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Find necessary and sufficient conditions that determine when the integral curve of two periodic functions kappa(s) and tau(s) with the same period L is a closed curve.

K=-e^2, where e is the eccentricity of a conic section.

A spiral that gives the solution to the central orbit problem under a radial force law r^..=-mu|r|^(-3)r^^, (1) where mu is a positive constant. There are three solution ...

There are no fewer than three distinct notions of curve throughout mathematics. In topology, a curve is a one-dimensional continuum (Charatonik and Prajs 2001). In algebraic ...

A curve created by starting with a circle, dividing it into six arcs, and flipping three alternating arcs. The process is then repeated an infinite number of times.

For a given curve C, consider the locus of the point P from where the tangents from P to C meet at a fixed given angle. This is called an isoptic curve of the given curve. ...

A curve also known as Gutschoven's curve which was first studied by G. van Gutschoven around 1662 (MacTutor Archive). It was also studied by Newton and, some years later, by ...

Curves with Cartesian equation ay^2=x(x^2-2bx+c) with a>0. The above equation represents the third class of Newton's classification of cubic curves, which Newton divided into ...

The quadratrix was discovered by Hippias of Elias in 430 BC, and later studied by Dinostratus in 350 BC (MacTutor Archive). It can be used for angle trisection or, more ...

Let C be a curve and let O be a fixed point. Let P be on C and let Q be the curvature center at P. Let P_1 be the point with P_1O a line segment parallel and of equal length ...