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#1 2013-05-16 04:31:49

Dets65
Member
Registered: 2013-05-15
Posts: 7

X^2 + XY Hyperbolic paraboloid

WolframAlpha says that X^2 + XY is a Hyperbolic paraboloid. Is it correct? It looks more like a foldy piece of paper than the signature saddle to me.

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#2 2013-05-16 07:16:20

SteveB
Member
Registered: 2013-03-07
Posts: 595

Re: X^2 + XY Hyperbolic paraboloid

I have looked at the Wikipedia definition of a "hyperbolic paraboloid" and it gives an equation of:

My immediate intuitive thought is that the expression you gave cannot be rearranged into this.
However what about a rotated version of this ?

The equation that Wikipedia gives for this is:

If a = b then it is not exactly the same as your expression, but bares some resemblance with an extra x^2 term.

If a is not equal to b in magnitude then a y^2 term seems to be needed to make it a proper fit to the equation.

At the time of writing I am not sure whether the expression you gave is another rotation or not possible with any rotation.
My guess at the moment is that it is not possible with any rotation. I do not know how to rotate the equation by an angle
in the +z direction in general, but I would have thought the y^2 term would be non zero.
This is just a guess though so I wonder what other members think about this.

Last edited by SteveB (2013-05-16 18:46:17)

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#3 2013-05-17 07:13:41

SteveB
Member
Registered: 2013-03-07
Posts: 595

Re: X^2 + XY Hyperbolic paraboloid

It is possible to transform the original wikipedia equation to the equation

Wikipedia stated that:

Now let a=b=c=1

We need a transformation such that:


It can be shown that the following substitution can be used to achieve this:



Using this transformation:

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