how to convert B-form into piecewise bezier curves

I am following the mathlab example to get the smooth curves from discrete points in B-form. The curve looks quite smooth. I wonder if there is any way to convert the B-form into multiple pieces of quadratic or cubic Bezier curves that cover the whole curve.

x = linspace(0,2*pi,51);
noisy_y = cos(x) + .2*(rand(size(x))-.5);
plot(x,noisy_y,'x')
axis([-1 7 -1.2 1.2])
tol = (.05)^2*(2*pi)
fnplt( spaps(x, noisy_y,  tol), 'r', 2 )

Answer 1

[Abhaya]

Hi Wong,

To convert a B-form spline into multiple Bézier curves in MATLAB, you can use the MATLAB ‘spbrk’ function. The ‘spbrk’ function breaks the B-form in the given spline into parts of B-form and returns the knots, coefficients, number of breakpoints, and the order of the B-curve.

With the coefficients obtained, we can create the Bézier segments. These segments can then be plotted individually.

Below is a sample code snippet to convert B-form into piecewise Bezier curves.

x = linspace(0, 2*pi, 51); 
noisy_y = cos(x) + 0.2 * (rand(size(x)) - 0.5); 
figure; 
plot(x, noisy_y, 'x'); 
hold on; 
axis([-1 7 -1.2 1.2]); 
tol = (0.05)^2 * (2*pi); 
 
% Fit a smoothing spline 
sp = spaps(x, noisy_y, tol); 
 
% Plot the B-spline curve 
fnplt(sp, 'r', 2); 
 
% break the B-form in SP 
[knots, coefs, number, order] = spbrk(sp); 
 
% Insert knots to convert to Bezier form 
% Insert knots at each unique knot value to get Bezier segments 
unique_knots = unique(knots); 
for i = 1:length(unique_knots) 
    sp = fnrfn(sp, unique_knots(i)); 
end 
 
% Extract the refined B-form 
[refined_knots, refined_coefs, refined_number, refined_order] = spbrk(sp); 
 
% Plot the Bezier curves 
for i = 1:refined_number - refined_order 
    % Extract control points for each Bezier segment 
    bezier_coefs = refined_coefs(:, i:i+refined_order-1); 
    t = linspace(0, 1, 100); 
    bezier_curve = zeros(1, length(t)); 
    for j = 1:refined_order 
        bezier_curve = bezier_curve + bezier_coefs(j) * nchoosek(refined_order-1, j-1) * (1-t).^(refined_order-j) .* t.^(j-1); 
    end 
    plot(linspace(refined_knots(i), refined_knots(i+1), 100), bezier_curve, 'b-', 'LineWidth', 1); 
end 
hold off;

The result plot of the given code is attached below.

To explore more about MATLAB ‘spbrk’ function, execute following command on MATLAB command window

doc spbrk

Hope this helps.