NURBS

NURBS Fitting

Provides curve and surface fitting functions.

This module provides functions for working with Non-Uniform Rational B-Splines (NURBS) geometries. Some functions are based on the geomdl open-source library (MIT license), originally created by the geomdl authors: orbingol/NURBS-Python.

We also use as main reference Les Piegl and Wayne Tiller. The NURBS Book. Springer Science & Business Media, 1997

For more information on NURBs and their use in geometric modeling, please refer to the literature in the field. The authors of this module make no warranty or guarantee as to the accuracy or suitability of this code for any particular purpose.

volmdlr.nurbs.fitting.approximate_curve(points: np.ndarray[np.double_t, ndim == 2], degree: cython.int, **kwargs)

Curve approximation using least squares method with fixed number of control points.

Please refer to The NURBS Book (2nd Edition), pp.410-413 for details.

Keyword Arguments:

• centripetal: activates centripetal parametrization method. Default: False

• ctrlpts_size: number of control points. Default: len(points) - 1

Parameters:

• points (list, tuple) – data points

• degree (int) – degree of the output parametric curve

Returns:

approximated B-Spline curve

Return type:

BSpline.Curve

volmdlr.nurbs.fitting.approximate_surface(points: np.ndarray[np.double_t, ndim == 2], size_u: cython.int, size_v: cython.int, degree_u: cython.int, degree_v: cython.int, **kwargs)

Surface approximation using least squares method with fixed number of control points.

This algorithm interpolates the corner control points and approximates the remaining control points. Please refer to Algorithm A9.7 of The NURBS Book (2nd Edition), pp.422-423 for details.

Keyword Arguments:

• centripetal: activates centripetal parametrization method. Default: False

• ctrlpts_size_u: number of control points on the u-direction. Default: size_u - 1

• ctrlpts_size_v: number of control points on the v-direction. Default: size_v - 1

Parameters:

• points (list, tuple) – data points

• size_u (int) – number of data points on the u-direction, \(r\)

• size_v (int) – number of data points on the v-direction, \(s\)

• degree_u (int) – degree of the output surface for the u-direction

• degree_v (int) – degree of the output surface for the v-direction

Returns:

approximated B-Spline surface

Return type:

BSpline.Surface

volmdlr.nurbs.fitting.interpolate_curve(points: np.ndarray[np.double_t, ndim == 2], degree: cython.int, centripetal: cython.bint)

Curve interpolation through the data points.

Please refer to Algorithm A9.1 on The NURBS Book (2nd Edition), pp.369-370 for details.

Keyword Arguments:

• centripetal: activates centripetal parametrization method. Default: False

Parameters:

• points (list, tuple) – data points

• degree (int) – degree of the output parametric curve

Returns:

interpolated B-Spline curve

Return type:

BSpline.Curve

volmdlr.nurbs.fitting.interpolate_surface(points: np.ndarray[np.double_t, ndim == 2], size_u: cython.int, size_v: cython.int, degree_u: cython.int, degree_v: cython.int, **kwargs)

Surface interpolation through the data points.

Please refer to the Algorithm A9.4 on The NURBS Book (2nd Edition), pp.380 for details.

Keyword Arguments:

• centripetal: activates centripetal parametrization method. Default: False

Parameters:

• points (list, tuple) – data points

• size_u (int) – number of data points on the u-direction

• size_v (int) – number of data points on the v-direction

• degree_u (int) – degree of the output surface for the u-direction

• degree_v (int) – degree of the output surface for the v-direction

Returns:

interpolated B-Spline surface

Return type:

BSpline.Surface

NURBS Operations

Nurbs main operations algorithms.

volmdlr.nurbs.operations.construct_split_curve(obj, curve1_kv, curve2_kv, knot_span, insertion_count)

Instantiate split curve.

volmdlr.nurbs.operations.construct_split_surfaces(obj, knotvectors, direction, knot_span, insertion_count)

Construct split surfaces.

volmdlr.nurbs.operations.ctrlpts2d_to_ctrlpts(control_points_table)

Transform the control_points m x n.

volmdlr.nurbs.operations.decompose_curve(obj, return_params: bool = False, number_max_patches: int | None = None, **kwargs)

Decompose the curve into Bézier curve segments of the same degree.

Generator

Parameters:

• obj (BSplineCurve) – Curve to be decomposed

• return_params (bool) – If True, returns the parameters from start and end of each Bézier patch with respect to the input curve.

• number_max_patches – number max of patches, if limiting is needed.

Type:

int

Returns:

a generator element with a Bezier segment.

Return type:

Generator element.

volmdlr.nurbs.operations.decompose_surface(obj, return_params, **kwargs)

Decomposes the surface into Bezier surface patches of the same degree.

Parameters:

• obj (BSplineSurface3D) – surface

• return_params (bool) – If True, returns the parameters from start and end of each Bézier patch with respect to the input curve.

Returns:

a list of Bezier patches

Return type:

list

volmdlr.nurbs.operations.extract_surface_curve_u(obj, param, curve_class, **kwargs)

Extract an isocurve from the surface at the input parametric coordinate on the u-direction.

This method works by inserting knots to the surface so that from the new control points created we can extract the curve. Under the hood this method performs an incomplete process of splitting the surface.

Keyword Arguments:

• find_span_func: FindSpan implementation. Default: helpers.find_span_linear()

Parameters:

• obj (volmdlr.surfaces.BSplineSurface3D) – surface

• param (float) – parameter for the u-direction

• curve_class (volmdlr.edges.BSplineCurve3D) – BSpline curve object

Returns:

The bspline curve at the specified parameter

Return type:

volmdlr.surfaces.BSplineSurface3D

volmdlr.nurbs.operations.extract_surface_curve_v(obj, param, curve_class, **kwargs)

Extract an isocurve from the surface at the input parametric coordinate on the v-direction.

This method works by inserting knots to the surface so that from the new control points created we can extract the curve. Under the hood this method performs an incomplete process of splitting the surface.

Keyword Arguments:

• find_span_func: FindSpan implementation. Default: helpers.find_span_linear()

Parameters:

• obj (volmdlr.surfaces.BSplineSurface3D) – surface

• param (float) – parameter for the u-direction

• curve_class (volmdlr.edges.BSplineCurve3D) – BSpline curve object

Returns:

The bspline curve at the specified parameter

Return type:

volmdlr.surfaces.BSplineSurface3D

volmdlr.nurbs.operations.flip_ctrlpts_u(ctrlpts, size_u, size_v)

Flips a list of 1-dimensional control points from u-row order to v-row order.

u-row order: each row corresponds to a list of u values

v-row order: each row corresponds to a list of v values

Parameters:

• ctrlpts (list, tuple) – control points in u-row order

• size_u (int) – size in u-direction

• size_v (int) – size in v-direction

Returns:

control points in v-row order

Return type:

list

volmdlr.nurbs.operations.get_knots_and_multiplicities(knotvector)

Get knots and multiplicities from knotvector in u and v direction.

volmdlr.nurbs.operations.helper_decompose(srf, idx, split_func, return_params, other_direction_params=None, **kws)

Decompose surface.

volmdlr.nurbs.operations.helper_split_knot_vectors(degree, knotvector, num_ctrlpts, param, span_func)

Compute knot vectors to split object into two pieces.

volmdlr.nurbs.operations.insert_control_points_surface_u(obj, param, num, **kwargs)

Calculate the control points equivalent to inserts knot n-times to a spline geometry.

Keyword Arguments:

• check_num: enables/disables operation validity checks. Default: True

Parameters:

• obj – spline geometry

• param (list, tuple) – knot to be inserted in u direction

• num (list, tuple) – number of knot insertions

Returns:

a list with the control points

volmdlr.nurbs.operations.insert_control_points_surface_v(obj, param, num, **kwargs)

Calculate the control points equivalent to inserts knot n-times to a spline geometry.

Keyword Arguments:

• check_num: enables/disables operation validity checks. Default: True

Parameters:

• obj – spline geometry

• param (list, tuple) – knot to be inserted in v direction

• num (list, tuple) – number of knot insertions

Returns:

a list with the control points

volmdlr.nurbs.operations.insert_knot_curve(obj, param, num, **kwargs)

Insert knots n-times to a spline geometry.

Keyword Arguments:

• check_num: enables/disables operation validity checks. Default: True

Parameters:

• obj – spline geometry

• param (list, tuple) – knot(s) to be inserted in [u, v, w] format

• num (list, tuple) – number of knot insertions in [num_u, num_v, num_w] format

Returns:

updated spline geometry

volmdlr.nurbs.operations.insert_knot_surface(obj, param, num, **kwargs)

Insert knots n-times to a spline geometry.

Keyword Arguments:

• check_num: enables/disables operation validity checks. Default: True

Parameters:

• obj – spline geometry

• param (list, tuple) – knot(s) to be inserted in [u, v] format

• num (list, tuple) – number of knot insertions in [num_u, num_v] format

Returns:

updated spline geometry

volmdlr.nurbs.operations.knot_insertion(degree, knotvector, ctrlpts, u, **kwargs)

Compute the control points of the rational/non-rational spline after knot insertion.

Part of Algorithm A5.1 of The NURBS Book by Piegl & Tiller, 2nd Edition.

Keyword Arguments:

• num: number of knot insertions. Default: 1

• s: multiplicity of the knot. Default: computed via :func:`.find_multiplicity`

• span: knot span. Default: computed via :func:`.find_span_linear`

Parameters:

• degree (int) – degree

• knotvector (list, tuple) – knot vector

• ctrlpts (list) – control points

• u (float) – knot to be inserted

Returns:

updated control points

Return type:

list

volmdlr.nurbs.operations.knot_insertion_alpha(u, knotvector, span, idx, leg)

Compute alpha coefficient for knot insertion algorithm.

Parameters:

• u (float) – knot

• knotvector (tuple) – knot vector

• span (int) – knot span

• idx (int) – index value (degree-dependent)

• leg (int) – i-th leg of the control points polygon

Returns:

coefficient value

Return type:

float

volmdlr.nurbs.operations.knot_insertion_kv(knotvector, u, span, num_insertions)

Compute the knot vector of the rational/non-rational spline after knot insertion.

Part of Algorithm A5.1 of The NURBS Book by Piegl & Tiller, 2nd Edition.

Parameters:

• knotvector (list, tuple) – knot vector

• u (float) – knot

• span (int) – knot span

• num_insertions (int) – number of knot insertions

Returns:

updated knot vector

Return type:

list

volmdlr.nurbs.operations.link_curves(curves, tol: float = 1e-07, validate: bool = True)

Links the input curves together.

The end control point of the curve k has to be the same with the start control point of the curve k + 1.

Returns:

a tuple containing: knots, knots multiplicities, control points and weights vector

volmdlr.nurbs.operations.refine_knot_vector_surface(obj, u, v, **kwargs)

Split the surface at the input parametric coordinate on the u-direction.

This method splits the surface into two pieces at the given parametric coordinate on the u-direction, generates two different surface objects and returns them. It does not modify the input surface.

Keyword Arguments:

• find_span_func: FindSpan implementation. Default: helpers.find_span_linear()

• insert_knot_func: knot insertion algorithm implementation. Default: operations.insert_knot()

Parameters:

• obj (abstract.Surface) – surface

• param (float) – parameter for the u-direction

Returns:

a list of surface patches

Return type:

list

volmdlr.nurbs.operations.separate_ctrlpts_weights(ctrlptsw)

Divide weighted control points by weights to generate unweighted control points and weights vector.

This function is dimension agnostic, i.e. control points can be in any dimension but the last element of the array should indicate the weight.

Parameters:

ctrlptsw (list, tuple) – weighted control points

Returns:

unweighted control points and weights vector

Return type:

list

volmdlr.nurbs.operations.split_curve(obj, param, **kwargs)

Split the curve at the input parametric coordinate.

This method splits the curve into two pieces at the given parametric coordinate, generates two different curve objects and returns them. It does not modify the input curve.

Keyword Arguments:

• find_span_func: FindSpan implementation. Default: helpers.find_span_linear()

• insert_knot_func: knot insertion algorithm implementation. Default: operations.insert_knot()

Parameters:

• obj (abstract.Curve) – Curve to be split

• param (float) – parameter

Returns:

a list of curve segments

Return type:

list

volmdlr.nurbs.operations.split_surface_u(obj, param, **kwargs)

Split the surface at the input parametric coordinate on the u-direction.

This method splits the surface into two pieces at the given parametric coordinate on the u-direction, generates two different surface objects and returns them. It does not modify the input surface.

Keyword Arguments:

• find_span_func: FindSpan implementation. Default: helpers.find_span_linear()

• insert_knot_func: knot insertion algorithm implementation. Default: operations.insert_knot()

Parameters:

• obj (abstract.Surface) – surface

• param (float) – parameter for the u-direction

Returns:

a list of surface patches

Return type:

list

volmdlr.nurbs.operations.split_surface_v(obj, param, **kwargs)

Split the surface at the input parametric coordinate on the v-direction.

This method splits the surface into two pieces at the given parametric coordinate on the v-direction, generates two different surface objects and returns them. It does not modify the input surface.

Keyword Arguments:

• find_span_func: FindSpan implementation. Default: helpers.find_span_linear()

• insert_knot_func: knot insertion algorithm implementation. Default: operations.insert_knot()

Parameters:

• obj (abstract.Surface) – surface

• param (float) – parameter for the v-direction

Returns:

a list of surface patches

Return type:

list