- #1
Tchakra
- 13
- 0
Here is a question i am not sure how to tackle, I am not familiar with how to deal with family of curves and don't really have much time to look around for the definition as i am sitting the exam in two days.
(this link has an image of the problem)
img126.imageshack.us/img126/807/diffgeombs3.png
hmmm, anything from differential geometry ie fundamental forms, theory of surfaces ...
The question is divided into three parts: Here is my attempt any help appreciated.
1) I am have no idea, i think it is a case of knowing the definition and i don't.
2) It is simply constraining the local parametrization to the given function so:
[itex] xz-hy=> v*sin(u)=h(1-cos(u))=> h= v*sin(u)/(1-cos(u))[/itex] which is a constant.
3) [itex]\psi(u,v)=const[/itex] is like phi therefore the tangent vectors to the family defined by the psi are of the multiples of [itex]\psi_{v}x_{u}-\psi_{u}x_{v}[/itex]
So for the families to be orthogonal their tangent must be orthogonal and so[itex](\psi_{v}x_{u}-\psi_{u}x_{v}).(\phi_{v}x_{u}-\phi_{u}x_{v})=0[/itex]
Using the fundamental forms E=1=G and F=0 we get [itex]\psi_{v}\phi_{v}+\psi_{u}\phi_{u}=0[/itex] which after differentiating gives [itex]\psi_{v}\sin(u)-\psi_{u}v=0[/itex]And after that i am stuck ... any help would be appreciated.
Homework Statement
(this link has an image of the problem)
img126.imageshack.us/img126/807/diffgeombs3.png
Homework Equations
hmmm, anything from differential geometry ie fundamental forms, theory of surfaces ...
The Attempt at a Solution
The question is divided into three parts: Here is my attempt any help appreciated.
1) I am have no idea, i think it is a case of knowing the definition and i don't.
2) It is simply constraining the local parametrization to the given function so:
[itex] xz-hy=> v*sin(u)=h(1-cos(u))=> h= v*sin(u)/(1-cos(u))[/itex] which is a constant.
3) [itex]\psi(u,v)=const[/itex] is like phi therefore the tangent vectors to the family defined by the psi are of the multiples of [itex]\psi_{v}x_{u}-\psi_{u}x_{v}[/itex]
So for the families to be orthogonal their tangent must be orthogonal and so[itex](\psi_{v}x_{u}-\psi_{u}x_{v}).(\phi_{v}x_{u}-\phi_{u}x_{v})=0[/itex]
Using the fundamental forms E=1=G and F=0 we get [itex]\psi_{v}\phi_{v}+\psi_{u}\phi_{u}=0[/itex] which after differentiating gives [itex]\psi_{v}\sin(u)-\psi_{u}v=0[/itex]And after that i am stuck ... any help would be appreciated.