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