In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form (with ) in some Cartesian coordinate system. Solving for leads to the explicit form which imply that every real point satisfies . The exponent explains the term semicubical parabola. (A parabola can be described by the equation .) Solving the implicit equation for yields a second explicit form The parametric equation can also be deduced from the implicit equation by putting The semicubical parabolas have a cuspidal singularity; hence the name of cuspidal cubic. The arc length of the curve was calculated by the English mathematician William Neile and published in 1657 (see section History). Properties of semicubical parabolas Similarity Any semicubical parabola is similar to the semicubical unit parabola Proof: The similarity (uniform scaling) maps the semicubical parabola onto the curve with Singularity The parametric representation is regular except at point At point the curve has a singularity (cusp). The proof follows from the tangent vector Only for this vector has zero length. Tangents Differentiating the semicubical unit parabola one gets at point of the upper branch the equation of the tangent: This tangent intersects the lower branch at exactly one further point with coordinates (Proving this statement one should use the fact, that the tangent meets the curve at twice.) Arclength Determining the arclength of a curve one has to solve the integral For the semicubical parabola one gets (The integral can be solved by the substitution Example: For (semicubical unit parabola) and which means the length of the arc between the origin and point (4,8), one gets the arc length 9.073. Evolute of the unit parabola The evolute of the parabola is a semicubical parabola shifted by 1/2 along the x-axis: Polar coordinates In order to get the representation of the semicubical parabola in polar coordinates, one determines the intersection point of the line with the curve. For there is one point different from the origin: This point has distance from the origin. With and ( see List of identities) one gets Relation between a semicubical parabola and a cubic function Mapping the semicubical parabola by the projective map (involutoric perspectivity with axis and center yields hence the cubic function The cusp (origin) of the semicubical parabola is exchanged with the point at infinity of the y-axis. This property can be derived, too, if one represents