/usr/lib/python3/dist-packages/pygal/interpolate.py is in python3-pygal 2.4.0-1.
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# This file is part of pygal
#
# A python svg graph plotting library
# Copyright © 2012-2016 Kozea
#
# This library is free software: you can redistribute it and/or modify it under
# the terms of the GNU Lesser General Public License as published by the Free
# Software Foundation, either version 3 of the License, or (at your option) any
# later version.
#
# This library is distributed in the hope that it will be useful, but WITHOUT
# ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
# FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more
# details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with pygal. If not, see <http://www.gnu.org/licenses/>.
"""
Interpolation functions
These functions takes two lists of points x and y and
returns an iterator over the interpolation between all these points
with `precision` interpolated points between each of them
"""
from __future__ import division
from math import sin
def quadratic_interpolate(x, y, precision=250, **kwargs):
"""
Interpolate x, y using a quadratic algorithm
https://en.wikipedia.org/wiki/Spline_(mathematics)
"""
n = len(x) - 1
delta_x = [x2 - x1 for x1, x2 in zip(x, x[1:])]
delta_y = [y2 - y1 for y1, y2 in zip(y, y[1:])]
slope = [delta_y[i] / delta_x[i] if delta_x[i] else 1 for i in range(n)]
# Quadratic spline: a + bx + cx²
a = y
b = [0] * (n + 1)
c = [0] * (n + 1)
for i in range(1, n):
b[i] = 2 * slope[i - 1] - b[i - 1]
c = [(slope[i] - b[i]) / delta_x[i] if delta_x[i] else 0 for i in range(n)]
for i in range(n + 1):
yield x[i], a[i]
if i == n or delta_x[i] == 0:
continue
for s in range(1, precision):
X = s * delta_x[i] / precision
X2 = X * X
yield x[i] + X, a[i] + b[i] * X + c[i] * X2
def cubic_interpolate(x, y, precision=250, **kwargs):
"""
Interpolate x, y using a cubic algorithm
https://en.wikipedia.org/wiki/Spline_interpolation
"""
n = len(x) - 1
# Spline equation is a + bx + cx² + dx³
# ie: Spline part i equation is a[i] + b[i]x + c[i]x² + d[i]x³
a = y
b = [0] * (n + 1)
c = [0] * (n + 1)
d = [0] * (n + 1)
m = [0] * (n + 1)
z = [0] * (n + 1)
h = [x2 - x1 for x1, x2 in zip(x, x[1:])]
k = [a2 - a1 for a1, a2 in zip(a, a[1:])]
g = [k[i] / h[i] if h[i] else 1 for i in range(n)]
for i in range(1, n):
j = i - 1
l = 1 / (2 * (x[i + 1] - x[j]) - h[j] * m[j]) if x[i + 1] - x[j] else 0
m[i] = h[i] * l
z[i] = (3 * (g[i] - g[j]) - h[j] * z[j]) * l
for j in reversed(range(n)):
if h[j] == 0:
continue
c[j] = z[j] - (m[j] * c[j + 1])
b[j] = g[j] - (h[j] * (c[j + 1] + 2 * c[j])) / 3
d[j] = (c[j + 1] - c[j]) / (3 * h[j])
for i in range(n + 1):
yield x[i], a[i]
if i == n or h[i] == 0:
continue
for s in range(1, precision):
X = s * h[i] / precision
X2 = X * X
X3 = X2 * X
yield x[i] + X, a[i] + b[i] * X + c[i] * X2 + d[i] * X3
def hermite_interpolate(x, y, precision=250,
type='cardinal', c=None, b=None, t=None):
"""
Interpolate x, y using the hermite method.
See https://en.wikipedia.org/wiki/Cubic_Hermite_spline
This interpolation is configurable and contain 4 subtypes:
* Catmull Rom
* Finite Difference
* Cardinal
* Kochanek Bartels
The cardinal subtype is customizable with a parameter:
* c: tension (0, 1)
This last type is also customizable using 3 parameters:
* c: continuity (-1, 1)
* b: bias (-1, 1)
* t: tension (-1, 1)
"""
n = len(x) - 1
m = [1] * (n + 1)
w = [1] * (n + 1)
delta_x = [x2 - x1 for x1, x2 in zip(x, x[1:])]
if type == 'catmull_rom':
type = 'cardinal'
c = 0
if type == 'finite_difference':
for i in range(1, n):
m[i] = w[i] = .5 * (
(y[i + 1] - y[i]) / (x[i + 1] - x[i]) +
(y[i] - y[i - 1]) / (
x[i] - x[i - 1])
) if x[i + 1] - x[i] and x[i] - x[i - 1] else 0
elif type == 'kochanek_bartels':
c = c or 0
b = b or 0
t = t or 0
for i in range(1, n):
m[i] = .5 * ((1 - t) * (1 + b) * (1 + c) * (y[i] - y[i - 1]) +
(1 - t) * (1 - b) * (1 - c) * (y[i + 1] - y[i]))
w[i] = .5 * ((1 - t) * (1 + b) * (1 - c) * (y[i] - y[i - 1]) +
(1 - t) * (1 - b) * (1 + c) * (y[i + 1] - y[i]))
if type == 'cardinal':
c = c or 0
for i in range(1, n):
m[i] = w[i] = (1 - c) * (
y[i + 1] - y[i - 1]) / (
x[i + 1] - x[i - 1]) if x[i + 1] - x[i - 1] else 0
def p(i, x_):
t = (x_ - x[i]) / delta_x[i]
t2 = t * t
t3 = t2 * t
h00 = 2 * t3 - 3 * t2 + 1
h10 = t3 - 2 * t2 + t
h01 = - 2 * t3 + 3 * t2
h11 = t3 - t2
return (h00 * y[i] +
h10 * m[i] * delta_x[i] +
h01 * y[i + 1] +
h11 * w[i + 1] * delta_x[i])
for i in range(n + 1):
yield x[i], y[i]
if i == n or delta_x[i] == 0:
continue
for s in range(1, precision):
X = x[i] + s * delta_x[i] / precision
yield X, p(i, X)
def lagrange_interpolate(x, y, precision=250, **kwargs):
"""
Interpolate x, y using Lagrange polynomials
https://en.wikipedia.org/wiki/Lagrange_polynomial
"""
n = len(x) - 1
delta_x = [x2 - x1 for x1, x2 in zip(x, x[1:])]
for i in range(n + 1):
yield x[i], y[i]
if i == n or delta_x[i] == 0:
continue
for s in range(1, precision):
X = x[i] + s * delta_x[i] / precision
s = 0
for k in range(n + 1):
p = 1
for m in range(n + 1):
if m == k:
continue
if x[k] - x[m]:
p *= (X - x[m]) / (x[k] - x[m])
s += y[k] * p
yield X, s
def trigonometric_interpolate(x, y, precision=250, **kwargs):
"""
Interpolate x, y using trigonometric
As per http://en.wikipedia.org/wiki/Trigonometric_interpolation
"""
n = len(x) - 1
delta_x = [x2 - x1 for x1, x2 in zip(x, x[1:])]
for i in range(n + 1):
yield x[i], y[i]
if i == n or delta_x[i] == 0:
continue
for s in range(1, precision):
X = x[i] + s * delta_x[i] / precision
s = 0
for k in range(n + 1):
p = 1
for m in range(n + 1):
if m == k:
continue
if sin(0.5 * (x[k] - x[m])):
p *= sin(0.5 * (X - x[m])) / sin(0.5 * (x[k] - x[m]))
s += y[k] * p
yield X, s
INTERPOLATIONS = {
'quadratic': quadratic_interpolate,
'cubic': cubic_interpolate,
'hermite': hermite_interpolate,
'lagrange': lagrange_interpolate,
'trigonometric': trigonometric_interpolate
}
if __name__ == '__main__':
from pygal import XY
points = [(.1, 7), (.3, -4), (.6, 10), (.9, 8), (1.4, 3), (1.7, 1)]
xy = XY(show_dots=False)
xy.add('normal', points)
xy.add('quadratic', quadratic_interpolate(*zip(*points)))
xy.add('cubic', cubic_interpolate(*zip(*points)))
xy.add('lagrange', lagrange_interpolate(*zip(*points)))
xy.add('trigonometric', trigonometric_interpolate(*zip(*points)))
xy.add('hermite catmul_rom', hermite_interpolate(
*zip(*points), type='catmul_rom'))
xy.add('hermite finite_difference', hermite_interpolate(
*zip(*points), type='finite_difference'))
xy.add('hermite cardinal -.5', hermite_interpolate(
*zip(*points), type='cardinal', c=-.5))
xy.add('hermite cardinal .5', hermite_interpolate(
*zip(*points), type='cardinal', c=.5))
xy.add('hermite kochanek_bartels .5 .75 -.25', hermite_interpolate(
*zip(*points), type='kochanek_bartels', c=.5, b=.75, t=-.25))
xy.add('hermite kochanek_bartels .25 -.75 .5', hermite_interpolate(
*zip(*points), type='kochanek_bartels', c=.25, b=-.75, t=.5))
xy.render_in_browser()
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