Semicubical parabola

In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form (with a ≠ 0) in some Cartesian coordinate system. Solving for y leads to the explicit form which imply that every real point satisfies x ≥ 0. The exponent explains the term semicubical parabola. (A parabola can be described by the equation y = ax^2.) Solving the implicit equation for x 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). Any semicubical parabola is similar to the semicubical unit parabola . Proof: The similarity (uniform scaling) maps the semicubical parabola onto the curve with . 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. 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.) 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 a = 1 (semicubical unit parabola) and b = 2, which means the length of the arc between the origin and point (4,8), one gets the arc length 9.073. The evolute of the parabola is a semicubical parabola shifted by 1/2 along the x-axis: 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.