RADIAL CURVE OF A GIVEN CURVE

 Notion studied by Tucker in 1864.

 The radial curve of a curve , associated to a fixed point O, is the locus of the points P defined by  where  is the vector joining the current point on  to its centre of curvature; in other words, it is the locus of the end of the radius-of-curvature vector, attached to a fixed point.

 For an initial curve  with current point , the radial curve is the set of points . Cartesian parametrization of the radial curve: . Complex parametrization: . If the intrinsic equation 2 of the initial curve is , then the polar equation of its radial curve is .

The radial curve of an algebraic curve is an algebraic curve of the same degree as its evolute.

Examples:

 initial curve associated radial curve circle circle ellipse sextic  with equation i.e.  (Loria p. 308) parabola duplicatrix cubic cycloid (with rolling circle with radius R) circle with radius 2R deltoid regular trifolium astroid quatrefoil epicycloid with parameter q rose hypocycloid with parameter q rose catenary of equal strength line catenary kampyle of Eudoxus tractrix kappa involute of a circle Archimedean spiral clothoid lituus logarithmic spiral logarithmic spiral Ribaucourt curve of index k the cases k = -2, -1, and 2 amount to the above cases of the parabola, the catenary and the cycloid. Clairaut's curve of index k-1 pseudo-spiral of index n the cases n = 0 , 1, and -1/2 amount to the above cases of the circle, the clothoid and the involute of a circle. Archimedean spiral of index