Conic sections

The standard right circular cone may be defined as the set of points $(x,y,z)$ in three-dimensional space $\mathbb{R}^3$ satisfying the equation

$$x^2+y^2=z^2$$

This is a smooth surface with the exception of a conical singularity at the origin. Here is a picture of the basic cone.

A conic section is the intersection of a two-dimensional plane with this cone. All planes are of the form $ax+by+cz=d$, where $\mathbf{n}=(a,b,c)$ is a normal vector to the plane. We also assume that $d\neq0$, since the other cases are the degenerate conics, including a single point, a single line, or a pair of intersecting lines. We may assume that the normal vector has length 1, that is, it is a point on the unit sphere. Rotation of $\mathbb{R}^3$ about the $z$ axis does not change the cone, and therefore we may rotate so that the normal vector belongs to the $xz$-plane. That means $\mathbf{n}=(a,0,c)$. By replacing $x$ by $-x$, if necessary, we may assume that $a\ge 0$. If we write this in polar coordinates $(a,0,c)=(\cos\theta,0,\sin\theta)$ with $-\pi/2<\theta<\pi/2$, it is a simple matter to classify the conics as follows:

Ellipse:

These are the sections with $\vert\theta\vert>\pi/4$.

Hyperbola:

$\vert\theta\vert<\pi/4$.

Parabola:

$\vert\theta\vert=\pi/4$.

Here are some pictures of each of these situations.