The cardioid and the nephroid as the envelope of a pencil of reflected rays


The code below plots the cardioid and the nephroid as the envelope of a pencil of lines and shows how this pencil corresponds to reflected rays.

The parameter N, when it is a multiple of 4, means the number of points minus one.

cardiod_28
Cardiod     N = 28

nephroid_28
Nephroid     N = 28

Increasing N to N = 48 we get:

cardiod_48
Cardiod     N = 48

nephroid_48
Nephroid     N = 48

This was an early version plotted by hand long time ago with Xfig. Yes, it is not a good idea.

card
 nefr



The code

This is SAGE code producing the plots of the carioid:


N = 28

# Forces N even
N = 2*round(N/2)

def nephr():
  L = [( sin(t)*sin(3*t)+1/3*cos(t)*cos(3*t),-sin(t)*cos(3*t)+1/3*cos(t)*sin(3*t)) for t in srange(0,2*pi,0.03)]
  P = list_plot(L, plotjoined=True, color='green', thickness=2, zorder=120, linestyle='--')
  return P

def pcir(k):
  k -= N
  return ( cos(pi*k/N).n(), sin(pi*k/N).n() )
    
def rline(k):
  # Reflected line of length l = 0.9
  l = 0.9
  v = vector(pcir(2*k)) - vector(pcir(k))
  v = l*v/v.norm() + vector(pcir(k))
  return line([pcir(k), v], thickness=2)

P = arc((0,0),1,1,0,(pi/2,3*pi/2), linestyle='--', thickness=3)
P += arc((0,0),1,1,0,(-pi/2,pi/2), thickness=3)
P += nephr()

for k in srange(N/2, 3*N/2+1):
  P += line([(-1.0,0), pcir(k)], thickness=2, color='red', zorder=100)
  P += rline(k) + point([pcir(k)], size=40, zorder=110)

P.set_aspect_ratio(1)
P.axes(False)

P.save('cardiod_'+str(N)+'.png')

The analogue SAGE code for the nephroid is:


N = 28

# Forces N to be a multiple of 4
N = 4*round(N/4)

def nephr():
  L = [((3*sin(t)+sin(3*t))/4,(3*cos(t)+cos(3*t))/4) for t in srange(0,2*pi,0.03)]
  P = list_plot(L, plotjoined=True, color='green', thickness=2, zorder=120, linestyle='--')
  return P

def pcir(k):
  k -= 3*N/4
  return ( cos(2*pi*k/N/3).n(), sin(2*pi*k/N/3).n() )
    
def rline(k):
  # Reflected line of length l = 0.9
  l = 0.9
  v = vector(pcir(3*k)) - vector(pcir(k))
  v = l*v/v.norm() + vector(pcir(k))
  return line([pcir(k), v], thickness=2)

P = arc((0,0),1,1,0,(pi/2,3*pi/2), linestyle='--', thickness=3)
P += arc((0,0),1,1,0,(-pi/2,pi/2), thickness=3)
P += nephr()

for k in srange(N/4, 5*N/4+1):
  P += line([(-1.1,pcir(k)[1]), pcir(k)], thickness=2, color='red', zorder=100)
  P += rline(k) + point([pcir(k)], size=40, zorder=110)

P.set_aspect_ratio(1)
P.axes(False)

P.save('nephroid_'+str(N)+'.png')