bernsteinMatrix
Bernstein matrix
Syntax
Description
B = bernsteinMatrix(n,t)t is a vector, returns the length(t)-by-(n+1) Bernstein
matrix B, such that B(i,k+1)= nchoosek(n,k)*t(i)^k*(1-t(i))^(n-k).
Here, the index i runs from 1 to length(t),
and the index k runs from 0 to n.
The Bernstein matrix is also called the Bezier matrix.
Use Bernstein matrices to construct Bezier curves:
bezierCurve = bernsteinMatrix(n, t)*P
n+1 rows of the matrix P specify
the control points of the Bezier curve. For example, to construct
the second-order 3-D Bezier curve, specify the control points as:P = [p0x, p0y, p0z; p1x, p1y, p1z; p2x, p2y, p2z]
Examples
2-D Bezier Curve
Plot the fourth-order Bezier curve specified by the control points p0 = [0 1], p1 = [4 3], p2 = [6 2], p3 = [3 0], p4 = [2 4]. Create a matrix with each row representing a control point.
P = [0 1; 4 3; 6 2; 3 0; 2 4];
Compute the fourth-order Bernstein matrix B.
syms t
B = bernsteinMatrix(4,t);Construct the Bezier curve.
bezierCurve = simplify(B*P);
Plot the curve adding the control points to the plot.
fplot(bezierCurve(1), bezierCurve(2), [0, 1]) hold on scatter(P(:,1), P(:,2),'filled') title('Fourth-order Bezier curve') hold off
3-D Bezier Curve
Construct the third-order Bezier curve specified by the 4-by-3 matrix P of control points. Each control point corresponds to a row of the matrix P.
P = [0 0 0; 2 2 2; 2 -1 1; 6 1 3];
Compute the third-order Bernstein matrix.
syms t
B = bernsteinMatrix(3,t);Construct the Bezier curve.
bezierCurve = simplify(B*P);
Plot the curve adding the control points to the plot.
fplot3(bezierCurve(1), bezierCurve(2), bezierCurve(3), [0, 1]) hold on scatter3(P(:,1), P(:,2), P(:,3),'filled') hold off
3-D Bezier Curve with the Evaluation Point Specified as a Vector
Construct the third-order Bezier curve with the evaluation point specified by the following 1-by-101 vector t.
t = 0:1/100:1;
Compute the third-order 101-by-4 Bernstein matrix and specify the control points.
B = bernsteinMatrix(3,t); P = [0 0 0; 2 2 2; 2 -1 1; 6 1 3];
Construct and plot the Bezier curve. Add grid lines and control points to the plot.
bezierCurve = B*P; plot3(bezierCurve(:,1), bezierCurve(:,2), bezierCurve(:,3)) hold on grid scatter3(P(:,1), P(:,2), P(:,3),'filled') hold off
