Cubic polynomial as single Bezier cubic curve

Samuel Dagan

Last Update: 17 May 2012

Bezier cubic curve and cubic polynomial

A Bezier cubic curve is defined by four points  in the cartesian coordinate system: Pn(Pnx ,  Pny)  with   n=0,1,2,3 . The curve is expressed by the point  P(x,y)  defined by the following parametric relation:
P = t3(P3-3P2+3P1-P0) +3t 2(P2-2P1+P0) +3t (P1-P0) + P0 , where  t  is a parameter in the interval of  0 ≤ t ≤ 1 .

A cubic polynomial function is given in general by the relation  y=a0+a1x+a2x2+a3x3 , where the indexed a's are real constants. If the variable  x  is limited in an interval  x0 ≤ x ≤ x0   with   χ>0 , then the polynomial function is equivalent to a special case of the cubic Bezier curves.

Conditions to be applied on the Bezier curve

In order to express the polynomial by a Bezier curve, one have to apply the following restriction: the  x  component of  P  should be linear to  t  . Only that way the  y  component of  P  becomes a cubic polynomial of  x . This linear expression should be consistent with the intervals of  x  and  t , therefore  t=(x−x0)/χ  with its inverse  x=x0+χt . This implies the following relations:
P3x-3P2x+3P1x-P0x = 0
P2x-2P1x+P0x = 0
3(P1x-P0x) = χ
P0x = x0

The solution of these relations is:  P0x=x0 ,  P1x=x1=x0+(χ/3) ,  P2x=x2=x0+(2χ/3)  and  P3x=x3=x0 .
The corresponding  t  values are  t0=0 , t1=1/3 , t2=2/3  and  t3=1 .
The corresponding  y  coordinates on the curve are:  yn =y(xn)  with  n=0,1,2,3 .
By definition  P0y=y0  and  P3y=y3 .

y components of the control points

For convenience, the expression of  P  is rewritten in a different form:
P = t3P3 + 3t2(1-t)P2 + 3t(1-t)2P1 + (1-t)3P0 .

In the case of  t=t1=1/3  one obtains:  y1 = (1/3)3y3 + 3(1/3)2(2/3)P2y + 3(1/3)(2/3)2P1y + (2/3)3y0 .
In the case of  t=t2=2/3  one obtains:  y2 = (2/3)3y3 + 3(2/3)2(1/3)P2y + 3(2/3)(1/3)2P1y + (1/3)3y0 .
Both represent two linear equations with two unknowns  P1y  and  P2y .

The solution is:
P1y=-5y0 +18y1 -9y2 +2y3 ) / 6  and
P2y=2y0 -9y1 +18y2 -5y3 ) / 6 .