# Source code for sympy.functions.special.bsplines

from __future__ import print_function, division

from sympy.core import S, sympify, expand
from sympy.functions import Piecewise, piecewise_fold
from sympy.functions.elementary.piecewise import ExprCondPair
from sympy.core.sets import Interval

"""Construct c*b1 + d*b2."""
if b1 == S.Zero or c == S.Zero:
rv = piecewise_fold(d*b2)
elif b2 == S.Zero or d == S.Zero:
rv = piecewise_fold(c*b1)
else:
new_args = []
n_intervals = len(b1.args)
assert(n_intervals == len(b2.args))
new_args.append((c*b1.args[0].expr, b1.args[0].cond))
for i in range(1, n_intervals - 1):
new_args.append((
c*b1.args[i].expr + d*b2.args[i - 1].expr,
b1.args[i].cond
))
new_args.append((d*b2.args[-2].expr, b2.args[-2].cond))
new_args.append(b2.args[-1])
rv = Piecewise(*new_args)

return rv.expand()

[docs]def bspline_basis(d, knots, n, x, close=True): """The n-th B-spline at x of degree d with knots. B-Splines are piecewise polynomials of degree d [1]_. They are defined on a set of knots, which is a sequence of integers or floats. The 0th degree splines have a value of one on a single interval: >>> from sympy import bspline_basis >>> from sympy.abc import x >>> d = 0 >>> knots = range(5) >>> bspline_basis(d, knots, 0, x) Piecewise((1, And(x <= 1, x >= 0)), (0, True)) For a given (d, knots) there are len(knots)-d-1 B-splines defined, that are indexed by n (starting at 0). Here is an example of a cubic B-spline: >>> bspline_basis(3, range(5), 0, x) Piecewise((x**3/6, And(x < 1, x >= 0)), (-x**3/2 + 2*x**2 - 2*x + 2/3, And(x < 2, x >= 1)), (x**3/2 - 4*x**2 + 10*x - 22/3, And(x < 3, x >= 2)), (-x**3/6 + 2*x**2 - 8*x + 32/3, And(x <= 4, x >= 3)), (0, True)) By repeating knot points, you can introduce discontinuities in the B-splines and their derivatives: >>> d = 1 >>> knots = [0,0,2,3,4] >>> bspline_basis(d, knots, 0, x) Piecewise((-x/2 + 1, And(x <= 2, x >= 0)), (0, True)) It is quite time consuming to construct and evaluate B-splines. If you need to evaluate a B-splines many times, it is best to lambdify them first: >>> from sympy import lambdify >>> d = 3 >>> knots = range(10) >>> b0 = bspline_basis(d, knots, 0, x) >>> f = lambdify(x, b0) >>> y = f(0.5) See Also ======== bsplines_basis_set References ========== .. [1] http://en.wikipedia.org/wiki/B-spline """ knots = [sympify(k) for k in knots] d = int(d) n = int(n) n_knots = len(knots) n_intervals = n_knots - 1 if n + d + 1 > n_intervals: raise ValueError('n + d + 1 must not exceed len(knots) - 1') if d == 0: result = Piecewise( (S.One, Interval(knots[n], knots[n + 1], False, not close).contains(x)), (0, True) ) elif d > 0: denom = knots[n + d + 1] - knots[n + 1] if denom != S.Zero: B = (knots[n + d + 1] - x)/denom b2 = bspline_basis(d - 1, knots, n + 1, x, close) else: b2 = B = S.Zero denom = knots[n + d] - knots[n] if denom != S.Zero: A = (x - knots[n])/denom b1 = bspline_basis( d - 1, knots, n, x, close and (B == S.Zero or b2 == S.Zero)) else: b1 = A = S.Zero result = _add_splines(A, b1, B, b2) else: raise ValueError('degree must be non-negative: %r' % n) return result
[docs]def bspline_basis_set(d, knots, x): """Return the len(knots)-d-1 B-splines at x of degree d with knots. This function returns a list of Piecewise polynomials that are the len(knots)-d-1 B-splines of degree d for the given knots. This function calls bspline_basis(d, knots, n, x) for different values of n. Examples ======== >>> from sympy import bspline_basis_set >>> from sympy.abc import x >>> d = 2 >>> knots = range(5) >>> splines = bspline_basis_set(d, knots, x) >>> splines [Piecewise((x**2/2, And(x < 1, x >= 0)), (-x**2 + 3*x - 3/2, And(x < 2, x >= 1)), (x**2/2 - 3*x + 9/2, And(x <= 3, x >= 2)), (0, True)), Piecewise((x**2/2 - x + 1/2, And(x < 2, x >= 1)), (-x**2 + 5*x - 11/2, And(x < 3, x >= 2)), (x**2/2 - 4*x + 8, And(x <= 4, x >= 3)), (0, True))] See Also ======== bsplines_basis """ n_splines = len(knots) - d - 1 return [bspline_basis(d, knots, i, x) for i in range(n_splines)]