Computer Aided Engineering Design

DESIGN OF CURVES 111

(e) Variation Diminishing: For a planar Bézier segment, it can be verified geometrically that no
straight line on that plane intersects with the segment more times than it does with the corresponding
control polyline. Similarly, for a spatial Bézier curve, the property holds for a plane intersecting the
curve and its control polyline. Note that special cases may occur when one or more legs of the control
polyline may coincide with the intersecting line or a plane for which the property holds true. This is
because the number of intersection points of the curve with the intersecting line/plane would be
finite. However, with the control polyline, they would be infinitely many. The property suggests that
the shape of the curve is no more complex compared with its control polyline. In some sense, the
convex hull and variation diminishing properties, together, suggest that the shape of a Bézier segment
is predictable and is roughly depicted by the control polyline. In a singular case where the control
polyline is a straight line (control line), so is the Bézier segment from the convex hull property. Here,
cae, the variation diminishing property may not be used as it is inconclusive especially when the
intersecting line/plane happens to coincide with/contain the control line.
( f ) Symmetry: Due to the symmetry in of Bernstein polynomials (Eq. (4.36)), if the sequence of the
control points is reversed, i.e. PPnr*– = ,r the symmetry of the curve is preserved, that is

rΣΣ

n

rr

n

r

n

nr r

Bu Bn u

=0 ( ) = (1 – )PP=0 –

(g) Parameter Transformation: At times, we may have to express a Bézier segment as a non-
normalized parameter u′ between aand b. In such a case, set

u = uaba′ – – (4.41)

to use Eq. (4.39).
(h) No Local Control: The shape of a Bézier segment changes globally if any data point is moved to
a new location. To see this, let a control point Pk be moved along a specified vector v. The original
Bézier segment changes to

rPPPrnew() = u BuBui=0 () + ()( + ) = =0 BuBu uBu() + () = () + ()

ik

n

i

n

i k

n

k i

n

i

n

i k

n

k

ΣΣn

≠

vvv(4.42)

Foru between 0 and 1, every point on the old Bézier segment r(u) gets translated by Bukn()vimplying
that the shape of the entire curve is changed.

(i) Derivative of a Bézier Curve
Using Eq. (4.38)

dudu nB u B u n B u n B u

i

n

i

n

i

n

i i

n

i

n

i i

n

i

n

rPPP()/ = () – () = () – ()i

=0 –1

–1 –1

=0 –1

–1

=0

ΣΣΣ–1

[]

= () + ()–1–1 0 – () – ()

=1 –1

–1

–1 =0

–1

nBun nBu nBu nBu–1 –1

i

n

i

n

i i

n

i

n

in

n

PP PPΣΣ n

= () – ()

=0

–1

–1 +1

=0

–1

nBu n Bu–1

i

n

i

n i

i

n

i

ΣΣPPn i

= ( )(=0 – )

–1

–1

nB ui +l

n

i

n

Σ PPi i (4.43)