Parabola

Synthetically, a parabola is the locus of points in a plane that are equidistant from a given line (the directrix) and a given point (the focus). Alternatively, a parabola is a conic section obtained as the intersection of a right circular cone with a plane parallel to a generator of the cone.

Let $d$ be a line and $F$ a point. In the degenerate when $F$ is a point of $d$, the "parabola" with directrix $d$ and focus $F$ is the line through $F$ that is perpendicular to $d$. In the language of conic sections, this corresponds to the case when the plane contains a generator of the cone.

To avoid the degenerate case, assume that $F$ does not lie in $d$, let $\Pi$ be the unique plane containing $F$ and $d$ and let $P$ be the parabola with focus $F$ and directrix $d$. The line $s$ through $F$ and perpendicular to $d$ is called the axis of the the parabola $P$ and is the unique line of symmetry of $P$. The unique point $V$ of $s$ that is equidistant from $F$ and $d$ lies on $P$ and is known as the vertex of the parabola, and the distance $FV$ is called the focal distance of the parabola.

Now let $F'$ be a point in $\Pi$ and $d'$ a line in $\Pi$ such that the distance from $F'$ to $d'$ equals the distance from $F$ to $d$. Then there is a unique, orientation-preserving rigid motion of $\Pi$ taking $F$ to $F'$ and $d$ to $d'$ and therefore, the parabola $P$ to the parabola with focus $F'$ and directrix $d'$. In other words, any two parabolas with the same focal distance are congruent.