spcol (B-spline collocation matrix) for B-Spline Basis and Derivative Computation

Overview

This Python script provides a set of functions to compute B-spline basis functions, their derivatives, and construct a collocation matrix for a set of input points. The script can be used for applications involving B-splines, such as numerical methods, interpolation, or computer-aided design (CAD).

Features

• B-spline basis function computation: The N function computes the B-spline basis functions using Cox-de Boor recursion formula.
• First derivative of the B-spline basis functions: The dN function computes the first derivative of the B-spline basis functions.
• Higher-order derivatives: The ddN function computes higher-order derivatives of the B-spline basis functions.
• Knot span locator: The findspan function finds the knot span for a given value of the parameter.
• Collocation matrix construction: The spcol function constructs a collocation matrix for a set of input values (u) based on the B-spline basis functions and their derivatives.

Functions

N(x, d, i, t)

Calculates the B-spline basis function N_i^d(x) recursively:

• x: The input parameter.
• d: Degree of the B-spline.
• i: Index of the basis function.
• t: Knot vector.

dN(x, d, i, t)

Computes the first derivative of the B-spline basis function N_i^d(x):

• Same parameters as N(x, d, i, t).

ddN(x, d, i, t, k)

Calculates the kth derivative of the B-spline basis function N_i^d(x):

• x: The input parameter.
• d: Degree of the B-spline.
• i: Index of the basis function.
• t: Knot vector.
• k: Order of the derivative.

findspan(t, x)

Finds the knot span for a given input x in the knot vector t:

• t: Knot vector.
• x: The input parameter.

spcol(U, d, u)

Constructs the collocation matrix for a given set of input parameters u:

• U: Knot vector.
• d: Degree of the B-spline.
• u: Array of input points.

Installation

From GitHub:

pip install git+https://github.com/AkshayK325/spcol.git

Example Usage

Below is a simple example of how to use the script:

import numpy as np
import matplotlib.pyplot as plt

# Assuming the spcol is installed
import spcol as sp

# Knot vector
U = np.array([0, 0, 0, 0, 0.25, 0.5, 0.75, 1, 1, 1, 1])
# Degree of the B-spline
p = 3
# Input points where we want to evaluate the B-spline curve
u = np.linspace(0,1,100)

# Control points for the B-spline in 2D (x, y)
XY = np.array([[0.0, 0.0],   # Control point 1 (x1, y1)
               [1.0, 1.0],   # Control point 2 (x2, y2)
               [2.0, 1.0],   # Control point 3 (x3, y3)
               [3.0, 2.0],   # Control point 4 (x4, y4)
               [4.0, 1.0],   # Control point 5 (x5, y5)
               [5.0, 0.0],   # Control point 6 (x6, y6)
               [1.0, 0.0]])  # Control point 7 (x7, y7)

# Generate the collocation matrix M
M = sp.spcol(U, p, u)

# Calculate the xy coordinates of the B-spline at points u
xy = M @ XY  # Matrix multiplication (M * control points)

# Extract the x and y values
x_vals = xy[:, 0]
y_vals = xy[:, 1]


# Generate the collocation matrix M
M = sp.spcol(U, p, np.unique(U))

# Calculate the xy coordinates of the B-spline at points u
xy = M @ XY  # Matrix multiplication (M * control points)

# Extract the x and y values
x_valsKnots = xy[:, 0]
y_valsKnots = xy[:, 1]

# Plotting
plt.figure()
# Plot the control polygon (control points)
plt.plot(XY[:, 0], XY[:, 1], 'o--', label='Control Polygon', color='grey')
# Plot the B-spline curve (evaluated points)
plt.plot(x_vals, y_vals, 'r-', label='B-spline Curve')
plt.scatter(x_valsKnots, y_valsKnots,c='b',marker="s", label='Knots')
plt.title("B-spline Curve")
plt.xlabel("X")
plt.ylabel("Y")
plt.legend()
plt.grid(True)
plt.show()

Requirements

• Python 3.x
• NumPy

You can install the required dependencies via pip:

pip install numpy

License

This project is licensed under the GNU General Public License v3.0. See the LICENSE file for details.

Author

• [email protected]
  Created on: October 20, 2021