Definition:Degenerate Conic
Theorem
A degenerate conic is a conic section whose slicing plane passes through the apex of the cone.
There are three possibilities:
Degenerate Circle
A point-circle is the locus in the Cartesian plane of an equation of the form:
- $(1): \quad \paren {x - a}^2 + \paren {y - b}^2 = 0$
where $a$ and $b$ are real constants.
There is only one point in the Cartesian plane which satisfies $(1)$, and that is the point $\tuple {a, b}$.
It can be considered to be a circle whose radius is equal to zero.
Degenerate Ellipse
A degenerate ellipse is the conic section whose slicing plane passes through the apex of the cone.
Hence it consists of a single point.
Degenerate Parabola
A degenerate parabola is the conic section whose slicing plane passes through the apex of the cone and is thus tangent to the cone
Hence it consists of a single straight line.
Degenerate Hyperbola
Let $\phi < \theta$, that is: so as to make $K$ a hyperbola.
However, let $D$ pass through the apex of $C$.
Then $K$ degenerates into a pair of intersecting straight lines.
Also see
- Results about degenerate conics can be found here.
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $1$.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conic (conic section)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): degenerate conic
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conic (conic section)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): degenerate conic