Rhodonea Curves

Polar equation:
r = a sin(kθ)

Click below to see one of the Associated curves.

Definitions of the Associated curves	Evolute
Involute 1	Involute 2
Inverse curve wrt origin	Inverse wrt another circle
Pedal curve wrt origin	Pedal wrt another point
Negative pedal curve wrt origin	Negative pedal wrt another point
Caustic wrt horizontal rays	Caustic curve wrt another point

When k is an integer there are k or 2k petals depending whether k is odd or even. If k is irrational then the number of petals is infinite.

The Quadrifolium is the rhodonea curve with k = 2. It has polar equation r = a sin(2θ) and cartesian form (x²+ y²) ³ = 4 a²x²y².

Luigi Guido Grandi was a member of the order of the Camaldolites. He became professor of philosophy in 1700 and professor of mathematics in 1714 both post being at the University of Pisa.

Grandi was the author of a number of works on geometry in which he considered the analogies of the circle and equilateral hyperbola. He also considered curves of double curvature on the sphere and the quadrature of parts of a spherical surface.

Other Web site:

Xah Lee

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/Curves/Rhodonea.html