To use the embedded plugins in this page you need the Imagine plugin.
In all the following graphs you can move the green points.
Look at the following graph.
The black point lies on the line segment between the 2 green points.
The slider controls how far away from the first green point the black point lies.
Pressing the t-value button shows the proportion.
For t=0.5 the black point is half way between the two green points,
for t=0.25 the point is one quarter of the distance between the two green points from the first point, and so on.
Pressing the labels button labels all the points.
We will call the black point the t-midpoint of the two green points.
Now we use 4 (movable) green points and mark the t-midpoints for each consecutive pair.
In the next step, we connect the new points with line segments.
Now we mark the t-midpoints on these two new line segments:
Now we connect the two new points:
Finally, we construct the t-midpoint on the new line segment (and mark it red).
We can trace this point by pressing the curve-button and '); by varying t (with the slider)
The curve produced this way is the Bezier curve.
You can move the green points to see the shape of the curve for different point configurations.