nurbspy.nurbs_curve module

class nurbspy.nurbs_curve.NurbsCurve(control_points=None, weights=None, degree=None, knots=None)

Bases: object

Create a NURBS (Non-Uniform Rational Basis Spline) curve object

Parameters:

control_pointsndarray with shape (ndim, n+1)

Array containing the coordinates of the control points The first dimension of P spans the coordinates of the control points (any number of dimensions) The second dimension of P spans the u-direction control points (0,1,…,n)

weightsndarray with shape (n+1,)

Array containing the weight of the control points

degreeint

Degree of the basis polynomials

knotsndarray with shape (r+1=n+p+2,)

The knot vector in the u-direction Set the multiplicity of the first and last entries equal to p+1 to obtain a clamped NURBS

Methods

compute_bspline_coordinates(P, p, U, u)	Evaluate the coordinates of a B-Spline curve for the input u parametrization
compute_bspline_derivatives(P, p, U, u, ...)	Compute the derivatives of a B-Spline (or NURBS curve in homogeneous space) up to order derivative_order
compute_nurbs_coordinates(P, W, p, U, u)	Evaluate the coordinates of a NURBS curve for the input u parametrization
compute_nurbs_derivatives(P, W, p, U, u, ...)	Compute the derivatives of a NURBS curve in ordinary space up to the desired order
get_arclength([u1, u2])	Compute the arc length of a parametric curve in the interval [u1,u2] using numerical quadrature
get_binormal(u)	Evaluate the unitary binormal vector to the curve for the input u-parametrization
get_curvature(u)	Evaluate the curvature of the curve for the input u-parametrization
get_derivative(u, order)	Evaluate the derivative of the curve for the input u-parametrization
get_normal(u)	Evaluate the unitary normal vector to the curve for the input u-parametrization
get_normal_2D(u)	Evaluate the the unitary normal vector using the special formula 2D formula
get_tangent(u)	Evaluate the unitary tangent vector to the curve for the input u-parametrization
get_torsion(u)	Evaluate the torsion of the curve for the input u-parametrization
get_value(u)	Evaluate the coordinates of the curve for the input u parametrization
plot([fig, ax, curve, control_points, ...])	Create a plot and return the figure and axes handles
plot_control_points(fig, ax[, linewidth, ...])	Plot the control points of the NURBS curve
plot_curve(fig, ax[, linewidth, linestyle, ...])	Plot the coordinates of the NURBS curve
plot_frenet_serret(fig, ax[, frame_number, ...])	Plot some Frenet-Serret reference frames along the NURBS curve
project_point_to_curve(P)	Solve the point projection problem for the prescribed point P
rescale_plot(fig, ax)	Adjust the aspect ratio of the figure

PointToCurveProjectionProblem
plot_curvature
plot_torsion

Notes

This class includes methods to compute:

• Curve coordinates for any number of dimensions

• Analytic curve derivatives of any order and number of dimensions

• The unitary tangent, normal and binormal vectors (Frenet-Serret reference frame) in 2D and 3D

• The analytic curvature and torsion in 2D and 3D

• The arc length of the curve in any number of dimensions.

  The arc length is compute by numerical quadrature using analytic derivative information

The class can be used to represent polynomial and rational Bézier, B-Spline and NURBS curves The type of curve depends on the initialization arguments

• Polymnomial Bézier: Provide the array of control points

• Rational Bézier: Provide the arrays of control points and weights

• B-Spline: Provide the array of control points, degree and knot vector

• NURBS: Provide the arrays of control points and weights, degree and knot vector

In addition, this class supports operations with real and complex numbers The data type used for the computations is detected from the data type of the arguments Using complex numbers can be useful to compute the derivative of the shape using the complex step method

References

The NURBS Book. See references to equations and algorithms throughout the code L. Piegl and W. Tiller Springer, second edition

Curves and Surfaces for CADGD. See references to equations the source code G. Farin Morgan Kaufmann Publishers, fifth edition

All references correspond to The NURBS book unless it is explicitly stated that they come from Farin’s book

class PointToCurveProjectionProblem(C, dC, P)

Bases: object

Methods

fitness(x)	Evaluate the deviation between the prescribed point and the parametrized point
get_bounds()	Set the bounds for the optimization problem
gradient(x)	Compute the gradient of the fitness function analytically

fitness(x)

Evaluate the deviation between the prescribed point and the parametrized point

static get_bounds()

Set the bounds for the optimization problem

gradient(x)

Compute the gradient of the fitness function analytically

static compute_bspline_coordinates(P, p, U, u)

Evaluate the coordinates of a B-Spline curve for the input u parametrization

This function computes the coordinates of a B-Spline curve as given by equation 3.1. See algorithm A3.1

Parameters:

Pndarray with shape (ndim, n+1)

Array containing the coordinates of the control points The first dimension of P spans the coordinates of the control points (any number of dimensions) The second dimension of P spans the u-direction control points (0,1,…,n)

pint

Degree of the B-Spline basis polynomials

Undarray with shape (r+1=n+p+2,)

Knot vector in the u-direction Set the multiplicity of the first and last entries equal to p+1 to obtain a clamped spline

uscalar or ndarray with shape (N,)

Parameter used to evaluate the curve

Returns:

Cndarray with shape (ndim, N)

Array containing the coordinates of the curve The first dimension of C spans the (x,y,z) coordinates The second dimension of C spans the u parametrization sample points

static compute_bspline_derivatives(P, p, U, u, up_to_order)

Compute the derivatives of a B-Spline (or NURBS curve in homogeneous space) up to order derivative_order

This function computes the analytic derivatives of a B-Spline curve using equation 3.3. See algorithm A3.2

Parameters:

Pndarray with shape (ndim, n+1)

Array containing the coordinates of the control points The first dimension of P spans the coordinates of the control points (x,y,z) The second dimension of P spans the u-direction control points (0,1,…,n)

pint

Degree of the B-Spline basis polynomials

Undarray with shape (r+1=n+p+2,)

Knot vector in the u-direction Set the multiplicity of the first and last entries equal to p+1 to obtain a clamped spline

uscalar or ndarray with shape (N,)

Parameter used to evaluate the curve

up_to_orderinteger

Order of the highest derivative

Returns:

bspline_derivatives: ndarray of shape (up_to_order+1, ndim, Nu)

The first dimension spans the order of the derivatives (0, 1, 2, …) The second dimension spans the coordinates (x,y,z,…) The third dimension spans u-parametrization sample points

static compute_nurbs_coordinates(P, W, p, U, u)

Evaluate the coordinates of a NURBS curve for the input u parametrization

This function computes the coordinates of the NURBS curve in homogeneous space using equation 4.5 and then maps the coordinates to ordinary space using the perspective map given by equation 1.16. See algorithm A4.1

Parameters:

Pndarray with shape (ndim, n+1)

Array containing the coordinates of the control points The first dimension of P spans the coordinates of the control points (any number of dimensions) The second dimension of P spans the u-direction control points (0,1,…,n)

Wndarray with shape (n+1,)

Array containing the weight of the control points

pint

Degree of the B-Spline basis polynomials

Undarray with shape (r+1=n+p+2,)

Knot vector in the u-direction Set the multiplicity of the first and last entries equal to p+1 to obtain a clamped spline

uscalar or ndarray with shape (N,)

Parameter used to evaluate the curve

Returns:

Cndarray with shape (ndim, N)

Array containing the coordinates of the curve The first dimension of C spans the (x,y,z) coordinates The second dimension of C spans the u parametrization sample points

compute_nurbs_derivatives(P, W, p, U, u, up_to_order)

Compute the derivatives of a NURBS curve in ordinary space up to the desired order

This function computes the analytic derivatives of the NURBS curve in ordinary space using equation 4.8 and the derivatives of the NURBS curve in homogeneous space obtained from compute_bspline_derivatives()

The derivatives are computed recursively in a fashion similar to algorithm A4.2

Parameters:

Pndarray with shape (ndim, n+1)

Array containing the coordinates of the control points The first dimension of P spans the coordinates of the control points (x,y,z) The second dimension of P spans the u-direction control points (0,1,…,n)

Wndarray with shape (n+1,)

Array containing the weight of the control points

pint

Degree of the B-Spline basis polynomials

Undarray with shape (r+1=n+p+2,)

Knot vector in the u-direction Set the multiplicity of the first and last entries equal to p+1 to obtain a clamped spline

uscalar or ndarray with shape (N,)

Parameter used to evaluate the curve

up_to_orderinteger

Order of the highest derivative

Returns:

nurbs_derivatives: ndarray of shape (up_to_order+1, ndim, Nu)

The first dimension spans the order of the derivatives (0, 1, 2, …) The second dimension spans the coordinates (x,y,z,…) The third dimension spans u-parametrization sample points

get_arclength(u1=0.0, u2=1.0)

Compute the arc length of a parametric curve in the interval [u1,u2] using numerical quadrature

The definition of the arc length is given by equation 10.3 (Farin’s textbook)

Parameters:

u1scalar

Lower limit of integration for the arc length computation

u2scalar

Upper limit of integration for the arc length computation

Returns:

Lscalar

Arc length of NURBS curve in the interval [u1, u2]

get_binormal(u)

Evaluate the unitary binormal vector to the curve for the input u-parametrization

The definition of the unitary binormal vector is given by equation 10.5 (Farin’s textbook)

Parameters:

uscalar or ndarray with shape (N,)

Scalar or array containing the u-parameter used to evaluate the function

Returns:

binormalndarray with shape (ndim, N)

Array containing the unitary binormal vector The first dimension of binormal spans the (x,y,z) coordinates The second dimension of binormal spans the u parametrization sample points

get_curvature(u)

Evaluate the curvature of the curve for the input u-parametrization

The definition of the curvature is given by equation 10.7 (Farin’s textbook)

Parameters:

uscalar or ndarray with shape (N,)

Scalar or array containing the u-parameter used to evaluate the curvature

Returns:

curvaturescalar or ndarray with shape (N, )

Scalar or array containing the curvature of the curve

get_derivative(u, order)

Evaluate the derivative of the curve for the input u-parametrization

Parameters:

uscalar or ndarray with shape (N,)

Scalar or array containing the u-parameter used to evaluate the curve

orderinteger

Order of the derivative

Returns:

dCndarray with shape (ndim, N)

Array containing the derivative of the desired order The first dimension of dC spans the (x,y,z) coordinates The second dimension of dC spans the u parametrization sample points

get_normal(u)

Evaluate the unitary normal vector to the curve for the input u-parametrization

The definition of the unitary normal vector is given by equation 10.5 (Farin’s textbook)

Parameters:

uscalar or ndarray with shape (N,)

Scalar or array containing the u-parameter used to evaluate the function

Returns:

normalndarray with shape (ndim, N)

Array containing the unitary normal vector The first dimension of normal spans the (x,y,z) coordinates The second dimension of normal spans the u parametrization sample points

get_normal_2D(u)

Evaluate the the unitary normal vector using the special formula 2D formula

Parameters:

uscalar or ndarray with shape (N,)

Scalar or array containing the u-parameter used to evaluate the function

Returns:

normalndarray with shape (2, N)

Array containing the unitary normal vector The first dimension of normal spans the (x,y,z) coordinates The second dimension of normal spans the u parametrization sample points

get_tangent(u)

Evaluate the unitary tangent vector to the curve for the input u-parametrization

The definition of the unitary tangent vector is given by equation 10.5 (Farin’s textbook)

Parameters:

uscalar or ndarray with shape (N,)

Scalar or array containing the u-parameter used to evaluate the function

Returns:

tangentndarray with shape (ndim, N)

Array containing the unitary tangent vector The first dimension of tangent spans the (x,y,z) coordinates The second dimension of tangent spans the u parametrization sample points

get_torsion(u)

Evaluate the torsion of the curve for the input u-parametrization

The definition of the torsion is given by equation 10.8 (Farin’s textbook)

Parameters:

uscalar or ndarray with shape (N,)

Scalar or array containing the u-parameter used to evaluate the torsion

Returns:

torsionscalar or ndarray with shape (N, )

Scalar or array containing the torsion of the curve

get_value(u)

Evaluate the coordinates of the curve for the input u parametrization

Parameters:

uscalar or ndarray with shape (N,)

Parameter used to evaluate the curve

Returns:

Cndarray with shape (ndim, N)

Array containing the coordinates of the curve The first dimension of C spans the (x,y,z) coordinates The second dimension of C spans the u parametrization sample points

plot(fig=None, ax=None, curve=True, control_points=True, frenet_serret=False, axis_off=False, ticks_off=False)

Create a plot and return the figure and axes handles

plot_control_points(fig, ax, linewidth=1.25, linestyle='-.', color='red', markersize=5, markerstyle='o')

Plot the control points of the NURBS curve

plot_curvature(fig=None, ax=None, color='black', linestyle='-')

plot_curve(fig, ax, linewidth=1.5, linestyle='-', color='black', u1=0.0, u2=1.0)

Plot the coordinates of the NURBS curve

plot_frenet_serret(fig, ax, frame_number=5, frame_scale=0.1)

Plot some Frenet-Serret reference frames along the NURBS curve

plot_torsion(fig=None, ax=None, color='black', linestyle='-')

project_point_to_curve(P)

Solve the point projection problem for the prescribed point P

rescale_plot(fig, ax)

Adjust the aspect ratio of the figure

nurbspy.nurbs_curve.merge_nurbs_curves(nurbs_list)

Merge multiple NURBS curves into a single continuous curve with C⁰ continuity.

Parameters:

nurbs_listlist of NurbsCurve

List of NURBS curve instances to merge. All must have the same polynomial degree.

Returns:

mergedNurbsCurve

A new NURBS curve representing the concatenation of all input curves.

Notes

• Each curve is mapped to a subinterval of [0, 1] with equal length.

• Adjacent curves are connected with multiplicity p + 1 (C⁰ continuity).

• A small epsilon offset is applied at internal joins to avoid Gmsh issues related to repeated knots with multiplicity exactly equal to degree.