Bezier curves

Bezier curves

Some observations:

• Tangent line at the first point passes through the second.
• Tangent line at the fourth point passes through the third.

This can be proved by taking derivatives of $x, y$ against $t$ and get $dy/dx |_{t = 0 \text{ or } 1}$.

Bizier curves and its handles

t = linspace(0, 1, 300)
t1 = (1-t).^3
t2 = t.*(1-t).^2
t3 = t.^2.*(1-t)
t4 = t.^3
tmat = vertcat(t1, t2, t3, t4)

ymove = linspace(0, 1, 4)
ymove = ymove(2:end)
pointsmat = []
numel(ymove)
for idx = 1:numel(ymove)
    anchorsmat = [0 0.4 0.5 0.6;
                  0 ymove(idx) 0.5 0]
    pointsmat = vertcat(pointsmat, anchorsmat * tmat)
end
pointsmatx = pointsmat(1:2:end, :)
pointsmaty = pointsmat(2:2:end, :)
plot(pointsmatx', pointsmaty')
hold
xmove = repmat(0.4, size(ymove), 1)
scatter(xmove, ymove)
hold on
linesegX = vertcat(repmat(0, size(xmove), 1), xmove)
linesegY = vertcat(repmat(0, size(ymove), 1), ymove)
plot(linesegX, linesegY)