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@Gro-Tsen
Created December 15, 2022 16:34
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Graphical representation of the Riemann mapping theorem for a square (inside and out) and an ellipse (inside and out)
# Square of eccentricity (but not that of the drawn ellipse):
modparm = 3/4
# The size of the drawn ellipse (semimajor and semiminor) is computed
# below.
### Mapping the inside of the disk to the inside of the ellipse:
prescale = N(modparm^(-1/4))
postscale = N(pi/(2*elliptic_kc(modparm)))
# The conformal map taking the unit disk in CC to the inside of the
# ellipse, and 0 to 0:
def phi(z):
z = N(z)
t = N(asin(CC(z*prescale)))
u = N(elliptic_f(t, modparm))
return N(sin(postscale*u))
# A memoized version of phi, for efficiency:
phi_tab = {}
def memoize_phi(z):
z = N(z)
if z in phi_tab:
return phi_tab[z]
phi_tab[z] = phi(z)
return phi_tab[z]
# Convenience function for plotting phi(rad*exp(2*I*pi*ang)):
def plothelp(rad, ang):
return memoize_phi(rad*exp(2*I*pi*ang))
def plothelp_x(rad, ang):
return real(plothelp(rad,ang))
def plothelp_y(rad, ang):
return imag(plothelp(rad,ang))
circplots = [parametric_plot([lambda t: plothelp_x(i/10, t), lambda t: plothelp_y(i/10, t)], (0,1)) for i in range(11)]
radplots = [parametric_plot([lambda t: plothelp_x(t, i/24), lambda t: plothelp_y(t, i/24)], (0,1)) for i in range(24)]
# sum(circplots + radplots)
### Mapping the outside of the disk to the outside of the ellipse:
#semimajor = real(phi(1))
#semiminor = imag(phi(I))
semimajor = N(cosh((pi/4)*elliptic_kc(1-modparm)/elliptic_kc(modparm)))
semiminor = N(sinh((pi/4)*elliptic_kc(1-modparm)/elliptic_kc(modparm)))
# The conformal map taking the complement of the unit disk in CC to
# the complement of the ellipse, and infinity to infinity:
def phi2(z):
z = N(z)
return N(((semimajor+semiminor)/2)*z + ((semimajor-semiminor)/2)/z)
# Convenience function for plotting phi2(rad*exp(2*I*pi*ang)):
def plothelp2(rad, ang):
return phi2(rad*exp(2*I*pi*ang))
def plothelp2_x(rad, ang):
return real(plothelp2(rad,ang))
def plothelp2_y(rad, ang):
return imag(plothelp2(rad,ang))
circplots2 = [parametric_plot([lambda t: plothelp2_x(1+i/10, t), lambda t: plothelp2_y(1+i/10, t)], (0,1), color="red") for i in range(11)]
radplots2 = [parametric_plot([lambda t: plothelp2_x(t, i/24), lambda t: plothelp2_y(t, i/24)], (1,2), color="red") for i in range(24)]
sum(circplots + radplots + circplots2 + radplots2)
### Mapping the inside of the disk to the inside of the square:
cst = N(elliptic_kc(-1))
# In fact, cst is equal to:
# cst = N(gamma(1/4)^2/(4*sqrt(2*pi)))
# The conformal map taking the unit disk in CC to the square [-1,1]+I*[-1,1],
# and 0 to 0:
def phi(z):
z = N(z)
t = N(asin(CC(z*(1-I)/sqrt(2))))
u = N(elliptic_f(t, -1))
return N(u*(1+I)/cst)
# A memoized version of phi, for efficiency:
phi_tab = {}
def memoize_phi(z):
z = N(z)
if z in phi_tab:
return phi_tab[z]
phi_tab[z] = phi(z)
return phi_tab[z]
# Convenience function for plotting phi(rad*exp(2*I*pi*ang)):
def plothelp(rad, ang):
return memoize_phi(rad*exp(2*I*pi*ang))
def plothelp_x(rad, ang):
return real(plothelp(rad,ang))
def plothelp_y(rad, ang):
return imag(plothelp(rad,ang))
circplots = [parametric_plot([lambda t: plothelp_x(i/10, t), lambda t: plothelp_y(i/10, t)], (0,1)) for i in range(11)]
radplots = [parametric_plot([lambda t: plothelp_x(t, i/24), lambda t: plothelp_y(t, i/24)], (0,1)) for i in range(24)]
# sum(circplots + radplots)
### Mapping the outside of the disk to the outside of the square:
cst2 = N(2*elliptic_ec(-1) - 2*elliptic_kc(-1))
# The conformal map taking the complement of the unit disk in CC to
# the complement of the square [-1,1]+I*[-1,1], and infinity to infinity:
def phi2(z):
z = N(z)
zz = N(((1+I)/sqrt(2))/z)
t = N(asin(CC(zz)))
u = N(sqrt(1-zz^4)/zz) - 2*N(elliptic_f(t, -1)) + 2*N(elliptic_e(t, -1))
return N(u*(1+I)/cst2)
# A memoized version of phi2, for efficiency:
phi2_tab = {}
def memoize_phi2(z):
z = N(z)
if z in phi2_tab:
return phi2_tab[z]
phi2_tab[z] = phi2(z)
return phi2_tab[z]
# Convenience function for plotting phi2(rad*exp(2*I*pi*ang)):
def plothelp2(rad, ang):
return memoize_phi2(rad*exp(2*I*pi*ang))
def plothelp2_x(rad, ang):
return real(plothelp2(rad,ang))
def plothelp2_y(rad, ang):
return imag(plothelp2(rad,ang))
circplots2 = [parametric_plot([lambda t: plothelp2_x(1+i/10, t), lambda t: plothelp2_y(1+i/10, t)], (0,1), color="red") for i in range(11)]
radplots2 = [parametric_plot([lambda t: plothelp2_x(t, i/24), lambda t: plothelp2_y(t, i/24)], (1,2), color="red") for i in range(24)]
sum(circplots + radplots + circplots2 + radplots2)
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