Bezier's Curve

The parametric expression of a cubic Bezier[Artificial Intelligence]

Now, the de Casteljau algorithm by itself is not very useful in drawing a cubic Bezier curve, since this would require the computation of a large number of the points on the curve for obtain a certain precision. Nevertheless, for further practical exploitation of the Bezier curves, the parametric expression of the coordinates of points on a cubic Bezier curve is necessary.
Let's start by note that the point D that divides the EF segment in a t/(1-t) ratio has the coordinates given by the expressions:

D_x = (1-t)•E_x+t•F_x and D_y = (1-t)•E_y+t•F_y   (1)

Since the expressions for the x and y coordinates are similar, in the followings the non-indexed notation will be used as a shortcut for expressions (1) above:
D = (1-t)•E + t•F   (1')
With this notation:

• after the step 1, the A, B and C coordinates will be:
  A = (1-t)•P₁ + t•C₁
  B = (1-t)•C₁ + t•C₂
  C = (1-t)•C₂ + t•P₂
• after step 2, the M and N coordinates will be:
  M = (1-t)•A + t•B = (1-t)²•P₁ + 2•(1-t) •t•C₁ + t²•C₂
  N = (1-t)•B + t•C = (1-t)²•C₁ + 2•(1-t) •t•C₂ + t²•P₂
• after step 3, the coordinates of the P point will be:
  P = (1-t)•M + t•N = (1-t)³•P₁ +
      + 3•(1-t)²•t•C₁ + 3•(1-t)•t²•C₂ + t³•P₂

The expression:

P = (1-t)³•P₁ + 3•(1-t)²•t•C₁ + 3•(1-t)•t²•C₂ + t³•P₂    (3)
  with t taking values in the [0, 1] interval

is called the parametric expression of a cubic Bezier curve.
For example, applying the expression (3) for a 0.5 parameter value, results in:

mid_c = (P₁ + 3•C₁ + 3•C₂ + P₂)/8    (4)

for the location of the middle point of a Bezier curve.

[Note] The Coordinate A may be calculated for both Ax, Ay.
            (P1x,P1y) - Starting Point.
            (P2x,P2y) - Ending Point.
            (Cx,Cy) - Common Intermediate Points.