Essays/Gauss-Bonnet/Bonnet04

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11 Geodesic Curvature

... McCONNELL Chapter XIV Section 10 ...
... Sokolnikoff Section 63 ...


11.1 V-Coordinate System
GGB0401C.jpg


NB. ... script (ijs) ...

tgnt=:(0,mcv22^_0.5"_)"1


GGB0402C.jpg


GGB0403C.jpg


NB. ... script (ijs) ...

tgntdrv1=:((0{1{1{ch2k)%mcv22)"1
tgntdrv2=:((_0.5*(mcv22^_2:)*mcv22v2)+(1{1{1{ch2k)%mcv22)"1
tgntdrv =:(tgntdrv1,tgntdrv2)"1


GGB0404C.jpg


NB. ... execute (ijx) ...

   *./@(*./)@(0=])@ts0@:(+/@((<0 1)|:axs@((1;mcv),1;tgnt*/tgntdrv))"1) vpgen''
1


GGB0405C.jpg


NB. ... script (ijs) ...

geoc=:+/@((<0 1)|:axs@((1;ecv),1;tgnt*/tgntdrv))"1


12 Gauss-Bonnet Theorem

... McCONNELL Chapter XIV EXAMPLES XIV Problem 14 ...
... Sokolnikoff Section 74 ...


12.1 U-Coordinate System
GBplotD.jpg
Figure 2: Tangent to Curve


GGB0407C.jpg


GGB0408C.jpg


GGB0409C.jpg


GGB0410C.jpg


12.2 V-Coordinate System
GGB0411C.jpg


NB. ... script (ijs) ...

dS=:%:@m*((f1-s1)%i1)*(f2-s2)%i2
ds=:%:@mcv22*(f2-s2)%i2


GGB0412C.jpg


NB. ... script (ijs) ...

KSurface=:(+/%2:)@:(+/@(+/)@:((K*dS)"1)"3)@(}.,:}:)@:(}:"2)


GGB0413C.jpg


NB. ... script (ijs) ...

geocCurve=:-/@:(+/"1)@:((geoc*ds)"1)@(}:"2@({.,:{:))


GGB0414C.jpg


NB. ... execute (ijx) ...

   KSurface vpgen''
1.31588

   geocCurve vpgen''
1.31783



Download: File:LGB0401C.txt
Download: File:LGB0402C.txt
Download: File:LGB0403C.txt
Download: File:LGB0404C.txt
Download: File:LGB0405C.txt
Download: File:LGB0407C.txt
Download: File:LGB0408C.txt
Download: File:LGB0409C.txt
Download: File:LGB0410C.txt
Download: File:LGB0411C.txt
Download: File:LGB0412C.txt
Download: File:LGB0413C.txt
Download: File:LGB0414C.txt




Download MoinMoin source: File:Bonnet04.ijs




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Contributed by Tom Allen