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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 |
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© Robert FERRÉOL 2017