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WMDhamnekar

MHB

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How to prove that every quadric surface can be translated and/or rotated so that its equation matches one of the six types of quadric surfaces namely 1) Ellipsoid

2)Hyperboloid of one sheet

3) Hyperboloid of two sheet 4)Elliptic Paraboloid

5) Elliptic Cone 6) Hyperbolic Paraboloid

The equation of hyperbolic parabolid is given by $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{z}{c}\tag{1}$

Now, for example, z=2*x*y is a case of equation of mixed variables of the form $Ax^2+By^2+Cz^2+Dxy+ Exz +Fyz + Gx + Hy +Iz + J =0$ with e.g $D\not=0$ so that we get

That is z=2xy is a hyperbolic paraboloid as in equation (1) but rotated $45^\circ$ in the xy-plane.How to prove \(z=2*x*y\) is a hyperbolic paraboloid?

If any member knows answer to this question may reply.

2)Hyperboloid of one sheet

3) Hyperboloid of two sheet 4)Elliptic Paraboloid

5) Elliptic Cone 6) Hyperbolic Paraboloid

The equation of hyperbolic parabolid is given by $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{z}{c}\tag{1}$

Now, for example, z=2*x*y is a case of equation of mixed variables of the form $Ax^2+By^2+Cz^2+Dxy+ Exz +Fyz + Gx + Hy +Iz + J =0$ with e.g $D\not=0$ so that we get

*xy*term. This equation does not match any of the equations of quadric surfaces, we know.However, by rotating the x- and y- axes by $ 45^\circ $ in the xy-plane by means of coordinate transformation $x=\frac{(x'-y')}{\sqrt{2}},y=\frac{(x'+y')}{\sqrt{2}}, z=z'$ then $ z=2xy $ becomes hyperbolic paraboloid $z'=(x')^2-(y')^2$ in the**(x',y',z')**coordinate system.That is z=2xy is a hyperbolic paraboloid as in equation (1) but rotated $45^\circ$ in the xy-plane.How to prove \(z=2*x*y\) is a hyperbolic paraboloid?

If any member knows answer to this question may reply.

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