def boundaryFunction(parameter):
return 1 + 0.1 * np.cos(parameter)

def boundaryDerivative(parameter):
    return -0.1 * np.sin(parameter)

def trajectoryFunction(parameter):
    aux = np.sin(beta - phi) / np.sin(beta - parameter)
    return boundaryFunction(phi) * aux

def difference(parameter):
    return trajectoryFunction(parameter) - boundaryFunction(parameter)

def integrand(parameter):
    rr = boundaryFunction(parameter)
    dd = boundaryDerivative (parameter)
    return np.sqrt(rr ** 2 + dd ** 2)

##### Main #####

length_vals = np.array([], dtype=np.float64)
alpha_vals = np.array([], dtype=np.float64)

# nof initial phi angles, alpha angles, and nof collisions for each.
n_phi, n_alpha, n_cols, count = 10, 10, 10, 0
# Length of the boundary
total_length, err = integrate.quad(integrand, 0, 2 * np.pi)

for phi in np.linspace(0, 2 * np.pi, n_phi):
    for alpha in np.linspace(0, 2 * np.pi, n_alpha):
        for n in np.arange(1, n_cols):

            nu = np.arctan(boundaryFunction(phi) / boundaryDerivative(phi))
            beta = np.pi + phi + alpha - nu

            # Determines next impact coordinate.
            bnds = (0, 2 * np.pi)
            phi_new = optimize.minimize_scalar(difference, bounds=bnds, method='bounded').x

            nu_new =  np.arctan(boundaryFunction(phi_new) / boundaryDerivative(phi_new))
            # Reflection angle with relation to tangent.
            alpha_new = phi_new - phi + nu - nu_new - alpha
            # Arc length for current phi value.
            arc_length, err = integrate.quad(integrand, 0, phi_new)

            # Append values to list
            length_vals = np.append(length_vals, arc_length / total_length)
            alpha_vals = np.append(alpha_vals, alpha)


        count += 1
    print  "{}%" .format(100 * count / (n_phi * n_alpha))