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

 
 

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© Robert FERRÉOL 2017