= Conic section
{wiki}
= Conic
{synonym}
A conic section is obtained by intersecting a <plane> with a <double circular cone>. Cuts away from the vertex produce an <ellipse>, a <parabola> or a <hyperbola>; a perpendicular cut is the <circle> case of an <ellipse>. A cut through the vertex can instead produce a <degenerate conic>. One simple surface therefore connects several familiar <plane curves>.<Precalculus 2e by Jay Abramson/10.1 The Ellipse>[Precalculus 2e, §10.1, Writing Equations of Ellipses in Standard Form, Figure 2]{ref}
Consider the original family
$$
z^2=x^2+y^2,\qquad z=1+mx.
$$
Substituting the <plane equation> into the <double circular cone> equation gives
$$
(1-m^2)x^2+y^2-2mx-1=0.
$$
Thus $m=0$ gives a <circle>, $m=3/5$ an <ellipse>, $m=1$ a <parabola>, and $m=7/5$ a <hyperbola>. The last cut meets both sheets of the <double circular cone>, producing its two branches. These parameters specify a mathematical family, rather than measurements of a physical object.
\Image[/conic-sections-double-cone-plane-cuts.png]
{title=<Circle>, <ellipse>, <parabola> and <hyperbola> from four cuts of one <double circular cone>}
{description=Original three-dimensional intersections of $z^2=x^2+y^2$ with $z=1+mx$. The cone meshes and planes are finite displayed patches. The <circle> can look elliptical in the perspective view; its intrinsic geometry is circular.}
The <Cartesian coordinate system> descriptions of these curves are second-degree equations
$$
Ax^2+Bxy+Cy^2+Dx+Ey+F=0.
$$
A <rotation> and a <translation> expose their principal directions and offsets. The <conic type discriminant> identifies the ordinary nondegenerate real type, while a separate <degenerate conic> check prevents confusing two crossing <straight lines> with a <hyperbola>.
This is useful because a difficult-looking equation can be understood through a small set of geometric features: <foci of a conic>, <directrices of a conic>, <eccentricity of a conic> and the <semi-major and semi-minor axes>. The same <ellipse> can be traced by a constant-length string or parametrized by $x=a\cos t$, $y=b\sin t$. These are different descriptions of one curve, with different advantages for construction and calculation.
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