BROCARD TRANSFORMATION

 Homemade name, stemming from the fact that Brocard defined an original curve using this transformation (see multicardioid). Other name: fan transformation.

 Equation of the initial curve with f -periodic with respect to . Equation of the Brocard transformation

The Brocard transformation of centre O and parameter n is defined by the above formulas.

 For n  > 0, the points on the initial curve (in green, opposite) have their polar angle divided by n, and the curve obtained is duplicated by consecutive rotations by 360/n °. If n is an integer, then the curve obtained is invariant under rotations by 360/n °. Therefore, it is a Goursat curve if the initial curve also has an axis symmetry and if O is on this axis.

Examples:

 Initial curve pole Brocard transformation straight line outside of the line knots circle on the circle roses Pascal's limaçon pole of the limaçon conchoids of a rose conic focal point of the conic polygasteroids kappa curve centre of the kappa Cotes' spirals cardioid any point multicardioids

© Robert FERRÉOL 2017