Hyperboloid equation in different coordinates

repptilia · **Joined:** Mon, 14 Nov 2011 21:34:36 UTC **Posts:** 3

Hello,
I currently have an expression of an Hyperboloid equation :
u_1u_2-u_1c-u_2a-u_2d-u_1u_3d-u_2u_3b=0

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 x^2/a^2+y^2/b^2-z^2/c^2=1

.
(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

outermeasure · **Posted:** Tue, 15 Nov 2011 09:04:23 UTC

repptilia wrote:

Hello,
I currently have an expression of an Hyperboloid equation :
u_1u_2-u_1c-u_2a-u_2d-u_1u_3d-u_2u_3b=0

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 x^2/a^2+y^2/b^2-z^2/c^2=1

.
(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 -u_2a

and

?

Start by diagonalising the quadratic form Q(u_1,u_2,u_3)=u_1u_2-du_1u_3-bu_2u_3

, so you have an orthogonal matrix

such that $Q(\mathbf{u})=\Lambda(P\mathbf{u})$ , where $\Lambda(v_1,v_2,v_3)=\lambda_1 v_1^2+\lambda_2v_2^3+\lambda_3 v_3^2$ . Now if $\tilde{Q}(u_1,u_2,u_3)=u_1u_2-u_1c-u_2a-u_2d-u_1u_3d-u_2u_3b$ , so $\tilde{Q}(\mathbf{u})=\tilde{\Lambda}(P\mathbf{u})=\Lambda(P\mathbf{u})+\text{linear}$ , which because of the form of $\Lambda$ , it is easy to get rid of the linear terms, leaving you with $\tilde{Q}(\mathbf{u})=\Lambda(P\mathbf{u}-\mathbf{v}^{(0)})-c$ , some constant c. Thus $\tilde{Q}(\mathbf{u})=0\iff c^{-1}\Lambda(P\mathbf{u}-\mathbf{v}^{(0)})=1$ .

repptilia · **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 u_1u_2-u_1c-u_2a-u_1u_3d-u_2u_3b=0

.
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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Hyperboloid equation in different coordinates

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