## 11 Geodesic Curvature

### 11.1 V-Coordinate System

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

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

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

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

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

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

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

## 12 Gauss-Bonnet Theorem

### 12.1 U-Coordinate System

 Figure 2: Tangent to Curve

### 12.2 V-Coordinate System

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

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

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

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

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

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

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

KSurface vpgen''
1.31588

geocCurve vpgen''
1.31783

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Contributed by TomAllen

Essays/Gauss-Bonnet/Bonnet04 (last edited 2008-12-08 10:45:44 by anonymous)