kdyh wkh
vdph uvw qxpehu/ wklv htxdwlrq grhv qrw uhsuh0
vhqw d ixqfwlrq1 Wkh judsk lv wkh yhuwlfdo olqh zlwk
{0lqwhufhsw 71
841 7| . 8{ @ 43
Ilqg wkh lqwhufhswv1
881 | 7 @ 3
Li | @ 3> wkhq
7+3, . 8{ @ 43
8{ @ 43
{ @ 5>
vr wkh {0lqwhufhsw lv 51
Li { @ 3> wkhq
7| . 8+3, @ 43
7| @ 43
|@
vr wkh |lqwhufhsw lv 85 =
43
8
@ >
7
5
Wklv htxdwlrq pd| eh uhzulwwhq dv | @ 7/ ru/
htxlydohqwo|/ | @ 3{ 7= Wkh |0ydoxh lv 7 iru
dq| ydoxh ri {= Wkh judsk lv wkh krul}rqwdo olqh
zlwk |0lqwhufhsw 7=
65
Fkdswhu 4
8:1 | @ 8{
Wkuhh rughuhg sdluv wkdw vdwlvi| wklv htxdwlrq duh
+3> 3,> +4> 8,> dqg +4> 8,= Xvh wkhvh srlqwv wr
gudz wkh judsk1
8<1 { 6| @ 3
941 +d, {
3
4
5
6
7
8
9
OLQHDU IXQFWLRQV
|
6=53:
6=8<<
7=335
7=684
7=977
7=<54
8=4;5
Wkh ghew lv lqfuhdvlqj dqg wkh gdwd dsshdu wr eh
qhduo| olqhdu1
Li | @ 3> wkhq { @ 3> vr wkh {0lqwhufhsw lv 31 Li
{ @ 3> wkhq | @ 3> vr wkh |0lqwhufhsw lv 31 Erwk
lqwhufhswv jlyh wkh vdph rughuhg sdlu/ +3> 3,=
Wr jhw d vhfrqg srlqw/ fkrrvh vrph rwkhu ydoxh ri
{ +ru |,= Iru h{dpsoh li { @ 6> wkhq
+e, Ilqg wkh vorsh xvlqj +3> 6=53:, dqg +9> 8=4;5,1
p@
4=<:8
8=4;5 6=53:
@
=65<5
93
9
Iurp | @ p{ . e/ zlwk p @ =65<5 dqg e @ 6=53:/
wkh olqhdu htxdwlrq lv
| @ =65<5{ . 6=53:
{ 6|
6 6|
6|
|
@3
@3
@ 6
@ 4>
Wkh vorsh 165<5 lqglfdwhv wkdw wkh ihghudo ghew
zdv lqfuhdvlqj dw d udwh ri 165<5 wuloolrq grooduv
shu |hdu1
jlylqj wkh rughuhg sdlu +6> 4,= Judsk wkh olqh wkurxjk
+3> 3, dqg +6> 4,=
+f, Wkh |hdu 4<<: fruuhvsrqgv wr
{ @ 4<<: 4<<3 @ :1
| @ =65<:+:, . 6=53;
| 8=844
Lq 4<<: wkh ihghudo ghew lv suhglfwhg wr eh 81844
wuloolrq grooduv1
Vhfwlrq 414
Vorshv dqg Htxdwlrqv ri Olqhv
Wkh |hdu 4<<; fruuhvsrqgv wr { @ 4<<;4<<3 @ ;1
| @ =65<5+;, . 6=53:
| 8=;74
Lq 4<<; wkh ihghudo ghew lv suhglfwhg wr eh 81;74
wuloolrq grooduv1
Iru erwk |hduv/ wkh uhvxowv duh juhdwhu wkdq wkh
dfwxdo ghew1
+g,
66
Xvh srlqw0vorsh irup wr zulwh wkh htxdwlrq1
83
+{ =:8,
<
83
83
| 5 @ { +=:8,
<
<
:8
83
.5
| @ {
<
4;
83
6<
@ {
<
4;
Qrz ghwhuplqh |> wkh vshhg/ zkhq {> wkh irufh/ lv
41491
83
6<
| @ +4=49, <
4;
8; 6<
@
<
4;
::
@
7=6
4;
Wkh srq| vzlwfkhv iurp d wurw wr d jdoors dw ds0
sur{lpdwho| 716 phwhuv shu vhfrqg1
|5@
9:1 | @ 67{ . 563
Wkh olqhdu wuhqg grhv qrw frqwlqxh sdvw 4<<9 dv
wkh ghew lqfuhdvhg dw d vorzhu udwh1 D olqhdu ds0
sur{lpdwlrq lv qrw qhfhvvdulo| ydolg ryhu grpdlqv
wkdw h{whqg sdvw wkh ruljlqdo gdwd1
961 +d, Wkh olqh jrhv wkurxjk +7> =4:, dqg +:> =66,=
p@
| =66 @
=49
=66 =4:
@
=386
:7
6
=49
+{ :,
6
| =66 @ =386{ =6:6
| =386{ =376
+e, Ohw | @ =8> vroyh iru {=
=8 @ =386{ =376
=876 @ =386{
43=5 @ {
Lq derxw 4315 |hduv/ kdoi ri wkhvh sdwlhqwv zloo kdyh
DLGV1
981 Ohw { uhsuhvhqw wkh irufh dqg | uhsuhvhqw wkh vshhg1
Wkh olqhdu ixqfwlrq frqwdlqv wkh srlqwv +=:8> 5, dqg
+=<6> 6,=
65
4
4
p@
@
@ 4;
=<6 =:8
=4;
433
83
433
@
@
4;
<
Ohw | @ 4333=
4333 @ 67{ . 563
::3 @ 67{
{ 55=9
Dffruglqj wr wklv irupxod/ dssur{lpdwho| 56 dfruqv
shu vtxduh phwhu zrxog uhvxow lq 4333 ghhu wlfn
oduydh shu 733 vtxduh phwhuv1
9<1 +d, Wkh olqh jrhv wkurxjk +3> 4=4, dqg +53> 7=<,1
p@
7=< 4=4
6=;
@
53 3
53
p @ =4<
e @ 4=4
| @ =4<{ . 4=4
+e, Wkh |hdu 5343 fruuhvsrqgv wr { @ 661
| @ =4<+66, . 4=4
| @ :=6:
Zh suhglfw wkdw wkh qxpehu ri frkdelwlqj dgxowv
lq wkh |hdu 5343 zloo eh derxw :17 ploolrq1
:41 +d, Xvlqj wkh srlqwv +3> 837, dqg +47> 6:7,
+xvlqj 4<;3 dv { @ 3,>
p@
463
837 6:7
@
<=6=
3 47
47
e @ 837
| @ i+{, @ <=6{ . 837
67
Fkdswhu 4
+e, 4<;: fruuhvsrqgv wr { @ :=
:81 +d,
i+:, @ <=6+:, . 837
@ 98=4 . 837
@ 76;=< 76<
Wkh dyhudjh sd|phqw lq 4<;: zdv derxw '76<1
+f, Vlqfh wkh vorsh ri wkh olqhdu ixqfwlrq lv <=6>
wkh dyhudjh sd|phqw lv ghfuhdvlqj e| derxw '<163
shu prqwk1
{
4<;9
4<;;
4<<3
4<<5
4<<7
4<<9
4<<;
OLQHDU IXQFWLRQV
|
998;
;337
<673
43/77<
44/:4<
45/<<7
47/83;
:61 +d, Sorw wkh srlqwv +48> 4933,> +533> 48/333,/
+5<3> 57/333,/ dqg +853> 73/333,1
\hv/ wkh gdwd dsshdu wr olh urxjko| dorqj d vwudljkw
olqh1
+e, Xvh srlqwv +;9> 998;, dqg +<9> 45/<<7,
p@
Wkh srlqwv olh dssur{lpdwho| rq d olqh/ vr wkhuh
dsshduv wr eh d olqhdu uhodwlrqvkls ehwzhhq glv0
wdqfh dqg wlph1
| 998; @ 966=9+{ ;9,
| 998; @ 966=9{ 87/7;<19
| @ 966=9{ 7:/;6419
+e, Wkh judsk ri dq| htxdwlrq ri wkh irup | @ p{
jrhv wkurxjk wkh ruljlq/ vr wkh olqh jrhv wkurxjk
+853> 73/333, dqg +3> 3,=
p@
Wkh vorsh ri 96619 lqglfdwhv wkdw wxlwlrq dqg ihhv
kdyh lqfuhdvhg dssur{lpdwho| '967 shu |hdu1
+f, Wkh |hdu 5353 lv wrr idu lq wkh ixwxuh wr pdnh
dffxudwh suhglfwlrqv edvhg rq wkh jlyhq gdwd1 Pdq|
idfwruv frxog dhfw froohjh frvwv e| wkhq1
73/333 3
:9=<
853 3
e@3
| @ :9=<{ . 3
| @ :9=<{
+f, Ohw | @ 93/333> vroyh iru {=
93/333 @ :9=<{
:;3=56 {
K|gud lv derxw :;3 phjdsduvhfv iurp hduwk1
<=8 4344
> p @ :9=<
p
<=8 4344
D@
:9=<
+g, D @
@ 45=7 eloolrq |hduv
9669
45/<<7 998;
@
@ 966=9
<9 ;9
43
415
Olqhdu Ixqfwlrqv dqg
Dssolfdwlrqv
41 Wklv vwdwhphqw lv wuxh1
Zkhq zh vroyh | @ i+{, @ 3> zh duh qglqj wkh
ydoxh ri { zkhq | @ 3> zklfk lv wkh {0lqwhufhsw1
Zkhq zh hydoxdwh i+3,> zh duh qglqj wkh ydoxh
ri | zkhq { @ 3> zklfk lv wkh |0lqwhufhsw1
61 Wklv vwdwhphqw lv wuxh1
Rqo| d yhuwlfdo olqh kdv dq xqghqhg vorsh/ exw d
yhuwlfdo olqh lv qrw wkh judsk ri d ixqfwlrq1 Wkhuh0
iruh/ wkh vorsh ri d olqhdu ixqfwlrq fdqqrw eh gh0
qhg1
Vhfwlrq 415
Olqhdu Ixqfwlrqv dqg Dssolfdwlrqv
:1 '45 lv wkh {hg frvw dqg '4 lv wkh frvw shu krxu1
Ohw { @ qxpehu ri krxuv>
F+{, @ frvw ri uhqwlqj d vdz iru
{ krxuv1
Wkxv/
F+{, @ {hg frvw . +frvw shu krxu,
+qxpehu ri krxuv,
F+{, @ 45 . 4{
45 . {=
<1 832f lv wkh {hg frvw dqg 682f lv wkh frvw shu kdoi0
krxu1
Ohw { @ wkh qxpehu ri kdoi0krxuv>
F+{, @ wkh frvw ri sdunlqj d fdu iru
{ kdoi0krxuv1
68
8
G+t, @ 49 t
7
4:1
8
+d, G+3, @ 49 +3, @ 49 3 @ 49
7
Zkhq 3 fdq rshqhuv duh ghpdqghg/ wkh sulfh lv
'491
8
+e, G+7, @ 49 +7, @ 49 8 @ 44
7
Zkhq 733 fdq rshqhuv duh ghpdqghg/ wkh sulfh lv
'441
8
+f, G+;, @ 49 +;, @ 49 43 @ 9
7
Zkhq ;33 fdq rshqhuv duh ghpdqghg/ wkh sulfh lv
'91
+g, Ohw G+t, @ 9= Ilqg t=
8
9 @ 49 t
7
Wkxv/
F+{, @ 83 . 68{
@ 68{ . 83=
8
t @ 43
7
441 Il{hg frvw/ '433> 83 lwhpv frvw '4933 wr surgxfh1
Ohw F+{, @ frvw ri surgxflqj { lwhpv1
F+{, @ p{ . e> zkhuh e lv wkh {hg
frvw1
t@;
Zkhq wkh sulfh lv '9/ ;33 fdq rshqhuv duh ghpdqghg1
+h, Ohw G+t, @ 44= Ilqg t=
8
44 @ 49 t
7
8
t@8
7
F+{, @ p{ . 433
Qrz/
F+{, @ 4933 zkhq { @ 83> vr
4933 @ p+83, . 433
4833 @ 83p
63 @ p=
t@7
Zkhq wkh sulfh lv '49/ 733 fdq rshqhuv duh ghpdqghg1
+i , Ohw G+t, @ 49= Ilqg t=
Wkxv/ F+{, @ 63{ . 433=
8
49 @ 49 t
7
8
3@ t
7
461 Pdujlqdo frvw/ '<3> 483 lwhpv frvw '49/333 wr sur0
gxfh1
F+{, @ <3{ . e
Qrz/ F+{, @ 49/333 zkhq { @ 483=
49/333 @ <3+483, . e
49/333 @ 46/833 . e
5833 @ e
Wkxv/ F+{, @ <3{ . 5833=
481 Iru gdwd wkdw fdq eh prghohg zlwk d olqhdu ixqf0
wlrq/ wkh dyhudjh udwh ri fkdqjh zloo eh wkh vdph
dv wkh vorsh ri wkh olqh1 Lq fdvhv zkhuh wkh gdwd lv
qrw olqhdu/ wkh dyhudjh udwh ri fkdqjh lv wkh vorsh
ri wkh vhfdqw olqh frqqhfwlqj wkh ehjlqqlqj dqg
hqglqj srlqwv1
3@t
+j,
69
Fkdswhu 4
+k, V+t, @
6
t
7
OLQHDU IXQFWLRQV
5
4<1 s @ V+t,> s @ G+t, @ 433 t
8
+d,
Ohw V+t, @ 3= Ilqg t=
6
3@ t
7
3@t
Zkhq wkh sulfh lv '3/ 3 fdq rshqhuv duh vxssolhg1
+l, Ohw V+t, @ 43= Ilqg t=
+e,
6
43 @ t
7
73
@t
6
5
5
t @ 433 t
8
8
7
t @ 433
8
t @ 46=6
Zkhq wkh sulfh lv '43/ derxw 4666 fdq rshqhuv duh
vxssolhg1
+m, Ohw V+t, @ 53= Ilqg t=
6
53 @ t
7
;3
@t
6
t @ 59=9
Zkhq wkh sulfh lv '53/ derxw 599: fdq rshqhuv duh
ghpdqghg1
+n, Vhh wkh dqvzhu judsk iru sduw +j,1
+o,
G+t, @ V+t,
8
6
49 t @ t
7
7
49 @ 5t
;@t
6
V+;, @ +;, @ 9
7
Wkh htxloleulxp txdqwlw| lv ;33/ dqg wkh htxlole0
ulxp sulfh lv '91
V+t, @ G+t,
t @ 458
5
V+458, @ +458, @ 83
8
Wkh htxloleulxp txdqwlw| lv 458/ wkh htxloleulxp
sulfh lv '83
541 +d, F+{, @ p{ . e> p @ 6=83> F+93, @ 633
F+{, @ 6=83{ . e
Ilqg e=
633 @ 6=83+93, . e
633 @ 543 . e
<3 @ e
F+{, @ 6=83{ . <3
+e, U+{, @ <{
F+{, @ U+{,
6=83{ . <3 @ <{
<3 @ 8=8{
49=69 @ {
\rvkl pxvw surgxfh dqg vhoo 4: vkluwv1
+f, S +{, @ U+{, F+{,> S +{, @ 833
833 @ <{ +6=83{ . <3,
833 @ 8=8{ <3
8<3 @ 8=8{
43:=5: @ {
Wr pdnh d surw ri '833/ \rvkl pxvw surgxfh dqg
vhoo 43; vkluwv1
Vhfwlrq 415
Olqhdu Ixqfwlrqv dqg Dssolfdwlrqv
561 +d, Xvlqj wkh surwv +433> 44=35, dqg +733> 73=45,>
p@
73=45 44=35
5<=4
@
@ =3<:=
733 433
633
| 44=35 @ =3<:+{ 433,
| 44=35 @ =3<:{ <=:
| @ =3<:{ . 4=65
F+{, @ 3<:{ . 4=65
+e, F+4333, @ =3<:+4333, . 4=65 @ <: . 4=65
@ <;=65
Wkh wrwdo frvw ri surgxflqj 4333 fxsv lv '<;1651
+f, F+4334, @ =3<:+4334, . 4=65 @ <:=3<: . 4=65
@ <;=74:
Wkh wrwdo frvw ri surgxflqj 4334 fxsv lv '<;174:1
+g, Pdujlqdo frvw @ <;=74: <;=65
@ '=3<: ru <=:2f
+h, Wkh pdujlqdo frvw iru dq| fxs lv wkh vorsh/
'13<: ru <=:2f=
581 +d, +433/333,+83, @ 8/333/333
Vdohv lq 4<<9 zrxog eh 433/333 . 8/333/333 @
8/433/3331
+e, Wkh rughuhg sdluv duh +4> 433/333, dqg
+9> 8/433/333,=
+f, p @
8/433/333 433/333
8/333/333
@
94
8
@ 4> 333> 333
| 433/333 @ 4/333/333+{ 4,
| 433/333 @ 4/333/333{ 4/333/333
| @ 4/333/333{ <33/333
V+{, @ 4/333/333{ <33/333
+g, Li V+{, @ 53/333/333/ qg {=
53/333/333 @ 4/333/333{ <33/333
53/<33/333 @ 4/333/333{
53=< @ {
Vdohv zrxog uhdfk '53/333/333 lq 4<<4 . 531< @
53441</ wkdw lv/ lq wkh |hdu 53441
5:1 F+{, @ 8{ . 53> U+{, @ 48{
+d,
F+{, @ U+{,
8{ . 53 @ 48{
53 @ 43{
5@{
Wkh euhdn0hyhq txdqwlw| lv 5 xqlwv1
6:
+e, S +{, @ U+{, F+{,
S +{, @ 48{ +8{ . 53,
S +433, @ 48+433, +8 433 . 53,
@ 4833 853 @ <;3
Wkh surw iurp 433 xqlwv lv '<;31
+f,
S +{, @ 833
48{ +8{ . 53, @ 833
43{ 53 @ 833
43{ @ 853
{ @ 85
Iru d surw ri '833/ 85 xqlwv pxvw eh surgxfhg1
5<1 F+{, @ ;8{ . <33
U+{, @ 438{
Vhw F+{, @ U+{, wr qg wkh euhdn0hyhq txdqwlw|1
;8{ . <33 @ 438{
<33 @ 53{
78 @ {
Wkh euhdn0hyhq txdqwlw| lv 78 xqlwv1 \rx vkrxog
ghflgh qrw wr surgxfh vlqfh qr pruh wkdq 6; xqlwv
fdq eh vrog1
S +{, @ U+{, F+{, @ 438{ +;8{ . <33,
@ 53{ <33
Wkh surw ixqfwlrq lv S +{, @ 53{ <331
641 F+{, @ :3{ . 833
U+{, @ 93{
:3{ . 833 @ 93{
43{ @ 833
{ @ 83
Wklv uhsuhvhqwv d euhdn0hyhq txdqwlw| ri 83 xqlwv1
Lw lv lpsrvvleoh wr pdnh d surw zkhq wkh euhdn0
hyhq txdqwlw| lv qhjdwlyh1 Frvw zloo dozd|v eh
juhdwhu wkdq uhyhqxh1
S +{, @ U+{, F+{, @ 93{ +:3{ . 833,
@ 43{ 833
Wkh surw ixqfwlrq lv S +{, @ 43{ 8331
6;
Fkdswhu 4
661 Xvh wkh irupxodv ghulyhg lq H{dpsoh : lq wklv vhf0
wlrq ri wkh wh{werrn1
<
F . 65
8
8
F @ +I 65,
<
I @
+d, I @ 8;> qg F=
8
+8; 65,
<
8
F @ +59,
<
F@
F @ 47=7
Wkh whpshudwxuh lv 4717 F=
+e, I @ 53> qg F1
8
+I 65,
<
8
F @ +53 65,
<
8
F @ +85,
<
416
61
Wkh Ohdvw Vtxduhv Olqh
{
9=;
:=3
:=4
:=5
:=7
68=8
|
=;
4=5
=<
=<
4=8
8=6
+f, F @ 83> qg I=
<
F . 65
8
<
I @ +83, . 65
8
I @
I @ <3 . 65
I @ 455
Wkh whpshudwxuh lv 455 I1
681 Li wkh whpshudwxuhv duh qxphulfdoo| htxdo/ wkhq
I @ F=
<
I @ F . 65
8
<
F @ F . 65
8
7
F @ 65
8
F @ 73
Wkh Fhovlxv dqg Idkuhqkhlw whpshudwxuhv duh qx0
phulfdoo| htxdo dw 73 =
{|
8=77
;=7
9=6<
9=7;
44=4
6:=;4
{5
79=57
7<=3
83=74
84=;7
87=:9
585=58
|5
=97
4=77
=;4
=;4
5=58
8=<8
8+6:=;4, +68=8,+8=6,
s
u@s
8+585=58, +68=8,5 8+8=<8, +8=6,5
=9<;8
u @ +=9<;8,5 =8
5
Wkh dqvzhu lv fkrlfh +f,1
9e . 888p @ :87
888e . 84/688p @ 9<> ;9<
81 +d,
F@
F @ 5;=<
OLQHDU IXQFWLRQV
9e @ :87888p
:87888p
9
:87 888p
.84/688p @ 9</;9<
888
9
e@
888+:87 888p, . 63;/463p @ 74</547
74;/7:363;/358p.63;/463p @ 74</547
438p @ :77
p :=3<
:87 88+:=3<,
e@
863
9
\ @ :=3<{ 863
+e, Ohw { @ <:> qg \=
\ @ :=3<+<:, 863 48:=:6
Wkhuh zloo eh derxw 48;/333 mrev lq 4<<:1
+f, Ohw \ @ 4:3> qg {=
4:3 @ :=3<{ 863
:33 @ :=3<{
{ @ <;=:6
Wkh qxpehu ri qhz mrev zloo uhdfk 4:3/333 lq 4<<<1
9+9</;9<, +888,+:87,
s
+g, u @ s
9+84/688, 8885 9+<8/:73, :875
@ =<76
Wklv lqglfdwhv wkdw wkh srlqwv rq wkh olqh jlyh jrrg
dssur{lpdwlrqv ri wkh gdwd srlqwv1
Vhfwlrq 416
Wkh Ohdvw Vtxduhv Olqh
6<
:1 +d,
\hv/ wkh gdwd srlqwv olh lq d olqhdu sdwwhuq1
+e,
{
539
;35
4::4
44<;
456;
5:;9
453:
;<5
5744
5;;8
5:38
<7;
5:95
5;48
57/959
|
43<
457
487
477
477
4:<
477
477
4:<
4:<
4:<
477
4:<
4:<
54;4
{|
55/787
<</77;
5:5/:67
4:5/845
4:;/5:5
7<;/9<7
4:6/;3;
45;/77;
764/89<
849/748
7;7/4<8
469/845
7<7/6<;
836/;;8
7/446/677
{5
75/769
976/537
6/469/774
4/768/537
4/865/977
:/:94/:<9
4/789/;7<
:<8/997
8/;45/<54
;/656/558
:/64:/358
;<;/:37
:/95;/977
:/<57/558
87/:3;/<;5
|5
44/;44
48/6:9
56/:49
53/:69
53/:69
65/374
53/:69
53/:69
65/374
65/374
65/374
53/:69
65/374
65/374
679/;<<
47+7/446/677, +57/959,+54;4,
s
u@ s
=<:5
47+87/:3;/<;5, +57/959,5 47+679/;<<, 54;45
\hv/ wkhuh lv d srvlwlyh olqhdu fruuhodwlrq ehwzhhq wkh frvw ri wkh wlfnhw dqg wkh glvwdqfh rzq1
+f,
47e . 57/959p @ 54;4
57/959e . 87/:3;/<;5p @ 7/446/677
47e @ 54;4 57/959p
e@
57/959
54;4 57/959p
47
54;4 57/959p
47
. 87/:3;/<;5p @ 7/446/677
57/959+54;4 57/959p, . :98/<58/:7;p @ 8:/8;9/;49
86/:3</639 939/76</:;9p . :98/<58/:7;p @ 8:/8;9/;49
48</7;8/;:5p @ 6/;::/843
p =3576
e@
54;4 57/959+13576,
446
47
\ @ 446 . =3576{
Wkh pdujlqdo frvw shu ploh lv 51762f shu ploh1
73
Fkdswhu 4
{
;7
;8
;9
;:
;;
;<
<3
<4
<5
<6
<7
<8
<9
44:3
<1 +d,
|
45=6
46=5
46=9
46=5
47=8
47=8
46=;
45=6
45=;
46=<
48=3
47=:
47=9
4:;=7
{|
4366=5
4455
449<=9
447;=7
45:9
45<3=8
4575
444<=6
44::=9
45<5=:
4743
46<9=8
4734=9
49/3:<=7
{5
:389
:558
:6<9
:89<
::77
:<54
;433
;5;4
;797
;97<
;;69
<358
<549
438/7;5
|5
484=5<
4:7=57
4;7=<9
4:7=57
543=58
543=58
4<3=77
484==5<
496=;7
4<6=54
558=33
549=3<
546=49
578;=59
9e . 873p @ ;4=;
873e . 7;/9:3p @ :69<=7
9e . 873p @ ;4=;
9e @ ;4=; 873p
46e . 44:3p @ 4:;=7
46e @ 4:;=744:3p
44:3
4:;=7 44:3p
46
4:;=744:3p
e@
46
. 438/7;5p @ 49> 3:<=7
44:3 +4:;=744:3p, . 4/6:4/599p @ 53</36515
53;/:5: 4/69;/<33p . 4/6:4/599p @ 53</36515
5699p @ 637=5
p @ =45;8
e@
4:;=7 44:3+=45;8,
@ 5=48
46
\ @ =45<{ . 5=48
46+49> 3:<=7, +44:3,+4:;=7,
s
u@s
46+438> 7;5, 44:35 46+578;=59, +4:;=7,5
@ =87:
+e,
{
;8
;:
;<
<4
<6
<8
873
|
46=5
46=5
47=8
45=6
46=<
47=:
;4=;
{|
4455
447;=7
45<3=8
444<=6
45<5=:
46<9=8
:69<=7
{5
:558
:89<
:<54
;5;4
;97<
<358
7;> 9:3
|5
4:7=57
4:7=57
543=58
484=5<
4<6=54
549=3<
444<=65
;4=; 873p
9
73=< 5:3p
@
6
e@
73=< 5:3p
. 7;/9:3p @ :69<=7
6
873+73=< 5:3p, . 479/343p @ 55/43;15
55/3;9 478/;33 . 479/343p @ 55/43;15
543p @ 55=5
p @ =439
873
e@
46e . 44:3p @ 4:;=7
44:3e . 438> 7;5p @ 49> 3:<=7
OLQHDU IXQFWLRQV
73=< 5:3+=439,
6
7=45
\ @ =439{ . 7=45
9+:69<=7, +873,+;4=;,
s
u@s
9+7;/9:3, 8735 9+444<=65, ;4=;5
@ =769
441 +d,
{
;;=9
:4=9
<6=6
;7=6
;3=9
:8=5
9<=:
;5=3
9<=7
;6=6
:<=9
;5=9
;3=9
;6=8
:9=6
4533=9
|
{|
{5
|5
53=3
4::5
:;7<=<9 733=3
49=3
4478=9
8459=89 589=3
4<=;
4;7:=67
;:37=;< 6<5=37
4;=7
4884=45
:439=7< 66;=89
4:=4
46:;=59
97<9=69 5<5=74
48=8
4498=9
8988=37 573=58
47=:
4357=8<
7;8;=3< 549=3<
4:=4
4735=5
9:57
5<5=74
48=7
439;=:9
7;49=69 56:=49
49=5
467<=79
9<6;=;< 595=77
48=3
44<7
9669=49 558
4:=5
4753=:5
9;55=:9 5<8=;7
49=3
45;<=9
97<9=69 589=3
4:=3
474<=8
9<:5=58 5;<=3
47=7
43<;=:5
8;54=9< 53:=69
57<=; 53/45:17: <9/:581;9 7533=89
Vhfwlrq 416
Wkh Ohdvw Vtxduhv Olqh
74
48e . 4533=9p @ 57<=;
4533=9e . <9> :58=;9p @ 53/45:17:
48e @ 57<=;4533=9p
57<=;4533=9p
e@
48
57<=;4533=9p
.<9/:581;9p @ 53/45:17:
48
4533=9+57<=;4533=9p,.4/783/;;:1<p @ 634/<45138
5<</<3<1;;4/774/773=69p @ 634/<45138
.4/783/;;:1<p
<77:=87p @ 5335=4:
p =545
\hv/ wkhuh lv d srvlwlyh olqhdu fruuhodwlrq ehwzhhq
wkh lqfrph dqg wkh |hdu1
9e . :8p @ 8:=:<<
:8e . 46:8p @ <65=;;
9e @ 8:=:<< :8p
+f,
4533=9
e@
e@
8:=:<< :8p
. 46:8p @ <65=;;
9
:8+8:=:<< :8p, . ;583p @ 88<:=5;
7667=<58 8958p . ;583p @ 88<:=5;
5958p @ 4595=688
p =7;3<
:8
57<=; 4533=9+=545,
@ =63<
48
\ @ =545{ =63<
+e, Ohw { @ :6> qg \=
\ @ =545+:6, =63<
48=5
Li wkh whpshudwxuh zhuh :6 I/
|rx zrxog h{shfw wr khdu 4815
fklusv shu vhfrqg1
+f, Ohw \ @ 4;> qg {=
4; @ =545{ =63<
4;=63< @ =545{
;9=7 {
Zkhq wkh fulfnhwv duh fkluslqj 4;
wlphv shu vhfrqg/ wkh whpshudwxuh
lv ;917 I1
+g,
48+53/45:, +4533=9,+57<=;,
s
u@s
48+<9/:581;9,+4533=9,5 48+7533=89,57<=;,5
@ =;68
461 +d,
\hv/ wkh gdwd dsshdu wr olh dorqj d vwudljkw olqh1
9+<65=;;, :8+8:=:<<,
s
+e, u @ s
9+46:8, :85 9+98;=7384;6, 8:=:<<5
@ =<<:;
8:=:<< :8p
9
e@
8:=:<< :8+=7;3<,
9
e @ 6=955
\ @ =7;3<{ . 6=955
+g, Wkh |hdu 5348 fruuhvsrqgv wr
{ @ 5348 4<:3 @ 78
\ @ =7;3<+78, . 6=955
\ 58=59
Wkh suhglfwhg sryhuw| ohyho lq wkh |hdu 5348 lv
'58/5931
481 +d,
{
873
843
7<3
893
7:3
933
873
8;3
9;3
893
893
833
7:3
773
853
953
9;3
883
953
43/<73
|
53
49
43
;
45
44
43
;
48
;
46
47
43
43
44
44
;
;
:
543
{|
43> ;33
;493
7<33
77;3
8973
9933
8733
7973
43/533
77;3
:5;3
:333
7:33
7733
8:53
9;53
8773
7733
7673
448/733
{5
5<4/933
593/433
573/433
646/933
553/<33
693/333
5<4/933
669/733
795/733
646/933
646/933
583/333
553/<33
4<6/933
5:3/733
6;7/733
795/733
635/833
6;7/733
8/;:5/833
|5
733
589
433
97
477
454
433
97
558
97
49<
4<9
433
433
454
454
97
97
7<
5855
75
Fkdswhu 4
:e . 4:=8p @ 45=35
4:=8e . 83=:8p @ 6594
4<e . 43> 7<3p @ 543
43> 7<3e . 8/;:5/833p @ 448/733
4<e . 43/7<3p @ 543
4<e @ 54343/7<3p
e@
e@
:e . 4:=8p @ 45=35
:e @ 45=35 4:=8p
54343/7<3p
4<
54343/7<3p
. 8/;:5/833p @ 448/733
43/7<3
4<
43/7<3+54343/7<3p,.444/8::/833p @ 5/4<5/933
5/535/<33443/373/433p.444/8::/833p @ 5/4<5/933
4/86:/733p @ 43/633
p @ =339:
543 43/7<3+=339:,
@ 47=:8
4<
OLQHDU IXQFWLRQV
e@
4:=8
45=35 4:=8p
:
45=35 4:=8p
:
. 83=:8p @ 65=94
4:=8 +45=35 4:=8p, . 688=58p @ 55;=5:
543=68 639=58p . 688=58p @ 55;=5:
7<p @ 4:=<5
p =698:
45=35 4:=8+=698:,
=;36
:
\ @ =699{ . =;36
e@
\ @ =339:{ . 47=:8
Wkh olqh vhhpv wr qg wkh gdwd1
+e, Ohw { @ 753> qg \=
\ @ =339:+753, . 47=:8
\ @ 44=<69 45
+f, Ohw { @ 953> qg \=
\ @ =339:+953, . 47=:8
\ @ 43=8<9 44
+g,
4<+448/733, +43/7<3,+543,
s
u@s
4<+8/;:5/833 +43/7<3,5 4<+5855, 5435
:+65=94, +4:=8,+45=35,
s
+f, u @ s
:+83=:8, 4:=85 :+54=8;87, 45=355
@ =<<8> zklfk lqglfdwhv d jrrg w dqg frqupv
wkh frqfoxvlrq lq sduw +e,1
@ =4587 =46
+h, Wkhuh lv qr olqhdu uhodwlrqvkls ehwzhhq d vwx0
ghqw*v pdwk VDW dqg pdwkhpdwlfv sodfhphqw whvw
vfruhv1
4:1 +d,
43+6<<=49, +833,+53=99;,
s
4<1 +d, u @ s
43+66/583,8335 43+<4=<5:375,+53=99;,5
@ =<<8
\hv/ wkhuh grhv dsshdu wr eh d olqhdu fruuhodwlrq1
43e . 833p @ 53=99;
833e . 66/583p @ 6<<=49
43e @ 53=99; 833p
e @ 5=399; 83p
833+5=399; 83p, . 66/583p @ 6<<=49
4366=7 58/333p . 66/583p @ 6<<=49
;583p @ 967=57
p 3=3:9<
e @ 5=399; 83+=3:9<,
e 8=<4
+e,
+e,
O
4=3
4=8
5=3
5=8
6=3
6=8
7=3
4:=8
W
4=44
4=69
4=8:
4=:9
4=<5
5=3;
5=55
45=35
OW
4=44
5=37
6=47
7=7
8=:9
:=5;
;=;;
65=94
O5
4
5=58
7
9=58
<
45=58
49
83=:8
W5
4=5654
4=;7<9
5=797<
6=3<:9
6=9;97
7=6597
7=<5;7
54=8;87
\ @ 3=3:9<{ . 8=<4
+f, Ohw { @ 83
\ @ 3=3:9<+83, . 8=<4
\ 5=3:
Fkdswhu 4 Uhylhz H{huflvhv
76
Wkh suhglfwhg qxpehu ri srlqwv h{shfwhg zkhq d
whdp lv dw wkh 83 |dug olqh lv 513: srlqwv1
541 +d, Frqyhuw wkh wlph iurp krxuv dqg plqxwhv lqwr
krxuv1
S
|
98;
dyhudjh vshhg @ S @
@ 6=69
4<8=:3
{
Fkdswhu 4 Uhylhz H{huflvhv
61 Wkurxjk +5> 8, dqg +7> :,
8:
5 7
5
@
9
4
@
6
p@
Kluvw*v dyhudjh vshhg lv 6169 plohv shu krxu1
+e,
81 Wkurxjk wkh ruljlq dqg +44> 5,
p@
\hv/ wkh gdwd dsshdu wr olh dssur{lpdwho| rq d
vwudljkw olqh1
+f,
{
313333
51<833
813833
:1699:
<17666
431499:
441599:
491349:
541999:
5713333
591;333
5;1<;66
6513333
|
313
4515
4<13
5;13
6617
6919
761;
8<17
9<19
:;17
;813
<418
43414
Xvlqj d judsklqj fdofxodwru/
\ @ 6=39{ . 7=87
5
5 3
@
44 3
44
:1 5{ . 6| @ 48
6| @ 5{ . 48
5
| @ { . 48
6
5
p@
6
<1 | . 7 @ <
|@8
| @ 3{ . 8
p@3
441 | @ 6{
| @ 6{ . 3
p @ 6
461 Wkurxjk +8> 4,> vorsh
5
6
Xvh srlqw0vorsh irup1
+g, Dovr/ xvlqj d judsklqj fdofxodwru/
u @ =<<97
\hv/ wklv ydoxh lqglfdwhv d jrrg w ri wkh ohdvw
vtxduhv olqh wr wkh gdwd1
+h, D jrrg ydoxh iru Kluvw*v dyhudjh vshhg zrxog
eh
p @ 6=39 plohv shu krxu1
5
+{ 8,
6
5
| . 4 @ +{ 8,
6
| +4, @
6+| . 4, @ 5+{ 8,
6| . 6 @ 5{ 43
6| @ 5{ 46
|@
5
46
{
6
6
77
Fkdswhu 4
481 Wkurxjk +8> 5, dqg +4> 6,
p@
8
5 6
8
@
@
84
7
7
Xvh srlqw0vorsh irup1
8
+{ 8,
7
8
58
|.5@ {.
7
7
| +5, @
7| . ; @ 8{ . 58
7| @ 8{ . 4:
4:
8
| @ {.
7
7
OLQHDU IXQFWLRQV
581 | @ 9 5{
Ilqg wkh lqwhufhswv1
Ohw { @ 3=
| @ 9 5+3, @ 9
Wkh |0lqwhufhsw lv 91
Ohw | @ 3=
3 @ 9 5{
5{ @ 9
{@6
Wkh {0lqwhufhsw lv 61
Gudz wkh olqh wkurxjk +3> 9, dqg +6> 3,=
4:1 Wkurxjk +4> 7,> xqghqhg vorsh
Xqghqhg vorsh phdqv wkh olqh lv yhuwlfdo1 Wkh
htxdwlrq ri wkh yhuwlfdo olqh wkurxjk +4> 7, lv { @
4=
4<1 Wkurxjk +5> 4,> sdudooho wr 6{ | @ 4
Vroyh 6{ | @ 4 iru |=
| @ 4 6{
| @ 4 . 6{
p@6
5:1 5{ . :| @ 47
Ilqg wkh lqwhufhswv1
Zkhq { @ 3> | @ 5> vr wkh |0lqwhufhsw lv 51
Zkhq | @ 3> { @ :> vr wkh {0lqwhufhsw lv :1=
Gudz wkh olqh wkurxjk +3> 5, dqg +:> 3,=
Wkh ghvluhg olqh kdv wkh vdph vorsh1
Xvh wkh srlqw0vorsh irup1
| +4, @ 6+{ 5,
| . 4 @ 6{ 9
| @ 6{ :
541 Wkurxjk +5> 43,/ shushqglfxodu wr d olqh zlwk xq0
ghqhg vorsh
D olqh zlwk xqghqhg vorsh lv d yhuwlfdo olqh1 D
olqh shushqglfxodu wr d yhuwlfdo olqh lv d krul}rqwdo
olqh zlwk htxdwlrq ri wkh irup | @ n= Wkh ghvluhg
olqh sdvvhg wkurxjk +5> 43,> vr n @ 43= Wkxv/
dq htxdwlrq ri wkh ghvluhg olqh lv | @ 43=
561 Wkurxjk +:> 7,> shushqglfxodu wr | @ ;
Wkh jlyhq olqh/ | @ ;> lv d krul}rqwdo olqh1 D olqh
shushqglfxodu wr d krul}rqwdo olqh lv d yhuwlfdo olqh
zlwk htxdwlrq ri wkh irup { @ k=
Wkh ghvluhg olqh sdvvhv wkurxjk +:> 7,> vr k @
:= Wkxv/ dq htxdwlrq ri wkh ghvluhg olqh lv { @
:=
5<1 | @ 4
Wklv lv wkh krul}rqwdo olqh sdvvlqj wkurxjk +3> 4,=
Fkdswhu 4 Uhylhz H{huflvhv
78
641 { . 6| @ 3
Zkhq { @ 3> | @ 3=
Zkhq { @ 6> | @ 4=
Gudz wkh olqh wkurxjk +3> 3, dqg +6> 4,=
661 V+t, @ 9t . 6> G+t, @ 4< 5t
+d, V+t, @ G+t, @ 43
43 @ 9t . 6
: @ 9t
:
@t
9
+vxsso|,
43 @ 4< 5t
< @ 5t
<
@t
5
+e, V+t, @ G+t, @ 48
48 @ 9t . 6
45 @ 9t
5 @ t +vxsso|,
48 @ 4< 5t
7 @ 5t
5 @ t +ghpdqg,
Zkhq wkh sulfh lv '48 shu srxqg/ wkh vxsso| lv 5
srxqgv shu gd|/ dqg wkh ghpdqg lv 5 srxqgv shu
gd|1
+f, V+t, @ G+t, @ '4;
+h, Wkh judsk vkrzv wkdw wkh olqhv uhsuhvhqwlqj
wkh vxsso| dqg ghpdqg ixqfwlrqv lqwhuvhfw dw wkh
srlqw +5> 48,= Wkh |0frruglqdwh ri wklv srlqw jlyhv
wkh htxloleulxp sulfh1 Wkxv/ wkh htxloleulxp sulfh
lv '481
+i , Wkh {0frruglqdwh ri wkh lqwhuvhfwlrq srlqw jlyhv
wkh htxloleulxp txdqwlw|1 Wkxv/ wkh htxloleulxp
txdqwlw| lv 5/ uhsuhvhqwlqj 5 srxqgv ri fudephdw
shu gd|1
681 Xvlqj wkh srlqwv +83> 7:=83, dqg +;3> 65=83,>
p@
7:=83 65=83
48
4
@
@ =8=
@
83 ;3
5
63
s 7:=83 @ =8+t 83,
s 7:=83 @ =8t . 58
s @ =8t . :5=83
G+t, @ =8t . :5=83
6:1 Hljkw xqlwv frvw '633> {hg frvw lv '931
Wkh {hg frvw lv wkh frvw li }hur xqlwv duh pdgh1
+;> 633, dqg +3> 93, duh srlqwv rq wkh olqh1
p@
4; @ 9t . 6
48 @ 9t
+vxsso|,
4; @ 4< 5t
4 @ 5t
4
@t
5
+g,
+ghpdqg,
Zkhq wkh sulfh lv '43 shu srxqg/ wkh vxsso| lv :9
srxqgv shu gd|/ dqg wkh ghpdqg lv <5 srxqgv shu
gd|1
8
@t
5
Zkhq wkh sulfh lv 85 srxqgv shu gd|/ wkh ghpdqg
lv 45 srxqg shu gd|1
+ghpdqg,
93 633
@ 63
3;
Xvh vorsh0lqwhufhsw irup1
| @ 63{ . 93
F+{, @ 63{ . 93
Wkluw| xqlwv frvw '4833> 453 xqlwv frvw '89731
79
Fkdswhu 4
6<1 Wzhoyh xqlwv frvw '778> 83 xqlwv frvw '48;81 Srlqwv
rq wkh olqh duh +45> 778, dqg +83> 48;8,=
p@
48;8 778
@ 63
83 45
OLQHDU IXQFWLRQV
43+4/<99/346, +5:/748,+9<;,
s
u@s
43+:</483/748, +5:/748,5 43+7</:57, +9<;,5
u @ =;5;9
\hv/ wkh gdwd vhhp wr w d vwudljkw olqh/ exw qrw
dv forvh dv lq vrph rwkhu h{dpsohv1
Xvh srlqw0vorsh irup1
| 778 @ 63+{ 45,
| 778 @ 63{ 693
| @ 63{ . ;8
F+{, @ 63{ . ;8
+e,
741 F+{, @ 533{ . 4333
U+{, @ 733{
+d, F+{, @ U+{,
533{ . 4333 @ 733{
4333 @ 533{
8@{
\hv/ wkh gdwd vhhp wr w d vwudljkw olqh1
+f,
Wkh euhdn0hyhq txdqwlw| lv 8 fduwrqv1
43e . 5:/748p @ 9<;
5:/748e . :</483/748p @ 4/<99/346
43e @ 9<; 5:/748p
e @ 9<=; 5:7418p
5:/748+9<=;5:74=8p,.:</483/748p @ 4/<99/346
4/<46/89::8/48;/55518p.:</483/748p @ 4/<99/346
6/<<5/4<518p @ 85/779
p =3464
+e, U+8, @ 733+8, @ 5333
Wkh uhyhqxh iurp 8 fduwrqv ri FG*v lv '53331
761 Xvlqj wkh srlqwv +;;> ;, dqg +<8> 6<,>
6< ;
64
p@
@
7=76=
<8 ;;
:
| ; @ 7=76+{ ;;,
| ; @ 7=76{ . 6;<=;7
| @ 7=76{ . 6;4=;7
781 Xvlqj wkh srlqwv +5> 453/333, dqg +:> 498/333,>
p@
498/333 453/333
78/333
@
@ <333
:5
8
\ @ 3=464{ . 66=<
+g, Ohw { @ 647<
| 453/333 @ <333+{ 5,
| 453/333 @ <333{ 4;/333
| @ <333{ . 435/333
S +{, @ <333{ . 435/333
7:1 +d,
{
4856
59:3
5;66
6798
56<8
64;4
5;67
4;;6
5<93
69:4
5:/748
|
77
:7
:9
:<
94
:5
:4
9:
:;
:9
9<;
{|
9:/345
4<:/8;3
548/63;
5:6/:68
479/3<8
55</365
534/547
459/494
563/;;3
5:;/<<9
4/<99/346
{5
5/64</85<
:/45;/<33
;/358/;;<
45/339/558
8/:69/358
43/44;/:94
;/364/889
6/878/9;<
;/:94/933
46/7:9/574
:</483/748
e @ 9<=; 5:74=8+=3464,
e 66=<
\ @ =3464+647<, . 66=;
\ :8
|5
4<69
87:9
8::9
9574
6:54
84;7
8374
77;<
93;7
8::9
7</:57
Wkh suhglfwhg olih h{shfwdqf| lq wkh Xqlwhg Nlqj0
grp/ zlwk d gdlo| fdorulh vxsso| ri 647</ lv derxw
:8 |hduv1
Wklv lv voljkwo| orzhu wkdq wkh dfwxdo jxuh ri ::
|hduv1
7<1 Xvlqj wkh srlqwv +:3> 464=:, dqg +<5> 447=4,>
p@
4:=9
464=: 447=4
@
@ =;=
:3 <5
55
| 464=: @ =;+{ :3,
| 464=: @ =;{ . 89
| @ =;{ . 4;:=:
Fkdswhu 4 Uhylhz H{huflvhv
841 Xvlqj d judsklqj fdofxodwru1 Iroorz wkh vwhsv rxw0
olqhg lq H{dpsohv 608 lq Vhfwlrq 416 ri wkh wh{werrn1
+d, u @ =7<7
Wkh gdwd gr qrw vhhp wr w d vwudljkw olqh yhu|
forvho|1
+e,
+f, \ @ 4=:3{ . ;:=;
+g, Wkh vorsh lv 41:3 wkrxvdqg +ru 4:33,1 Rq dyhu0
djh/ wkh jryhuqru*v vdodu| lqfuhdvhv '4:33 iru hdfk
dgglwlrqdo ploolrq lq srsxodwlrq1
7:
7;
Fkdswhu 4
OLQHDU IXQFWLRQV
Fkdswhu 4 Whvw
^414`
Ilqg wkh vorsh ri hdfk olqh wkdw kdv d vorsh1
41 Wkurxjk +5> 8, dqg +4> :,
61 6{ :| @ <
51 Wkurxjk +<> 8, dqg +<> 5,
71 Shushqglfxodu wr 5{ . 8| @ :
Ilqg dq htxdwlrq lq wkh irup | @ p{ . e +zkhuh srvvleoh, iru hdfk olqh1
81 Wkurxjk +4> 9, dqg +8> 6,
;1 Wkurxjk +4> 7, > shushqglfxodu wr 6{ . | @ 4
91 {lqwhufhsw 9/ |lqwhufhsw 8
<1 Wkurxjk +5> 8, > shushqglfxodu wr { @ 8
:1 Wkurxjk +3> 6, > sdudooho wr 5{ 7| @ 4
^415`
Judsk hdfk ri wkh iroorzlqj1
431 6{ . 8| @ 48
441 5{ . | @ 3
451 { 6 @ 3
461 Ohw wkh vxsso| dqg ghpdqg ixqfwlrqv iru d fhuwdlq surgxfw eh jlyhq e| wkh iroorzlqj htxdwlrqv1
Vxsso|= s @ =53t 8
Ghpdqg= s @ 433 =48t>
zkhuh s uhsuhvhqwv wkh sulfh +lq grooduv, dw d vxsso| ru ghpdqg/ uhvshfwlyho|/ ri t xqlwv1
+d, Judsk wkhvh htxdwlrqv rq wkh vdph d{hv1
+e, Ilqg wkh htxloleulxp sulfh1
+f, Ilqg wkh htxloleulxp txdqwlw|1
^416`
471 Iru d jlyhq surgxfw/ hljkw xqlwv frvw '783/ zkloh iruw| xqlwv frvw '::31
+d, Ilqg wkh dssursuldwh olqhdu frvw ixqfwlrq1
+e, Zkdw lv wkh {hg frvwB
+f, Zkdw lv wkh pdujlqdo frvw shu lwhpB
481 Iru d jlyhq surgxfw/ wkh yduldeoh frvw lv '433/ zkloh 483 lwhpv frvw '49/333 wr surgxfh1
+d, Ilqg wkh dssursuldwh olqhdu frvw ixqfwlrq1
+e, Zkdw lv wkh {hg frvwB
+f, Zkdw lv wkh pdujlqdo frvw shu lwhpB
Fkdswhu 4 Whvw
7<
491 Surgxflqj { kxqguhg xqlwv ri zlgjhwv frvwv F +{, @ 7{ . 49> uhyhqxh lv 45{> zkhuh U +{, dqg F +{, duh lq
wkrxvdqgv ri grooduv1
+d, Zkdw lv wkh euhdn0hyhq txdqwlw|B
+e, Zkdw lv wkh surw iurp 633 xqlwvB
+f, Krz pdq| xqlwv zloo surgxfh d surw ri '73/333B
^417`
4:1 Dq hohfwurqlfv up zdv sodqqlqj wr h{sdqg lwv surgxfw olqh dqg zdqwhg wr jhw dq lghd ri wkh vdodu| slfwxuh iru
whfkqlfldqv lw zrxog kluh lq wklv hog1 Wkh iroorzlqj gdwd zdv froohfwhg1
S 5
q @ 45
{ @ 6495
S
S 5
{ @ 4:9
| @ 43/;:3
S
| @ 689
S
{| @ 895<
+d, Ilqg dq htxdwlrq iru wkh ohdvw vtxduhv olqh1
+e, Ilqg wkh frh!flhqw ri fruuhodwlrq1
4;1 Dq hfrqrplvw zdv lqwhuhvwhg lq wkh surgxfwlrq frvwv iru frpsdqlhv vxsso|lqj fkhplfdov iru xvh lq ihuwlol}huv1
Wkh gdwd ehorz uhsuhvhqwv wkh uhodwlrqvkls ehwzhhq wkh qxpehu ri wrqv surgxfhg gxulqj d jlyhq |hdu +{, dqg
wkh surgxfwlrq frvw shu wrq +|, iru vhyhq frpsdqlhv1
Qxpehu ri Wrqv
+lq wkrxvdqgv,
Frvw shu Wrq
+lq grooduv,
613
713
517
813
519
713
818
73
83
83
68
88
68
63
+d, Ilqg wkh htxdwlrq iru wkh ohdvw vtxduhv olqh1
+e, Ilqg wkh frh!flhqw ri fruuhodwlrq1
83
Fkdswhu 4
OLQHDU IXQFWLRQV
Fkdswhu 4 Whvw Dqvzhuv
41 7
451
51 Xqghqhg
61
6
:
71
8
5
81 | @ 65 { .
<
5
461 +d,
91 | @ 89 { 8
:1 | @ 45 { . 6
;1 | @ 46 { . 6
<1 | @ 8
431
+e, '88 +f, 633 xqlwv
471 +d, | @ 43{ . 6:3
+e, '6:3
+f, '43
481 +d, | @ 433{ . 4333
+e, '4333
+f, '433
441
491 +d, 533 +e, ';333 +f, :33
4:1 +d, \ @ =:3{ . 4<=6: +e, u @ =<9
4;1 +d, \ @ 9=6:{ . 99=57 +e, u @ =:<