Closed b-spline

I want to write a closed b spline function which takes [k (degree is k-1), x coordinate and y coordinates of control points] as inputs. I am unable to generate a basis function which will calculate N0,k value. Any help on this matter will be appreciated.

Answer 1

[Abhaya]

Hi Priyabrata,

To create a closed B-spline curve of degree ‘k-1’, you need to append the initial ‘k-1’ points of the x and y vectors to the end of their respective vectors.

For instance, if the vector x is [0, 1, 2, 4, 3] and the vector y is [0, 2, 4, 2, 0], and the degree is 3, you should transform these vectors to [0, 1, 2, 4, 3, 0, 1, 2] and [0, 2, 4, 2, 0, 0, 2, 4], respectively. This transformation can be achieved using the following commands:

 x = [x, x(1:k-1)]; 
 y = [y, y(1:k-1)];

Once you have the updated vectors, you can utilize the basis function equation to generate the spline curve.

Below is a sample code snippet for the basis function, followed by an example demonstrating how to call this function.

 function N = basisFunction(i, k, t, knots) 
 % Base case for degree 0 
     if k == 1 
         if knots(i) <= t && t < knots(i+1) 
             N = 1; 
         else 
             N = 0; 
         end 
     else 
         % Recursive definition 
         if knots(i+k-1) - knots(i) == 0 
             term1 = 0; 
         else 
             term1 = ((t - knots(i)) / (knots(i+k-1) - knots(i))) * basisFunction(i, k-1, t, knots); 
         end 
         if knots(i+k) - knots(i+1) == 0 
             term2 = 0; 
         else 
             term2 = ((knots(i+k) - t) / (knots(i+k) - knots(i+1))) * basisFunction(i+1, k-1, t, knots); 
         end 
         N = term1 + term2; 
     end 
 end 
 
 function [splineX, splineY] = closedBSpline(k, x, y, numPoints) 
     % To ensure the control points form a closed loop 
     x = [x, x(1:k-1)]; 
     y = [y, y(1:k-1)]; 
     
     % Number of control points 
     n = length(x); 
     
     % Knot vector for a closed B-spline 
     knot = [0:(n + k - 1)]; 
     
     % Parameter values 
     tParam = linspace(k-1, n, numPoints); 
     
     % Initialize the spline points 
     splineX = zeros(1, numPoints); 
     splineY = zeros(1, numPoints); 
     
     % Calculate spline points 
     for j = 1:numPoints 
         for i = 1:n 
             N = basisFunction(i, k, tParam(j), knot); 
             splineX(j) = splineX(j) + N * x(i); 
             splineY(j) = splineY(j) + N * y(i); 
         end 
     end 
     
     % Plot the spline 
     figure; 
     plot(x, y, 'o-', 'DisplayName', 'Control Points'); 
     hold on; 
     plot(splineX, splineY, 'r-', 'DisplayName', 'B-Spline'); 
     legend('show'); 
     title('Closed Parametric B-Spline'); 
     xlabel('x'); 
     ylabel('y'); 
     hold off; 
 end 
 

By applying the code provided above, you can create a cubic B-spline that results in the following graph. The subsequent code details the steps required to achieve this implementation.

 % Define the degree of the B-spline (k = degree + 1) 
 k = 4;  
 
 % Define the control points (ensure they form a closed loop) 
 x = [0, 1, 2, 5, 3]; 
 y = [0, 2, 4, 0.5, 0]; 
 
 % Number of points to generate along the spline 
 numPoints = 200; 
 
 [splineX, splineY] = closedBSpline(k, x, y, numPoints); 

To explore more about basis function, please follow the given link.

https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/B-spline/bspline-basis.html

I hope this helps in solving the query.