A conic section (or simply conic) can be described as the intersection of a plane and a double-napped cone (i.e. two cones; one resting on the other, point to point).
Sometimes called a double cone, it is formed by revolving the line $y=x$ around the $y$-axis. The point where the two cones meet (at the tips) is called the vertex.
Notice in the image below that for the four basic conics, the intersecting plane does not pass through the vertex of the cone.
When the plane passes through the vertex, the resulting figure is a degenerate conic, as shown in bottom half of the image below.
For a more detailed / rigorous definition of conic sections, see George Simmons' Calculus with Analytic Geometry, Second Edtion.
While you may laugh, these are not naughty or feisty conic sections (ok, you can laugh).
Degenerate conic sections are formed when a plane through the vertex forms either a single point, line, or pair of intersecting lines, as depicted in the illustration above.
Most, if not all, of your work will be concerned with conic sections that are not degenerate.
Circles, like all other conic sections, can be modeled in Cartesian coordinates using rectangular or parametric equations. They can also be modeled in Polar Coordinates.
| Rectangular Equation | Parametric Equation | Polar Equation |
|---|---|---|
| $(x-h)^2 + (y-k)^2 = r^2$ | $x=h+r\cos(t) \\ y=k+r\sin(t)$ |
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