116. lib.nurbs_e — Python equivalents of the functions in lib.nurbs_c

The functions in this module should be exact emulations of the external functions in the compiled library. Currently however this module only contains a few of the functions in lib.nurbs_c, making nurbs functionality in pyFormex currently only available when using the compiled lib.

116.1. Functions defined in module lib.nurbs_e

lib.nurbs_e.length(A)[source]

Return the length of an ndim vector.

lib.nurbs_e.bernstein(i, n, u)[source]

Compute the value of the Bernstein polynomial B(i,n) at u.

Parameters:

  • i (int) – The index of the polynomial

  • n (int) – The degree of the polynomial

  • u (float) – The parametric value where the polynomial is evaluated

Returns:

  • float – The value of the i-th Bernstein polynomials of degree n at parameter value u.

  • >>> bernstein(2, 5, 0.4)

  • 0.3456

lib.nurbs_e.allBernstein(n, u)[source]

Compute the value of all n-th degree Bernstein polynomials.

Parameters:

  • n: int, degree of the polynomials

  • u: float, parametric value where the polynomials are evaluated

Returns: an (n+1,) shaped float array with the value of all n-th degree Bernstein polynomials B(i,n) at parameter value u.

Algorithm A1.3 from ‘The NURBS Book’ p20.

lib.nurbs_e.find_span(U, u, p, n)[source]

Find the knot span index of the parametric point u.

Parameters:

  • U (float array (m+1,)) – The non-descending knot sequence: U[0] .. U[m]

  • u (float) – The parametric value U[0] <= u <= U[m] for which to find the span

  • p (int) – The degree of the B-spline basis functions

  • n (int) – The number of control points - 1 = m - p - 1

Returns:

int – The index of the knot span.

Notes

Algorithm A2.1 from ‘The NURBS Book’ p.68.

lib.nurbs_e.basis_funs(U, u, p, i)[source]

Compute the nonvanishing B-spline basis functions for index span i.

Parameters:

  • U (float array (m+1,)) – The knot sequence: U[0] .. U[m], non-descending.

  • u (float) – The parametric value U[0] <= u <= U[m] where to compute the functions.

  • p (int) – Degree of the B-spline basis functions

  • i (int) – Index of the knot span for value u (from find_span())

Returns:

float array (p+1) – The (p+1) values of nonzero basis functions at u.

Notes

Algorithm A2.2 from ‘The NURBS Book’ p.70.

lib.nurbs_e.basis_derivs(U, u, p, i, n)[source]

Compute the nonvanishing B-spline basis functions and derivatives.

Parameters:

  • U (float array (m+1,)) – The knot sequence: U[0] .. U[m], non-descending.

  • u (float) – The parametric value U[0] <= u <= U[m] where to compute the functions.

  • p (int) – Degree of the B-spline basis functions

  • i (int) – Index of the knot span for value u (from find_span())

  • n (int) – Number of derivatives to compute (n <= p)

Returns:

float array (n+1, p+1) – The (n+1, p+1) values of the nonzero basis functions and their first n derivatives at u

Notes

Algorithm A2.3 from ‘The NURBS Book’ p.72.

lib.nurbs_e.curveDegreeElevate(Pw, U, t)[source]

Elevate the degree of the Nurbs curve.

Parameters:

  • Pw: float array (nk,nd): nk=n+1 control points

  • U: int array(nu): nu=m+1 knot values

  • t: int: how much to elevate the degree

Returns a tuple:

  • Qw: nh+1 new control points

  • Uh: mh+1 new knot values

  • nh: highest control point index

  • mh: highest knot index

This is based on algorithm A5.9 from ‘The NURBS Book’ pg206.

lib.nurbs_e.BezDegreeReduce(Q, return_errfunc=False)[source]

Degree reduce a Bezier curve.

Parameters:

  • Q (float array (nk, nd)) – The control points of a Bezier curve of degree p = nk-1

  • return_errfunc (bool) – If True, also returns a function to evaluate the error along the parametric values.

Returns:

  • P (float array (nk-1, nd)) – The control points of a Bezier curve of degree p-1 that is as close as possible to the original curve.

  • maxerr (float) – An upper bound on the error introduced by the degree reduction.

  • errfunc (function) – A callable to evaluate the error as function of the parameter u. Only returned if return_errfunc is True.

Notes

Based on The NURBS Book 5.6.

lib.nurbs_e.curveDegreeReduce(Qw, U, tol=1.0)[source]

Reduce the degree of the Nurbs curve.

Parameters:

  • Qw (float array (nc, nd)) – The nc control points of the Nurbs curve

  • U (float array (nu)) – The nu knot values of the Nurbs curve

Returns:

  • Pw (float array (nctrl, nd)) – The new control points

  • U (float array (nknots)) – The new knot vector

  • err (float array (nerr)) – The error vector

  • This is algorithm A5.11 from ‘The NURBS Book’ pg223.

lib.nurbs_e.curveUnclamp(P, U)[source]

Unclamp a clamped curve.

Input: P,U Output: P,U

Note: this changes P and U inplace.

Based on algorithm A12.1 of The NURBS Book.

lib.nurbs_e.curveGlobalInterpolationMatrix(u, p, t0, t1)[source]

Compute the global curve interpolation matrix.

Parameters:

  • u (float array (nc)) – The parameter values at the nc points Q to be interpolated

  • p (int) – The degree of the B-spline to construct.

  • t0 (0 | 1) – 1 if the tangent at the start of the curve is specified

  • t1 (0 | 1) – 1 if the tangent at the end of the curve is specified

Returns:

  • U (float array (nU)) – The knot sequence, with nU = nu + p + 1, nu = nc + t0 + t1

  • A (float array (nu, nu)) – The coefficient matrix for the interpolation. The control points P can be found by solving the system of linear equations: A * P = Q.

See also

plugins.nurbs.globalInterpolationCurve()

the normal way to use this

Notes

Modified algorithm A9.1 from ‘The NURBS Book’ p.369.

lib.nurbs_e.curveGlobalInterpolationMatrix2(Q, D, u, p)[source]

Compute the global curve interpolation matrix for all tangents given.

Parameters:

  • Q (float array (nc)) –

  • D (float array(nc)) –

  • u (float array (nc)) – The parameter values at the nc points Q to be interpolated

  • p (2 | 3) – The degree of the B-spline to construct.

Returns:

  • U (float array (nU)) – The knot sequence, with nU = 2*nc + p + 1

  • A (float array (2*nc, 2*nc)) – The coefficient matrix for the interpolation. The control points P can be found by solving the system of linear equations: A * P = Q.

See also

plugins.nurbs.globalInterpolationCurve()

the normal way to use this

Notes

Modified algorithm A9.1 from ‘The NURBS Book’ p.369.

lib.nurbs_e.cubicSplineInterpolation(Q, t0, t1, U)[source]

Compute the control points of a cubic spline interpolate.

Parameters:

  • Q (float array (nc, 3)) – The nc points where the curve should pass through.

  • t0 (float array (3,)) – The tangent to the curve at the start point Q[0]

  • t1 (float array (3,)) – The tangent to the curve at the end point Q[nc-1]

  • U (float array (nc+6,)) – The clamped knot vector: 3 zeros, the nc parameter values for the points, 3 ones.

Returns:

  • float array (nc+2, 3) – The control points of the curve. With the given knots they will create a 3-rd degree NURBS curve that passes through the points Q and has derivatives t0 and t1 at its end.

  • Based on algorithm A9.2 of ‘The Nurbs Book’, p. 373