Graphics Reference
In-Depth Information
(2) They take more storage. For example, a circle takes seven control points and ten
knots.
(3) They can produce bad parameterizations.
Finally, we point out to the interested reader some topics that were omitted in this
chapter (see, for example, [HosL93]):
(1) Splines in tension: These are interpolating splines that try to dampen unwanted
oscillations and extraneous inflection points. See also [Niel74] or [BaBB87].
(2) Shape preserving splines: If data is monotone or convex, then so should be the
spline.
(3) Various “geometric” spline curves
11.16
E
XERCISES
Section 11.2.1
11.2.1.1
Determine the Lagrange polynomials through the following sequences of points:
(a)
(0,5), (1,2), (3,2)
(b)
(-1,1), (0,-1), (1,-1), (2,1)
Section 11.2.2
11.2.2.1
Prove the validity of equation (11.21) by a direct argument, that is, let
2
3
px
i
()
=+ +
a bx cx
+
dx
and solve for the coefficients a, b, c, and d as in the proof of Lemma 11.2.2.1.
11.2.2.2
Prove equation (11.24).
Section 11.2.3
11.2.3.1
Use equations (11.25) and (11.33) to find the cubic spline curve p(u) that has knots
0, 1, 2, and 3 and that passes through the points (0,0), (2,2), (4,2), and (6,4). Assume
that p¢(0) = (1,3) and p¢(3) = (1,6).
Section 11.3
11.3.1
Express the curve
(
)
()
=+-+
3
3
2
pu
32
u
,
u
52 7 3
,
u
+ -
u
in the form of equation (11.34).
