SC Tools

This package is based on our research to calculated Schwarz-Christoffel (SC) parameters and conformal modulus of a quadrilateral.

Installation

Package can be installed using pip

pip install --user git+https://github.com/giorgi94/sc-tools.git

Usage

To start using this package first you install it using pip or directly download it from github. To start working you can copy and paste following code

from sctools import Quadrilateral, phi


"""
    After initializing Quadrilateral
    with clockwise oriented vertices
    it will automatically calculate
    its inner angles and side lengths.
"""
quad = Quadrilateral((1, 0), (-10, -15), (-8, 10), (15, 1))


# print its angles in degree
print(quad.get_angles_degree())

# print tau's, where pi*tau is inner angle
print(quad.tau1, quad.tau2, quad.tau3, quad.tau4)

# print side lengths
print(quad.len1, quad.len2, quad.len3, quad.len4)

"""
    find r, A, C parameters for the quadrilateral,
    which come from following mapping

    f(x) = A + C int_{-infty}^{x}
        (1-u)^(tau1-1) (1-u/2)^(tau2-1) (1-u/(2+r))^(tau3-1) du
"""
r, A, C = quad.sc.calc()

# print vertices calculated by SC mapping
print(quad.sc.get_vertices(A, C, r))

# print error between real and calculated vertices
print(quad.sc.error_calc(A, C, r))

# print conformal modulus of the quadrilateral
print(quad.sc.conformal_modulus(r))

Generalized modulus

from sctools import phi
import sctools.series as sc_series

# set some values for tau
tau1, tau2, tau3 = 1.27589191934767, 0.175999535986355, 0.406684994391987

# caclulate phi power series coefficients
phi_coeffs = sc_series.phi_series_coeffs(40, tau1, tau2, tau3)


"""
    following function prints real
    and approximate values for generalized modulus.
    Approximation is good when x is close to 1.
"""
def print_real_and_approx(x):

    print(phi(x, tau1, tau2, tau3))
    print(sc_series.phi_series(x, phi_coeffs))
    print()

print_real_and_approx(0.534)
print_real_and_approx(1.534)

print_real_and_approx(0.0534)
print_real_and_approx(3.534)

Modular equation

A modular equation reduces to the generalized modular equation with signature $1/\tau$ and degree $p$ which can be rewritten in the form

$$ \frac{_{2}F_{1}(1-\tau,\tau;1-\tau;1-s^2)}{_{2}F_{1}(1-\tau,\tau;1-\tau;s^2)}=p \frac{_{2}F_{1}(1-\tau,\tau;1-\tau;1-l^2)}{_{2}F_{1}(1-\tau,\tau;1-\tau;l^2)} $$

which can be written using generalized modulus and we can solve it for $s$

from mpmath import hyp2f1
import sctools.series as sc_series
from sctools import phi

tau = 0.43

phi_coeffs = sc_series.phi_series_coeffs(25, 1 - tau, tau, 1 - tau)
psi_coeffs = [sc_series.psi_coeff(i, phi_coeffs) for i in range(15)]

phi_tau = lambda x: phi(x, 1 - tau, tau, 1 - tau)
psi_tau = lambda y: sc_series.psi_series(y, psi_coeffs)

F = lambda x: hyp2f1(tau, 1 - tau, 1, 1 - x**2) / hyp2f1(tau, 1 - tau, 1, x**2)

l = 0.4
p = 0.5

s = (1 + psi_tau(p * phi_tau(1 / l**2 - 1))) ** (-0.5)

print(F(s))
print(p * F(l))