A third degree Bezier curve is defined by four points: P0,P1,P2 and P3.
The curve is constructed as follows:
On the segments P0P1, P1P2 and P2P3 the points Q0,Q1 and Q2 are drawn such that P0Q0/P0P1=P1Q1/P1P2=P2Q2/P2P3=t (t in [0,1]).
On the segments Q0Q1 and Q1Q2 the points R0 and R1 are drawn such that Q0R0/Q0Q1=Q1R1/Q1Q2=t for the same value of t.
On the segment R0R1 the point B is drawn such that R0B/R0R1=t for the same value of t. The Bezier curve defined by the points P0, P1, P2, P3 is the locus of B as Q0 takes all possible positions on the segment P0P1. (Please note that for all points the value of t is the same.)
In the applet to the right you can drag the points P0, P1, P2 and P3 to see what the Bezier curve (green curve) defined by those points looks like. You can also drag the point Q0 along the segment P0P1.
From the construction it is clear that the bezier curve will be tangent to the segments P0P1 in P0 and P2P3 in P3.