Definition:Conic Section/Reduced Form

Definition

Let $K$ be a conic section.

Let $K$ be embedded in a cartesian plane such that:

$(1)$ the center is at the origin
$(2)$ the foci are at $\left({\pm c, 0}\right)$

for some $c \in \R_{\ge 0}$.

This can be interpreted in the contexts of the specific classes of conic section as follows:

Reduced Form of Ellipse

Let $K$ be an ellipse embedded in a cartesian plane.

$K$ is in reduced form if and only if:

$(1)$ its major axis is aligned with the $x$-axis
$(2)$ its minor axis is aligned with the $y$-axis.

Reduced Form of Hyperbola

Let $K$ be a hyperbola embedded in a cartesian plane.

$K$ is in reduced form if and only if:

$(1)$ its major axis is aligned with the $x$-axis
$(2)$ its minor axis is aligned with the $y$-axis.

Reduced Form of Circle

Let $K$ be a circle embedded in a cartesian plane.

$K$ is in reduced form if and only if its center is located at the origin.

Reduced Form of Parabola

Let $K$ be a parabola embedded in a cartesian plane.

As a Parabola has no Center, it is not possible to define the reduced form of a parabola in the same way as for the other classes of conic section.

Instead, $K$ is in reduced form if and only if:

$(1)$ its focus is at the point $\tuple {c, 0}$
$(2)$ its directrix is aligned with the line $x = -c$

for some $c \in \R_{> 0}$.