View Single Post
 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 $$S(u,v)$$ can be defined parametrically by: $$x = (c + a \ cos(v) ) \ cos(u)$$ $$y = (c + a \ cos(v) ) \ sin(u)$$ $$z = a \ sin(v)$$ where u and v $$\in [0, 2 \pi ]$$ The coefficients E, F, and G of the first fundamental form (Line Element) are : $$S_u = \frac{\partial S}{\partial u} = ( \ -(c + a \ cos(v) ) sin(u) \ , \ (c + a \ cos(v) ) \ cos(u) \ , \ 0 \ )$$ $$S_v = \frac{\partial S}{\partial v} = ( \ -(a \ cos(u) \ sin(v) ) \ , \ -(a \ sin(u) \ sin(v) ) \ , \ a \ cos(v) \ )$$ Therefore, $$E = \frac{\partial S}{\partial u} \ . \ \frac{\partial S}{\partial u} = ( - (c + a \ cos(v) ) \ sin(u) )^2 \ + \ (( c + a \ cos(v) ) cos(u) )^2 \ + \ 0 = ( c + a \ cos(v) )^2$$ $$F = \frac{\partial S}{\partial u} \ . \ \frac{\partial S}{\partial v} = ( - (c + a \ cos(v) ) \ sin(u) ) \ -(a \ cos(u) \ sin(v) ) \ + \ (( c + a \ cos(v) ) cos(u) ) \ -(a \ sin(u) \ sin(v) ) \+ \ (0)\ a \ cos(v) \ = 0$$ $$G = \frac{\partial S}{\partial v} \ . \ \frac{\partial S}{\partial v} = ( \ -(a \ cos(u) \ sin(v) ) \ )^2 \ + \ (\ -(a \ sin(u) \ sin(v) ) \ )^2 \ + \ ( \ a \ cos(v) \ )^2 = a^2$$ The line element ds^2 (s here is an arc length) is : $$ds^2 = E \ du^2 \ + \ 2 \ F \ du \ dv \ + G \ dv^2 \$$ $$ds^2 = ( c + a \ cos(v) )^2 \ du^2 \ + a^2 \ dv^2 \$$ The metric is $$g_{ij}$$ is : $$g_{ij} = \left [ \begin{array}{ccc} ( c + a \ cos(v) )^2 & 0 \\ 0 & a^2 \end{array}\right ]$$ $$g^{ij} = \left [ \begin{array}{ccc} \frac{1}{( c + a \ cos(v) )^2} & 0 \\ 0 & \frac{1}{a^2} \end{array}\right ]$$