Hello Amina,
Creating a triangular Bézier patch involves defining a Bézier surface using a set of control points arranged in a triangular grid. A triangular Bézier patch is a type of Bézier surface that is defined over a triangular domain, typically parameterized by barycentric coordinates.
Here's a basic example of how you can create and visualize a triangular Bézier patch:
function triangularBezierPatch()
% Define control points for a quadratic triangular Bézier patch
controlPoints = [0, 0, 0; 1, 0, 0; 0, 1, 0; 0.5, 0.5, 1];
degree = 2;
% Generate barycentric coordinates
[u, v] = meshgrid(linspace(0, 1, 20));
w = 1 - u - v;
valid = (w >= 0);
u = u(valid); v = v(valid); w = w(valid);
% Compute Bézier patch points
points = computeBezierPatch(controlPoints, degree, u, v, w);
% Plot Bézier patch
trisurf(delaunay(u, v), points(:, 1), points(:, 2), points(:, 3), ...
'FaceColor', 'cyan', 'EdgeColor', 'none');
hold on;
plot3(controlPoints(:, 1), controlPoints(:, 2), controlPoints(:, 3), ...
'ro', 'MarkerSize', 8, 'LineWidth', 2);
title('Triangular Bézier Patch');
axis equal; grid on; view(3);xlabel('X');ylabel('Y');zlabel('Z');
end
function points = computeBezierPatch(controlPoints, degree, u, v, w)
% Compute points on the Bézier patch given barycentric coordinates
numPoints = length(u);
numControlPoints = size(controlPoints, 1);
points = zeros(numPoints, 3);
% Compute Bernstein polynomials for the triangular Bézier patch
B = zeros(numPoints, numControlPoints);
index = 1;
for i = 0:degree
for j = 0:(degree - i)
k = degree - i - j;
B(:, index) = nchoosek(degree, i) * nchoosek(degree - i, j) .* ...
(u .^ i) .* (v .^ j) .* (w .^ k);
index = index + 1;
end
end
% Compute the Bézier surface points
for i = 1:numControlPoints
points = points + B(:, i) * controlPoints(i, :);
end
end
- Define a set of control points arranged in a triangular grid. Here a quadratic patch is used, which requires 6 control points.
- Use barycentric coordinates (u, v, w) to parameterize the triangular domain.
- Compute the Bernstein basis polynomials for the triangular Bézier patch.
- Calculate the points on the Bézier patch using the Bernstein polynomials and control points.
- Use 'trisurf' function to visualize the Bézier patch and plot the control points.
This code provides a simple framework for generating and visualizing a triangular Bézier patch. You can modify the control points and degree to create different patches.
