- #1
Mr.Brown
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Tangent space of single layered hyperboloid
Ok i´m given a single layered hyperboloid given by [tex]\left(\frac{x}{a}-\frac{z}{c}\right)\cdot\left(\frac{x}{a}+\frac{z}{c}\right)-\left(1-\frac{y}{b}\right)\cdot\left(1+\frac{y}{b}\right)=0[/tex]
Now the Problem asks me to take this as a vanishing determinant, i guess they mean
[tex]\begin{vmatrix} \frac{x}{a}-\frac{z}{c} &1-\frac{y}{b} \\ 1+\frac{y}{b} & \frac{x}{a}+\frac{z}{c} \end{vmatrix}=0[/tex]
Now out of the knowledge that the surface is parametrized by a vanishing determinant i should find 2 streight lines through every point p in the surface.
Now I am totally lost how to do it by the determinant.
Im perfectly fine with just taking the tangent space at p and the just look for the direction vectors that produce lines entirely in the surface but how to do it this way.
I thought of i as a Jacobian of some function (of a,b,c ?? )for whitch the hyperboloid is sort of a level set but i don´t know how to go on sorry some hint would be fine :)
Thanks
Ok i´m given a single layered hyperboloid given by [tex]\left(\frac{x}{a}-\frac{z}{c}\right)\cdot\left(\frac{x}{a}+\frac{z}{c}\right)-\left(1-\frac{y}{b}\right)\cdot\left(1+\frac{y}{b}\right)=0[/tex]
Now the Problem asks me to take this as a vanishing determinant, i guess they mean
[tex]\begin{vmatrix} \frac{x}{a}-\frac{z}{c} &1-\frac{y}{b} \\ 1+\frac{y}{b} & \frac{x}{a}+\frac{z}{c} \end{vmatrix}=0[/tex]
Now out of the knowledge that the surface is parametrized by a vanishing determinant i should find 2 streight lines through every point p in the surface.
Now I am totally lost how to do it by the determinant.
Im perfectly fine with just taking the tangent space at p and the just look for the direction vectors that produce lines entirely in the surface but how to do it this way.
I thought of i as a Jacobian of some function (of a,b,c ?? )for whitch the hyperboloid is sort of a level set but i don´t know how to go on sorry some hint would be fine :)
Thanks
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