There are a lot of equations for a cone, such as
- $x^2 + y^2=z^2$
- $(x-a)^2+ (y-b)^2=((z-c)R)^2$
- $\dfrac{x^2}{a^2} + \dfrac{y^2}{c^2}= \dfrac{z^2}{c^2}$
- $ax^2 + by^2 + cz^2 +2fyz + 2gzx + 2hxy=0$
But then comes this equation that I found in the book Maths in Minutes.
$$x^2 + y^2= |z|$$
The book says that this is the cone that is used for the conic section, and this equation is slightly different because the z variable is not squared.
When I input the equation in the 3d plot it shows this
And the equation with $z^2$ gives us a graph like this:
Which one is suitable for the conic section?
If you intersect a cone, or a paraboloid, or a hyperboloid, or a sphere, or an ellipsoid with a plane, then the curve of intersection, seen as a curve in the intersecting plane, is the solution to a quadratic equation. And those are always conic sections (unless they are degenerate, e.g. the union of two lines, or a single point).
In particular, you can use any cone you want, and get the same qualitative results. The actual resulting equations will be different, of course, and some cone equations might be easier to use for calculation than others.