Curves in 3D - 3D Math Primer for Graphics and Game Development - page 682

section of the curve that looks how we want, we would prefer that editing
of some other distant control point not disturb the section that was shaped
the way we liked it. This envious situation, known as local support, occurs
when we move a particular control point and only the part of the curve
near that control point is affected, for some definition of “near.”
Local support means that the basis function is nonzero only in some
interval, and outside this interval it is zero. Unfortunately, such a ba-
sis function cannot be described as a polynomial, and thus no polynomial
curve can achieve local control. However, local support is possible by piec-
ing together small curves that fit together just right to form a spline, as
Section 13.6 discusses.

One local maximum. Although each control point exercises influence over
the entire curve, each exerts the most influence at one particular point
along the curve. Each Bernstein polynomial B i (t), which serves as the
blend weight for the control point b i , has one maximum at the auspicious
time t = i/n. Furthermore, at that time, b i exerts more weight than any
other control point.
Thus, although every point on the interior of the curve is influenced to
some degree by all the control points (because Bezier control points have
global support), the nearest control point has the most influence.

13.4.3 B ezier Derivatives and Their Relationship
to the Hermite Form
Let's take a look at the derivatives of a Bezier curve. Since we like to use
the cubic curve as our example, we're talking about the velocity and accel-
eration of the curve. Remember that the velocity is related to the tangent
(direction) of the curve, and the acceleration is related to its curvature.
Section 13.1.6 showed how to get the velocity function of a curve from
the monomial coe cients:

p (t) = c 0 + c 1 t + c 2 t 2 + c 3 t 3 ,
v (t) = p (t) = c 1 + 2 c 2 t + 3 c 3 t 2 .

Position and velocity of a
cubic curve

(13.29)

And Section 13.4.1 showed how to extract the monomial coe cients from
a cubic Bezier curve:

c 0 = b 0 ,
c 1 = 3 b 1 − 3 b 0 ,
c 2 = 3 b 0 − 6 b 1 + 3 b 2 ,
c 3 = − b 0 + 3 b 1 − 3 b 2 + b 3 .