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Mehdi_
Mehdi_ is offline
#32
Oct19-06, 04:31 PM
P: 65
The Line Element and Metric of a torus

If the major radius of this torus is c and the minor radius a ; with c>a .
The torus [tex]S(u,v)[/tex] can be defined parametrically by:

[tex] x = (c + a \ cos(v) ) \ cos(u) [/tex]
[tex] y = (c + a \ cos(v) ) \ sin(u) [/tex]
[tex] z = a \ sin(v) [/tex]

where u and v [tex] \in [0, 2 \pi ] [/tex]

The coefficients E, F, and G of the first fundamental form (Line Element) are :

[tex] S_u = \frac{\partial S}{\partial u} = ( \ -(c + a \ cos(v) ) sin(u) \ , \ (c + a \ cos(v) ) \ cos(u) \ , \ 0 \ ) [/tex]
[tex] S_v = \frac{\partial S}{\partial v} = ( \ -(a \ cos(u) \ sin(v) ) \ , \ -(a \ sin(u) \ sin(v) ) \ , \ a \ cos(v) \ ) [/tex]

Therefore,

[tex] E = \frac{\partial S}{\partial u} \ . \ \frac{\partial S}{\partial u} = ( - (c + a \ cos(v) ) \ sin(u) )^2 \ + \ (( c + a \ cos(v) ) cos(u) )^2 \ + \ 0 = ( c + a \ cos(v) )^2 [/tex]

[tex] F = \frac{\partial S}{\partial u} \ . \ \frac{\partial S}{\partial v} = ( - (c + a \ cos(v) ) \ sin(u) ) \ -(a \ cos(u) \ sin(v) ) \ + \ (( c + a \ cos(v) ) cos(u) ) \ -(a \ sin(u) \ sin(v) ) \+ \ (0)\ a \ cos(v) \ = 0 [/tex]

[tex] G = \frac{\partial S}{\partial v} \ . \ \frac{\partial S}{\partial v} = ( \ -(a \ cos(u) \ sin(v) ) \ )^2 \ + \ (\ -(a \ sin(u) \ sin(v) ) \ )^2 \ + \ ( \ a \ cos(v) \ )^2 = a^2 [/tex]

The line element ds^2 (s here is an arc length) is :

[tex] ds^2 = E \ du^2 \ + \ 2 \ F \ du \ dv \ + G \ dv^2 \ [/tex]
[tex] ds^2 = ( c + a \ cos(v) )^2 \ du^2 \ + a^2 \ dv^2 \ [/tex]

The metric is [tex] g_{ij} [/tex] is :

[tex] g_{ij} = \left [ \begin{array}{ccc} ( c + a \ cos(v) )^2 & 0 \\ 0 & a^2 \end{array}\right ] [/tex]

[tex] g^{ij} = \left [ \begin{array}{ccc} \frac{1}{( c + a \ cos(v) )^2} & 0 \\ 0 & \frac{1}{a^2} \end{array}\right ] [/tex]