Equation for a cone that is "upside down"?
The equation of a cone who was its tip on the xy plane is $x^{2}+y^{2}=z^{2}$. How can I describe a cone whose base is on the xy plane, and tip on a point on the z-axis, i.e. the cone $x^{2}+y^{2}=z^{2}$ but upside down?
$\endgroup$ 31 Answer
$\begingroup$For a right circular cone having height $H$, radius $R$ on the $z = 0$ plane, and apex at $z = H$:$$\left( x^2 + y^2 \right) \left( \frac{H}{R} \right)^2 = \left( z - H \right)^2, \quad 0 \le z \le H$$i.e.$$\left( x^2 + y^2 \right) \left( \frac{H}{R} \right)^2 - \left( z - H \right)^2 = 0, \quad 0 \le z \le H$$or as a function $z = f(x, y)$ (solving above for $z$),$$z = H \left( 1 - \sqrt{\frac{ x^2 + y^2 }{R^2}} \right), \quad x^2 + y^2 \le R^2$$
$\endgroup$ 6
