Butterflies in Polar Coordinates

I was looking for mathematical functions I could use to shape guitar bodies, and I came across Hubpages’ user calculus-geometry‘s beautiful page on how to generate butterfly outlines using functions in polar coordinates.

The butterfly above was generated using the function:

r(θ) = 12 – sin(θ) + 2 sin(3θ) + 2 sin(5θ) – sin(7θ) + 3 cos(2θ) – 2 cos(4θ)

The code I used (using VPython) is:

from visual import *

''' the main function '''
def r(theta):
    #r = 1+cos(theta)
    
    #Archimides' sprial
    #r = 0.5*(theta) 
    
    #heart: http://jwilson.coe.uga.edu/EMT669/Essay.ideas/Heart/Hearts.html
    #r = 5*sin(theta) - sin(5*theta)
    
    #butterfly: http://calculus-geometry.hubpages.com/hub/Butterfly-Curves-in-Polar-Coordinates-on-a-Graphing-Calculator
    #r = 8-sin(theta)+2*sin(3*theta)+2*sin(5*theta)-sin(7*theta)+3*cos(2*theta)-2*cos(4*theta)
    r = 12-sin(theta)+2*sin(3*theta)+2*sin(5*theta)-sin(7*theta)+3*cos(2*theta)-2*cos(4*theta)

    return r

'''convert to rectangular coordinates'''
def xy(r, theta):
    x = r * cos(theta)
    y = r * sin(theta)
    return vector(x, y)


path = curve(color=color.green, radius=.2)


theta = 0.0

print pi, theta, r(theta) , xy(r(theta), theta)

while theta <= 2*pi:
    rate(100)
    theta += 0.01
    path.append(pos=xy(r(theta), theta))