Draw Bezier curve through 4 control points

Control points P0(0,0), P1(3,3), P2(−2,−2), P3(2,3). Four points give a cubic Bezier curve.

P(u) = (1−u)³P0 + 3u(1−u)²P1 + 3u²(1−u)P2 + u³P3

Power form: a = −P0 + 3P1 − 3P2 + P3 = (9 + 6 + 2, 9 + 6 + 3) = (17, 18); b = 3P0 − 6P1 + 3P2 = (−18 − 6, −18 − 6) = (−24, −24); c = −3P0 + 3P1 = (9, 9).

x(u) = 17u³ − 24u² + 9u, y(u) = 18u³ − 24u² + 9u

Blending values and points:

Example, u = 0.5: x = 0.375·3 + 0.375·(−2) + 0.125·2 = 1.125 − 0.75 + 0.25 = 0.625; y = 0.375·3 + 0.375·(−2) + 0.125·3 = 1.125 − 0.75 + 0.375 = 0.75.

Drawing steps: plot the four control points and join them to form the control polygon. Plot the points in the table and join them smoothly in order of u.

 3 |                                o P3    o P1
   |
 2 |                        e
   |
 1 |                   d c  b
   |                     a
 0 |----------------o P0----------------------> x
   |
-1 |
   |
-2 |o P2
    -2              0       1       2       3

 a, b, c, d, e = curve points at u = 0.1, 0.25, 0.5, 0.75, 0.9
 curve order: P0 -> a -> b -> c -> d -> e -> P3

The curve leaves P0 along P0P1, goes out to about (1,1), turns back because of the pull of P2, and then moves up towards P3. It passes only through P0 and P3, not through P1 or P2.