Semi-cubic parabola
A semi-cubic parabola , or Neil's parabola , is a plane algebraic curve described by the equation y 2 = ax 3 in some rectangular coordinate system. It is named for Neil , who in 1657 calculated the length of its arc.
Equations
- Algebraic equation: y 2 = ax 3 ( a ≠ 0).
- Parametric equation: x = t 2 , y = at 3 ( a ≠ 0).
Properties
A semi-cubic parabola is a caustic of the Chirnhausen curve . Moreover, any dovetail caustic near the summit is well approximated by a semi-cubic parabola, which makes this curve a reference in catastrophe theory .
The radius of curvature of the semi-cubic parabola at the origin is zero.