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Thread: guassian curvature over zero

  1. #1
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    Unhappy guassian curvature over zero

    The guassian curvature of a surface is $\displaystyle (LM - N^2) / (EG - F^2)$. Basically

    i am doing a question where i found that $\displaystyle (EG - F^2)=0$ , so i got $\displaystyle (LN - M^2)/ 0 $.

    i was wondering if anyone knows what this means for guassian curvature or the surface as it is divided by zero.
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  2. #2
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    Quote Originally Posted by dopi View Post
    i was wondering if anyone knows what this means for guassian curvature or the surface as it is divided by zero.
    I would say this means you made a mistake, or that your surface is not smooth... If the surface is given locally by $\displaystyle (u,v)\mapsto X(u,v)$, then applying Cauchy-Schwarz inequality to $\displaystyle \partial_u X\cdot \partial_v X$ gives $\displaystyle EG\geq F^2$, and the equality implies that the vectors $\displaystyle \partial_u X,\ \partial_v X$ are colinear. However, this is supposed not to happen for a parameterization of a surface (the differential must be injective: the partial derivatives span the tangent plane).
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  3. #3
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    Quote Originally Posted by Laurent View Post
    I would say this means you made a mistake, or that your surface is not smooth... If the surface is given locally by $\displaystyle (u,v)\mapsto X(u,v)$, then applying Cauchy-Schwarz inequality to $\displaystyle \partial_u X\cdot \partial_v X$ gives $\displaystyle EG\geq F^2$, and the equality implies that the vectors $\displaystyle \partial_u X,\ \partial_v X$ are colinear. However, this is supposed not to happen for a parameterization of a surface (the differential must be injective: the partial derivatives span the tangent plane).
    here is my original quesiton

    alpha(u) is a unit speed space curve, and use this curve to construct a tangent developable surface with $\displaystyle chart (x,U) $where

    $\displaystyle x(u,v) = alpha(u) + v*(alpha'(u)), (u,v) in U$

    where $\displaystyle U = {(u,v) in R^2 : -infinity < u < infinity, v>0}$

    so i want to work out the guassian curvature of this surface.

    my solution just for E, F , G

    $\displaystyle E = Xu . Xu = 1 + (v^2)*(alpha''(u)^2) +2*v*t*alpha''(u)$

    $\displaystyle F = Xu . Xv = ! + v*t*alpha''(u)$

    $\displaystyle G = t.t =1$
    and
    $\displaystyle F^2 = 1 + (v^2)*(alpha''(u)^2) +2*v*t*alpha''(u)$

    so $\displaystyle EG - F^2 = 0$

    where i used t = alpha'(u) from frenet serret equation, as txt =0 and t.t =1

    im not what i done is right but thats how i got EG - f^2 for the guassian curvature as zero
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  4. #4
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    Quote Originally Posted by dopi View Post
    here is my original quesiton

    alpha(u) is a unit speed space curve, and use this curve to construct a tangent developable surface with $\displaystyle chart (x,U) $where

    $\displaystyle x(u,v) = alpha(u) + v*(alpha'(u)), (u,v) in U$

    where $\displaystyle U = {(u,v) in R^2 : -infinity < u < infinity, v>0}$

    so i want to work out the guassian curvature of this surface.

    my solution just for E, F , G

    $\displaystyle E = Xu . Xu = 1 + (v^2)*(alpha''(u)^2) +2*v*t*alpha''(u)$

    $\displaystyle F = Xu . Xv = ! + v*t*alpha''(u)$

    $\displaystyle G = t.t =1$
    and
    $\displaystyle F^2 = 1 + (v^2)*(alpha''(u)^2) +2*v*t*alpha''(u)$

    so $\displaystyle EG - F^2 = 0$

    where i used t = alpha'(u) from frenet serret equation, as txt =0 and t.t =1

    im not what i done is right but thats how i got EG - f^2 for the guassian curvature as zero
    There are mistakes in your final computation. Here are the details:
    You have $\displaystyle X_u=T+v\kappa N$ and $\displaystyle X_v=T$. Since $\displaystyle (T,N,B)$ is orthonormal, $\displaystyle \|X_u\|^2=1+(v\kappa)^2$, $\displaystyle \|X_v\|^2=1$ and $\displaystyle X_u\cdot X_v=1$.
    Then $\displaystyle EG-F^2=(1+\kappa^2v^2)-1=\kappa^2v^2$.


    Note: you can notice that $\displaystyle EG-F^2=\|X_u\times X_v\|^2$. This is always true, so that your previous question already gave you the result...
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  5. #5
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    Question

    Quote Originally Posted by Laurent View Post
    There are mistakes in your final computation. Here are the details:
    You have $\displaystyle X_u=T+v\kappa N$ and $\displaystyle X_v=T$. Since $\displaystyle (T,N,B)$ is orthonormal, $\displaystyle \|X_u\|^2=1+(v\kappa)^2$, $\displaystyle \|X_v\|^2=1$ and $\displaystyle X_u\cdot X_v=1$.
    Then $\displaystyle EG-F^2=(1+\kappa^2v^2)-1=\kappa^2v^2$.


    Note: you can notice that $\displaystyle EG-F^2=\|X_u\times X_v\|^2$. This is always true, so that your previous question already gave you the result...
    thanks for that it makes more sense now, could u check my answer for L,N,M and M^2

    L= n.Xuu = n.(alpha''(u) + v*alpha'''(u))
    M= n.Xuv = n.alpha''(u)
    N= n.Xvv = n.0 = 0

    so therefore LN - M^2= -M^2 , as L=0

    therefore -M^2 = -(n.alpha''(u))^2

    so im assuming n.n =1 and therefore -M^2 = -alpha''(u)^2

    thanks for the response
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  6. #6
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    Quote Originally Posted by dopi View Post
    thanks for that it makes more sense now, could u check my answer for L,N,M and M^2

    L= n.Xuu = n.(alpha''(u) + v*alpha'''(u))
    M= n.Xuv = n.alpha''(u)
    N= n.Xvv = n.0 = 0

    so therefore LN - M^2= -M^2 , as L=0

    therefore -M^2 = -(n.alpha''(u))^2

    so im assuming n.n =1 and therefore -M^2 = -alpha''(u)^2

    thanks for the response
    Hi,
    I don't understand how you got $\displaystyle L=0$. You didn't give your formula for $\displaystyle n$, and that's probably where there's a problem.
    Using Serret-Frenet coordinates (like in my previous post), we see that the tangent vectors $\displaystyle \partial_u X$ and $\displaystyle \partial_v X$ are linear combinations of $\displaystyle T$ and $\displaystyle N$. Therefore, $\displaystyle n=\pm B$ (the sign depends on the sign of the curvature, I let you find out).

    After computations, I find $\displaystyle L=v\kappa\tau$ (where $\displaystyle \tau$ is the torsion), and $\displaystyle N=M=0$. Can you find the same?
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