nurbspy.jax.nurbs_basis_functions module

nurbspy.jax.nurbs_basis_functions.compute_all_basis_polynomials_derivatives(n, p, U, u)

Evaluate the analytic derivatives of B-spline basis functions of degree p.

The function implements the recursive derivative relations for B-spline basis functions (following the same recursion pattern as the basis definition). Computations are vectorized across multiple u-values using jax.vmap.

Parameters:

nint

Highest index of the basis functions (number of functions = n+1).

pint

Degree of the basis polynomials.

Uarray_like

Knot vector of length n+p+2 defining the B-spline basis.

ufloat or array_like

Scalar or array of parameter values where the derivatives are evaluated.

derivative_orderint

Order of the derivative to compute (0 ≤ derivative_order ≤ p).

Returns:

N_dersndarray of shape (n+1, Nu)

Array containing the analytic derivative of each basis function at all u-values.

nurbspy.jax.nurbs_basis_functions.compute_basis_polynomials(n, p, U, u)

Evaluate B-spline basis functions of degree p for a set of parameter values u.

The function implements the Cox-de Boor recursion formula using JAX and vectorizes the scalar evaluation over multiple u-values using jax.vmap.

Parameters:

nint

Highest index of the basis functions (number of functions = n+1).

pint

Degree of the basis polynomials.

Uarray_like

Knot vector of length n+p+2 defining the B-spline basis. The first and last knots should typically have multiplicity p+1 for clamped splines.

ufloat or array_like

Scalar or array of parameter values where the basis functions are evaluated.

Returns:

Nndarray of shape (n+1, Nu)

Array containing all basis functions evaluated at each u value. The first axis spans the basis index i, the second spans the parameter samples.

nurbspy.jax.nurbs_basis_functions.compute_basis_polynomials_derivatives(n, p, U, u, derivative_order)

Return only the specified derivative order of B-spline basis functions.

nurbspy.jax.nurbs_basis_functions.compute_basis_polynomials_derivatives_jax(n, p, U, u, derivative_order)

Evaluate derivatives of B-spline basis functions using JAX automatic differentiation.

This method uses nested applications of jax.jacfwd on the scalar evaluation function _basis_single_u() to compute derivatives of arbitrary order. It is mathematically exact within floating-point precision and provides a useful verification of the analytic recursive formulation.

Parameters:

nint

Highest index of the basis functions (number of functions = n+1).

pint

Degree of the basis polynomials.

Uarray_like

Knot vector of length n+p+2 defining the B-spline basis.

ufloat or array_like

Scalar or array of parameter values where the derivatives are evaluated.

derivative_orderint

Order of the derivative to compute (0 ≤ derivative_order ≤ p).

Returns:

N_dersndarray of shape (n+1, Nu)

Array containing the derivatives of each basis function evaluated at all u-values.