###############
# Authored by Weisheng Jiang
# Book 4  |  From Basic Arithmetic to Machine Learning
# Published and copyrighted by Tsinghua University Press
# Beijing, China, 2022
###############

# Bk4_Ch14_03.py

import sympy
import numpy as np
import matplotlib.pyplot as plt
from numpy import linalg as L


def mesh_circ(c1, c2, r, num):
    
    theta = np.linspace(0, 2*np.pi, num)
    r     = np.linspace(0,r, num)
    theta,r = np.meshgrid(theta,r)
    xx1 = np.cos(theta)*r + c1
    xx2 = np.sin(theta)*r + c2
    
    return xx1, xx2


#define symbolic vars, function
x1,x2 = sympy.symbols('x1 x2')

A = np.array([[0.5, -0.5],
              [-0.5, 0.5]])

Lambda, V = L.eig(A)

x = np.array([[x1,x2]]).T

f_x = x.T@A@x
f_x = f_x[0][0]

f_x_fcn = sympy.lambdify([x1,x2],f_x)

xx1, xx2 = mesh_circ(0, 0, 1, 50)

ff_x = f_x_fcn(xx1,xx2)

if Lambda[1] > 0:
    levels = np.linspace(0,Lambda[0],21)
else:
    levels = np.linspace(Lambda[1],Lambda[0],21)

t = np.linspace(0,np.pi*2,100)

# 2D visualization
fig, ax = plt.subplots()

ax.plot(np.cos(t), np.sin(t), color = 'k')

cs = plt.contourf(xx1, xx2, ff_x,
                  levels=levels, cmap = 'RdYlBu_r')
plt.show()
ax.set_aspect('equal')
ax.xaxis.set_ticks([])
ax.yaxis.set_ticks([])
ax.set_xlabel('$x_1$')
ax.set_ylabel('$x_2$')
ax.set_xlim(-1,1)
ax.set_ylim(-1,1)
clb = fig.colorbar(cs, ax=ax)
clb.set_ticks(levels)

#%% 3D surface of f(x1,x2)

x1_ = np.linspace(-1.2,1.2,31)
x2_ = np.linspace(-1.2,1.2,31)

xx1_fine, xx2_fine = np.meshgrid(x1_,x2_)

ff_x_fine = f_x_fcn(xx1_fine,xx2_fine)

f_circle = f_x_fcn(np.cos(t), np.sin(t))

# 3D visualization

fig, ax = plt.subplots()
ax = plt.axes(projection='3d')

ax.plot(np.cos(t), np.sin(t), f_circle, color = 'k')
# circle projected to f(x1,x2)

ax.plot_wireframe(xx1_fine,xx2_fine,ff_x_fine,
                  color = [0.8,0.8,0.8],
                  linewidth = 0.25)

ax.contour3D(xx1_fine,xx2_fine,ff_x_fine,15,
             cmap = 'RdYlBu_r')

ax.view_init(elev=30, azim=60)
ax.xaxis.set_ticks([])
ax.yaxis.set_ticks([])
ax.zaxis.set_ticks([])
ax.set_xlim(xx1_fine.min(),xx1_fine.max())
ax.set_ylim(xx2_fine.min(),xx2_fine.max())
plt.tight_layout()
ax.set_proj_type('ortho')
plt.show()