High-Precision Bezier Curves

Fast, robust and accurate arbitrary-dimension and precision Bezier curve generation from its control points and/or knots.

Syntax

y = bspl(x)

y = bspl(x, t)

y = bspl(x, t, dim)

y = bspl(x, t, dim, bin)

Description

y = bspl(x, t, dim, bin) Returns the Bézier curve 'y' that results from evaluating the control points 'x' with a knot vector 't' along the dimension specified in 'dim'. An optional parameter 'bin' could parse the binomial coefficient terms to be used. In the event of parsing a scalar value as knot, then a uniform knot vector with the parsed resolution would be used.

Please refer to the code section for a detailed description of the functions used. It is recommended that the case-allocation coefficients are adjusted based on the specifications of the machine that runs this code for maximum speed, using speed routines as the one shown in the last example.

Examples

% Bezier curve evaluation
x = rand(randi(4) + 2, randi(2) + 1);
a = bspl(x);
b = bspl(x, rand(10, 1));
figure
hold on
if size(x, 2) > 2
    plot3(a(:, 1), a(:, 2), a(:, 3))
    plot3(x(:, 1), x(:, 2), x(:, 3))
    plot3(b(:, 1), b(:, 2), b(:, 3), 'o')
    view(3)
else
    plot(a(:, 1), a(:, 2))
    plot(x(:, 1), x(:, 2))
    plot(b(:, 1), b(:, 2), 'o')
end
title('Generalised Curve')
legend('curve', 'points', 'knots')

% Piece-wise input
x = [0.73, 0.72; 0.48, 0.85; 0.24, 0.97; 0.23, 0.89; 0.23, 0.89; 0.23, 0.80;
     0.46, 0.51; 0.56, 0.42; 0.56, 0.42; 0.67, 0.33; 0.65, 0.45; 0.64, 0.57];
a = bspl(x, 50);
b = [x(1, :); x(diff(x(:, 1)) == 0, :); x(end, :)];
figure
hold on
plot(a(:, 1), a(:, 2))
plot(x(:, 1), x(:, 2))
plot(b(:, 1), b(:, 2), 'o')
title('Piece-Wise Curve')
legend('curve', 'points', 'nodes')

% High-order approximation
x = 2 * pi * (0 : 1 / (5e3 - 1) : 1)';
x = [x, exp(-x / 2) .* sin(2 * x) + 5e-3 * (rand(5e3, 1) - 0.5)];
a = bspl(x, 1e4);
figure
hold on
b = plot(a(:, 1), a(:, 2), 'LineWidth', 7);
b.Color(4) = 0.1;
plot(x(:, 1), x(:, 2), '.', 'MarkerSize', 1)
title('Data Approximation')
legend('Fitted', 'Data')

% Iterative workflow
b = [1 4 6 4 1];
figure
hold on
h = cell(4, 1);
for i = 1 : 4
    x = rand(1, 5);
    plot(bspl(x, [], 1, b))
    h{i} = strcat('curve_', num2str(i));
end
title('Workflow Example')
legend(h)

% Speed test
n = 500;
t = zeros(n, 1);
bspl(rand(n));
for i = 1 : n
    x = rand(i, 2);
    tic
    for j = 1 : 10
        bspl(x, 10);
    end
    t(i) = toc / j;
end
figure
plot(smooth(t))
title('Speed Test')