Bernstein Polynomial and B´ezier-Bernstein Spline - Bezier and Splines in Image Processing and Machine Vision - page 24

for quadrant II.

X Y = y 1 −

= x 1 hor,

X Y = y 1 ,

= x 1 − h

for quadrant III and,

X = x 1 + h Y = y 1 or,

X = x 1 Y = y 1 − h,

for quadrant IV, corresponding to the two possible senses.
Having determined the point ( X ,Y ), the next task is to construct the
line passing through ( X ,Y ) and parallel to P 1 P 2 so that it meets the line
P 2 P 3 at some point P 4 . The midpoint of this line, together with the pairs of
points ( X ,Y ), ( X 1 ,Y 1 ), and (( X 2 ,Y 2 ) ,P 4 ) then constitutes the Bezier
characteristic triangles for the arc.

1.6.3 Recursive Computation Algorithm

The recursive algorithm for computation of values for the second-order Bezier
approximation curve uses the forward difference scheme. Let

y = at 2 + bt + c

be a polynomial representation of (1.17), where the constant parameters
a, b, c are determined by the vertices of the Bezier characteristic triangle.
Suppose a number of points (values of y) on the arc are to be evaluated for
equispaced value of the independent variable t . The usual Newton's method
of evaluating the polynomial results in multiplications and does not make use
of the previously computed values to compute new values.
Assume that the parameter t ranges from 0 to 1. Let the incremental value
be q . Then the corresponding y values will be c , aq 2 + bq + c ,4 aq 2 +2 bq + c ,
9 aq 2 +3 bq + c,

. The difference Table 1.1 for recursive computation of points
for Bezier curve then takes the following form. Observe that

···

Table 1.1. Difference table for recursive computation of points.
t

2 y (2nd difference)

y y (1st difference)

aq 2 + bq 2 aq 2 0 c

aq 2 + bq + c 3 aq 2 + bq 2 aq 2 q

2q 4 aq 2 +2 bq + c 5 aq 2 + bq 2 aq 2

3q 9 aq 2 +3 bq + c 7 aq 2 + bq

4q 16 aq 2 +4 bq + c