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C"l c,,lus qllô!{ti.*
The Shoctos'f Porlh B¿ftec" Tto P"n{s
Consìder tto poì;ts (X^, yJ qncl (Xt, y!
in *he (x,y) Plane
Y¡
Yr
Whot curve Y (x)
shor*est pclh le.gth 5ttne "ndr #oi"*s (år7¡)
hos *hehdtw"e"t
"rå Cx",y.Í
How cqn v{e Prove Thisl
G, qlcul%, qf Voriotlo"s
The Sl'orTe$ N+l- sogr"ait P"{h B"-f.,o", Jto Pd"ts
Lonsiåer tt" poin-ts (Xar)^) qnd (Xe,yJìn {he (X,y)
!tone
Whdt vqlues of fy,¡z,!,,*,yJ ]eaå ù, {heShortast N+l- segnen* po+|" l."bl,h^ 5 between*he end poln'ts (n, Y^) ä"¿ (Xg, yr)1
Flow co¡n ïúg Pcove Ìhìs'l
C.^lculos of V"ritìons
The -|o+"1 pa{L, ler'g{h s in *he N+t - segwrent
cqse îst!t
S = 5,+ä{ ea'; ç5u *Sr.tr = u7 tt
whece, 5¡ =
En¿ Pa¡;ts (&, y)= (Xa, ye)C**,, y*ù = Cir, yù
hl, L0, G $re ¿o\n Sgt f,o = " = O
Xr+¡ c' ! I*, . h
v{e, tqKe X¡ -X.¡-, = AX = }J:íi .[ V;of1,-ru+lJ
rr+l
-
|
+ g= 5(X,y,.-,yr)=;t
5 ìa q f"""tìon dF Ï,, Í", -, Yr(of .Ñ' ./qcìo,bhJ
C q \ culus oî V"ci¿tio"?
Ouc Aool0c cuthen,rnlnìrnìzpå
ls -to rninì¡trìze S dt S,nìn.
-fr ênst¡re +ht S ls
"t 5(y,,Yr,- yn) = s'i.r.
As w¿ dìÅ Êcevìooslv. we öre fcee-l! choose '*heninitrul,ri dt (y,,lz-,-
, yr)pcovtåeâ we p"q^o*fize qrbì*rqcfVqrio*ìon5 owqy from {he mìnìrnurn-
Tl'r"1 ls v'rs wrìtet f = !*q?;
5o +hfrt 5 ( Y', Y., .,, , Y") = 5(y,+ q?, Y.+"(%¿ r
for ct Vori,itìon 'l,,alr,,- ,qln wïlh f;xeJ ,ofr,r' ' Y" + AfJn)
ìrúc Øn oonsidec $ = 5(d)
S(d)=g9+8å1",* +.i..5da
A nec¿ssc,ry c!ndîtion foc 5(y,,y',-,Y")= 5(¿=o)= 5¡n
*Lro
Cqlcrlrs of Vqri"lions
e
?o
s(q)
By+r^e chor,, rure *[ = Z&1"* = å&Lr,*1" = o =,?#,1, 1
ocbirrary! ffi|" = o Jj" 'tã,o.ny 'I
N Al ge brorlc E ttlon.f, foc
æ1":åLffiW)t+l f yi- yi-l-¡Í"try
solv-tlon + y; -li-t = /t*t-)i VioL.,r¡{
l" V (xi,yt)
iltt=2
¿ul
Y,,Y\ "., rYt
ñowYí-X-t. 3/r,
rJ¡x\ cyi-X.)t 37i
ã;,;
Y=
C"l c,,lus qllô!{ti.*
The Shoctos'f Porlh B¿ftec" Tto P"n{s
Consìder tto poì;ts (X^, yJ qncl (Xt, y!
in *he (x,y) Plane
Y¡
Yr
Whot curve Y (x)
shor*est pclh le.gth 5ttne "ndr #oi"*s (år7¡)
hos *hehdtw"e"t
"rå Cx",y.Í
How cqn v{e Prove Thisl
CqÌcrlrs of Vqríqtions.
The toto,\ pnlh length s is
s = l^ rt = f^rtsìnce ds = JímfEd Pot;ls (Xr,y^) onô (y¡,y¡) qr¿ fix"d
W.L, O, C we c^oy, ggt X¡ = Y^ = O
Xs=l )e =h
We cqn choose c,ny p"th y(x) \^re l¡ t<e
qnd 5 drepends oh *he eritìre fun¿1iony(r) (o.1".tV /'rx) ln *hfs corse). We shoold
sovnehow de no{e {åts qnd hrg wrì{e
g= slvd = liår.l[tæ
Sis Yrx)fof I f.lrrc,tiô
q func+ïonql of