from sympy import prod, cos, sin
from sympy.abc import t
from sympy.plotting import plot, plot_parametric

t_i = [0, 0.8976, 1.7952, 2.6928, 3.5904, 4.4880, 5.3856, 6.2832]
x_i = [1, 1.5984, -0.6564, -1.6828, -0.1191, 0.2114, -0.3514, 1]
y_i = [0, 0.7818, 0.9750, 0.4339, -0.4339, -0.975, -0.7818, 0]
n = len(x_i)


def cardinal(x, i, data_x):
    return prod([(x - data_x[j]) / (data_x[i] - data_x[j]) for j in range(n) if i != j])


def lagrange_polynomial(x, k, data_x, data_y):
    return sum(data_y[i] * cardinal(x, i, data_x) for i in range(k))


n = len(x_i)
domain = (t, t_i[0], t_i[-1])

# a)
x_t = lagrange_polynomial(t, n, t_i, x_i)
plot(
    x_t,
    domain,
    title="x(t) – interpolated",
    xlabel="t",
    ylabel="x",
    show=False,
).save("x.png")

# b)
y_t = lagrange_polynomial(t, n, t_i, y_i)
plot(
    y_t,
    domain,
    title="y(t) – interpolated",
    xlabel="t",
    ylabel="y",
    show=False,
).save("y.png")

# c) & d)
interpolated = (x_t, y_t)
actual = (cos(t) + sin(2 * t), sin(t))
plot_parametric(
    (x_t, y_t, domain, "interpolated"),
    (cos(t) + sin(2 * t), sin(t), domain, "actual"),
    title="helicopter path - approx. vs actual",
    xlabel="x",
    ylabel="y",
    legend=True,
    show=False,
).save("path.png")