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Thread: Points on the surface

  1. #1
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    Question Points on the surface

    Points on a surface can be: elliptic, hyperbolic, parabolic, (flat or planar)
    I have the surface
    \vec{r}(u,v)=( sh(u) , ch(u)*cos(v) ,ch(u) *sin(v) )
    sh(u)=sinh(u), ch(u)=cosh(u)
     \vec{r'_{u}}(u,v)=( ch(u) , sh(u)*cos(v) ,sh(u) *sin(v) )
     \vec{r'_{v}}(u,v)=( 0 , -ch(u)*sin(v) ,ch(u) *cos(v) )
     \vec{r''_{u^2}}(u,v)=( sh(u) , ch(u)*cos(v) ,ch(u) *sin(v) )
     \vec{r''_{u*v}}(u,v)=( 0 , -sh(u)*sin(v) ,sh(u) *cos(v) )
     \vec{r''_{v^2}}(u,v)=( 0 , -ch(u)*cos(v) ,-ch(u)*sin(v) )
    E=ch^2(u)+sh^2(u)*cos^2(v)+sh^2(u)*sin^2(v)=ch^2(u  )+sh^2(u);
    F=0;G=ch^2(u);H=ch(u)*{(ch^2(u)+sh^2(u))}^{(1/2)}
     D=(\frac{1}{H})*(-ch(u)) ; D'=0 ; D''=(\frac{1}{H})*{(ch^3(u))};
    Nature: D*D''-D'^2=(\frac{(-ch^4(u))}{(H^2)})=(\frac{-ch^2(u)}{(ch^2(u)+sh^2(u))})<0\forall{u}
    So all points are hyperbolic?
    Here is a graph : Imgur: The most awesome images on the Internet
    Last edited by Valer; Dec 30th 2016 at 02:43 PM.
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  2. #2
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    Re: Points on the surface

    All points are hyperbolic? Yes or No. An answer is greatly appreciated.
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