# conics - Python library for dealing with conics
#
# Copyright 2026 Sergiu Deitsch <sergiu.deitsch@gmail.com>
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#   http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

"""
The importance of data normalization
====================================

Standardizing point coordinates before performing algebraic fitting can
noticeably improve the numerical conditioning of the resulting estimate. This
example fits a parabola to points whose coordinate ranges differ substantially
between the horizontal and vertical axes, once without scaling and once with
scaling enabled, and compares the two results.
"""

from conics.fitting import fit_nievergelt
import matplotlib.pyplot as plt
import numpy as np

y = [-2, -0.1, 0, 1, 2, 3, 4, 5, 6, 7, 8]
x = [1, 0.5, 0.1, 0, 0, 0, 0.2, 0.3, 0.4, 0.5, 0.6]

pts = np.column_stack((x, y))

C1 = fit_nievergelt(pts, type='parabola', scale=False)
C2 = fit_nievergelt(pts, type='parabola', scale=True)

X, Y = np.meshgrid(
    np.linspace(np.min(x) - 1, np.max(x) + 1),
    np.linspace(-1 + np.min(y), np.max(y) + 1),
)
Z1 = C1(np.dstack([X, Y]))
Z2 = C2(np.dstack([X, Y]))

fig = plt.figure()

ax1, ax2 = fig.subplots(1, 2)

ax1.contour(X, Y, Z1, levels=0)

ax2.contour(X, Y, Z2, colors='red', levels=0)

ax1.scatter(x, y, label='observations')
ax2.scatter(x, y)

ax1.set_title('without scaling')
ax2.set_title('with scaling')

fig.legend(loc='upper center')
plt.show()