Can a paraboloid become cone under limiting conditions?

[gikiian]

What will be the limiting conditions?

 

[HallsofIvy]

Your question does not really make sense. A paraboloid doesn't "become" anything. I presume you mean a family of paraboloids depending on one or more parameters. Yes, the limit as the parameters go to some value could be a cone. Exactly how depends upon the parametric equations.

 

[arildno]

If you start out with:
$$z=(x^{2}+y^{2})^{n}$$
then n=1 gives you a paraboloid, whereas n=1/2 a cone, and other values of n something in between or beyond.

 

[LCKurtz]

I think the OP means his question in the same sense that in 2-D the family of hyperbolas

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}= k$$

becomes the intersecting asymptotes when k = 0. And as such, the answer to his question is no. In 2D the parabola y = kx² becomes a straight line. These are degenerate forms of their corresponding conics.

In 3D the corresponding situation arises with hyperboloids:

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}= k$$

degenerates into a cone if k = 0. Whether it is an elliptical or circular cone depends on whether a = b. The paraboloid has no conical degenerate form.