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numerical_methods.py
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numerical_methods.py
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import numpy as np
import math
from math import factorial
import functools
# import matplotlib.pyplot as plt
# prolly wise to memoize
def nCk(n, k):
numerator = factorial(n)
denominator = factorial(k) * factorial(n - k)
return numerator/denominator
class NumericalMethods:
@classmethod
def lagrange(cls, pts):
k = len(pts) - 1
# Lagrange polynomials are not defined when any two of the x coords over the set of points are the same
if len(set(pt[0] for pt in pts)) < len(pts):
raise ValueError('Lagrange polynomial cannot be defined.')
def lj_of_x(j, x):
ret = 1
for m in range(0, k + 1):
xm = pts[m][0]
xj = pts[j][0]
if m != j:
denom = xj - xm
ret *= (x - xm) / denom
return ret
def L(x):
summ = 0
for j in range(0, k + 1):
yj = pts[j][1]
summ += yj * lj_of_x(j, x)
return summ
if len(pts):
xmin = min(pts, key=lambda pt: pt[0])[0]
xmax = max(pts, key=lambda pt: pt[0])[0]
return [(xi, L(xi)) for xi in np.arange(xmin, xmax + 1, 0.01)]
else:
return []
@classmethod
def bezier(cls, pts):
# formula taken from hearn, baker, carithers pg 422/429
points = np.array(pts)
def P(u):
summ = np.array([0, 0], dtype = 'float')
n = len(points) - 1
for k in range(0, n + 1):
prod = points[k] * nCk(n, k) * u ** k * (1-u) ** (n-k)
summ += prod
return summ
# calculating a 1000 output points for the bezier curve
out_pts = []
for u in np.linspace(0, 1, 1000):
out_pts.append(P(u))
return out_pts
@classmethod
def cardinal(cls, pts):
if len(pts) < 4:
raise ValueError('Insufficient points for Cardinal spline.')
t = 0 # this Cardinal spline is an Overhauser spline
s = (1 - t) / 2
cardinal_matrix = np.array([
[-s, 2 - s, s - 2, s],
[2 * s, s - 3, 3 - 2 * s, -s],
[-s, 0, s, 0],
[0, 1, 0, 0]
])
def P(iu, out_pts):
pos_u, u = iu
ret = np.array([u**3, u**2, u, 1]).reshape([1, 4])
ret = np.matmul(ret, cardinal_matrix)
modified_pts = pts[-1:] + pts + pts[:2]
for k in range(len(pts)):
four_pts = np.array([
modified_pts[k], # p_k-1
modified_pts[k + 1], # p_k
modified_pts[k + 2], #p_k+1
modified_pts[k + 3], #p_k+2
]).reshape([4, 2])
out_pts[k,pos_u,:] = np.matmul(ret, four_pts)
nf_pieces_per_pair = 100
out_pts = np.zeros([len(pts), nf_pieces_per_pair, 2])
for i, u in enumerate(np.linspace(0, 1, nf_pieces_per_pair)): P((i, u), out_pts)
out_pts = out_pts.reshape(len(pts) * nf_pieces_per_pair, 2)
# plt.scatter(out_pts[:,0], out_pts[:, 1], color='blue')
# pts = np.array(pts)
# plt.scatter(pts[:,0], pts[:, 1], color='red')
# plt.show()
return out_pts
if __name__ == '__main__':
## figure 16
# NumericalMethods.cardinal([
# [0, 0],
# [3, 6],
# [7, 6],
# [10, 0],
# ])
## figure 17
# NumericalMethods.cardinal([
# [0, 0],
# [5, 6],
# [5, 6],
# [10, 0],
# ])
## figure 18
# NumericalMethods.cardinal([
# [0, 0],
# [4.9, 6],
# [5.1, 6],
# [10, 0],
# ])
pass