the semicubical parabola by homogeneous coordinates: In equation (A) the replacement (the line at infinity has equation and the multiplication by is performed. One gets the equation of the curve in homogeneous coordinates: Choosing line as line at infinity and introducing yields the (affine) curve Isochrone curve An additional defining property of the semicubical parabola is that it is an isochrone curve, meaning that a particle following its path while being pulled down by gravity travels equal vertical intervals in equal time periods. In this way it is related to the tautochrone curve, for which particles at different starting points always take equal time to reach the bottom, and the brachistochrone curve, the curve that minimizes the time it takes for a falling particle to travel from its start to its end. History The semicubical parabola was discovered in 1657 by William Neile who computed its arc length. Although the lengths of some other non-algebraic curves including the logarithmic spiral and cycloid had already been computed (that is, those curves had been rectified), the semicubical parabola was the first algebraic curve (excluding the line and circle) to be rectified. References August Pein: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten , 1875, Dissertation Clifford A. Pickover: The Length of Neile's Semicubical Parabola External links Plane curves