Graphics Reference
In-Depth Information
Y
R
min
R
maj
X
Fig. 9.2.
An ellipse showing the major and minor radii.
9.3 Bezier Curves
Two people, working for competing French car manufacturers, are associated
with what are now called Bezier curves: Paul de Casteljau, who worked at
Citroen, and Pierre Bezier, who worked at Renault. De Casteljau's work was
slightly ahead of Bezier, but because of Citroen's policy of secrecy it was
never published, so Bezier's name has since been associated with the theory
of polynomial curves and surfaces. Casteljau started his research work in 1959,
but his reports were only discovered in 1975, by which time Bezier had already
become known for his special curves and surfaces.
9.3.1 Bernstein Polynomials
Bezier curves employ
Bernstein polynomials
, which were described by S. Bern-
stein in 1912. They are expressed as follows:
B
i
(
t
)=
n
t
i
(1
t
)
n−i
−
(9.5)
i
where
n
i
is shorthand for the number of selections of
i
different items from
n
distinguishable items when the order of selection is ignored, and equals
n
!
(9.6)
(
n
−
i
)!
i
!
where, for example, 3! (factorial 3) is shorthand for 3
1.
When (9.6) is evaluated for different values of
i
and
n
, we discover the
pattern of numbers shown in Table 9.1. This pattern of numbers is known as
Pascal's triangle
. In western countries they are named after a 17th century
French mathematician, even though they had been described in China as early
as 1303 in
Precious Mirror of the Four Elements
by the Chinese mathemati-
cian Chu Shih-chieh. The pattern represents the coe
cients found in binomial
expansions. For example, the expansion of (
x
+
a
)
n
for different values of
n
is
×
2
×
