Christopher_Symbols

from sympy import *
import numpy as np
import math

def det(l):
  return (l[0][0]*l[1][1]-l[0][1]*l[1][0])

def solve(a,b):
    if det(a) == 0:
        print("\nSistema sem solução.\n\n")
    else:
        return [det([[b[0],a[0][1]],[b[1],a[1][1]]])/det(a),det([[a[0][0],b[0]],[a[1][0],b[1]]])/det(a)]

x, y, z, t, r = symbols('x y z t r', real=True)

def first_form_para(func):
    Fx=[]
    Fy=[]
    for k in range(3):
        Fx.append(diff(func[k],x))
        Fy.append(diff(func[k],y))
        E = simplify(np.dot(Fx,Fx))
        F = simplify(np.dot(Fx,Fy))
        G = simplify(np.dot(Fy,Fy))
    return (E,F,G)

def christopher_symbols_para(func):
    E = first_form_para(func)[0]
    F = first_form_para(func)[1]
    G = first_form_para(func)[2]
    M = [[E,F],[F,G]]
    i = [[simplify(diff(E,x)/2),simplify(diff(F,x)-diff(E,y)/2)],
        [simplify(diff(E,y)/2),simplify(diff(G,x)/2)],
        [simplify(diff(F,y)-diff(G,x)/2),simplify(diff(G,y)/2)]]
    cs = []
    for k in range(3):
        cs.append(simplify(solve(M,i[k])))
    return cs

def curvature_para(func):
    cs = christopher_symbols_para(func)
    K = simplify((diff(cs[1][1],x)-diff(cs[0][1],y)+cs[1][0]*cs[0][1]+cs[1][1]**2-cs[0][1]*cs[2][1]-cs[0][0]*cs[1][1])/-E)
    return K

def first_form(func):
    E = simplify(np.dot(diff(func,x),diff(func,x)))
    F = simplify(np.dot(diff(func,x),diff(func,y)))
    G = simplify(np.dot(diff(func,y),diff(func,y)))
    return (E, F, G)



f = (sin(x)*cos(y),sin(x)*sin(y),cos(x)) # Esfera
print(first_form_para(f))
# print(christopher_symbols_para(f))

g = (y*cos(x),y*sin(x),t*x) # Helicóide
print(first_form_para(g))
# print(christopher_symbols_para(g))

h = ((t+r*cos(x))*cos(y),(t+r*cos(x))*sin(y),r*sin(x)) #
print(first_form_para(h))
# print(christopher_symbols_para(h))

# i = ((2-y*sin(x/2))*sin(x), (2-y*sin(x/2)*cos(x)), cos(x/2)) # Möbius
# print(first_form_para(i))
# print(christopher_symbols_para(i))

j = (x,y,x*y)
print(first_form_para(j))