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 Post subject: Hyperboloid equation in different coordinatesPosted: Mon, 14 Nov 2011 21:49:34 UTC
 S.O.S. Newbie

Joined: Mon, 14 Nov 2011 21:34:36 UTC
Posts: 3
Hello,
I currently have an expression of an Hyperboloid equation :

Where a,b,c,d are some real constants.

I would like to know what is the best way to transform it into the standard Hyperboloid, i.e .
(which can be written as v.Av=1 considering v=(x,y,z))

I was thinking of putting u=(u1,u2,u3), so we have to transform u in v, i.e we have to find some P invertible matrix, and a u0 ( shift ) so we have u=Pv+u0 and then we can identify to my equation
but this method requires so many variables, i have a system with way too many variables (12, 9 from the matrix and 3 form the shift) and i can obtain only 10 equations from identification, so i have to choose some coefficients.

Is there a better way to find this matrix P and this u0 / an other way of considerating things?

Thanks again, and sorry for my bad english
E.F

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 Post subject: Re: Hyperboloid equation in different coordinatesPosted: Tue, 15 Nov 2011 09:04:23 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
repptilia wrote:
Hello,
I currently have an expression of an Hyperboloid equation :

Where a,b,c,d are some real constants.

I would like to know what is the best way to transform it into the standard Hyperboloid, i.e .
(which can be written as v.Av=1 considering v=(x,y,z))

I was thinking of putting u=(u1,u2,u3), so we have to transform u in v, i.e we have to find some P invertible matrix, and a u0 ( shift ) so we have u=Pv+u0 and then we can identify to my equation
but this method requires so many variables, i have a system with way too many variables (12, 9 from the matrix and 3 form the shift) and i can obtain only 10 equations from identification, so i have to choose some coefficients.

Is there a better way to find this matrix P and this u0 / an other way of considerating things?

Thanks again, and sorry for my bad english
E.F

Are you sure you have both and ?

Start by diagonalising the quadratic form , so you have an orthogonal matrix such that , where . Now if , so , which because of the form of , it is easy to get rid of the linear terms, leaving you with , some constant c. Thus .

_________________

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 Post subject: Re: Hyperboloid equation in different coordinatesPosted: Tue, 15 Nov 2011 12:40:17 UTC
 S.O.S. Newbie

Joined: Mon, 14 Nov 2011 21:34:36 UTC
Posts: 3
Okay, thank you very much, i will try to compute these equations.
You were right, the equation is .
Can you be more precise when you say i have to find a orthogonal matrix?
Should i solve the equations manually? I want a litteral expression.

Anyway it's much more clear now thank you

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