Curves in 3D - 3D Math Primer for Graphics and Game Development - page 684

p 0 = b 0 ,

(13.32)

v 0 = 3( b 1 − b 0 ), (13.33)

Converting cubic curve
from Bezier form to
Hermite form

v 1 = 3( b 3 − b 2 ), (13.34)

p 1 = b 3 . (13.35)

Or, we can convert from Hermite to Bezier:

Converting cubic curve
from Hermite form to
Bezier form

b 0 = p 0 ,
b 1 = p 0 + (1/3) v 0 ,
b 2 = p 1 − (1/3) v 1 ,
b 3 = p 1 .

Thus, Hermite and Bezier forms are very closely related, and it is very
easy to convert between them. Their relationship is depicted graphically in
Figure 13.14.

Figure 13.14
Relationship between B ezier and Hermite
forms

We've said that the first derivative at either endpoint is completely de-
termined by the nearest two Bezier control points. We can actually make a
more general statement. The nth derivative at either endpoint is completely
determined by the nearest n + 1 control points. The “0th derivative” (the
position of the curve) is completely determined by the interpolated con-
trol point. The first derivative has been discussed. The second derivative
(acceleration) at the end of the curve is determined by the closest three
control points. In fact, let's see exactly what the acceleration is in terms
of the Bezier control points for a cubic curve. Converting the accelera-
tion function (Equation (13.6)) from monomial to Bezier form, we get