From Pythagoras to Fourier and From Geometry to Nature


Chapter 7. Grandi (Rhodonea) Curves

Full-Text Views:

The curves with polar equation ρ = cos() (0 ≤ θ ≤ 2π) are also known as Grandi’s roses, in honor of Guido Grandi who communicated his discovery to Gottfried Wilhelm Leibniz in 1713. Curves with polar equation ρ = sin() (0 ≤ θ ≤ 2π) are equivalent to the preceding ones, up to a rotation of π/(2n) radians.

As can be seen in Figure 10, Grandi’s roses display n petals if n is odd and 2n petals if n is even. By using these polar equations it is impossible to obtain roses with 4n + 2 (nN ∪ {0}) petals. Roses with 4n + 2 petals can be obtained by using the Bernoulli Lemniscate and its extensions. More precisely:

  • The trigonometric function y=cos2x(π4+kπxπ4+kπ) becomes the so-called Bernoulli Lemniscate ρ=cos1/2(2θ)(π4+kπθπ4+kπ) (kN) that is a rose with two petals (Figure 11).

  • The functions y=cos(4n+2)x(n>1)(π4(2n+1)+kπ2n+1xπ4(2n+1)+kπ2n+1) become the polar equations ρ=cos1/2[(4n+2)θ](π4(2n+1)+kπ2n+1θπ4(2n+1)+kπ2n+1) (kN) which give roses with 4n + 2 petals.

Figure 10

Rhodonea cos(2θ) and cos(5θ).

Figure 11

Bernoulli Lemniscate.

A few graphs of Rhodonea curves with fractional indices are shown in Figures 1214.

Figure 12

Rhodonea cos(pqθ).

Figure 13

Rhodonea cos(14θ) and cos(54θ).

Figure 14

Rhodonea cos(18θ) and cos(38θ) .