This preview shows pages 1–2. Sign up to view the full content.
Angel: Interactive Computer Graphics, Fourth Edition
Chapter 11 Solutions
11.1 (
m
+1)
3
11.3 As
u
varies over (
a, b
),
v
=
u
−
a
b
−
a
varies over (0
,
1). Substituting into
the polynomial
p
(
u
)=
∑
n
k
=0
c
k
u
k
,wehave
q
(
v
)=
∑
v
i
=0
d
i
v
i
=
∑
n
k
=0
c
k
((
b
−
a
)
v
+
a
)
k
.
We can expand the products on
the right and match powers of
v
to obtain
{
d
i
}
.
11.5 Consider the Bernstein polynomial
b
kd
(
u
)=
d
k
!
u
k
(1
−
u
)
d
−
k
.
For
k
=0or
k
=
d
, the maximum value of 1 is at one end of the interval
(0,1) and the minimum is at the other because all the zeros are at 1 or 0.
For other values of
k
, the polynomial is 0 at both ends of the interval and
we can di±erentiate to ²nd that the maximum is at
u
=
k/d.
Substituting
into the polynomial, the maximum value is
d
!
d
d
k
k
k
!
(
d
−
k
)
d
−
k
(
d
−
k
)!
which is always
between 0 and 1.
11.7 Proceeding as in the text, we have the interpolating control point
array
q
and can form the interpolating polynomial
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 01/25/2011 for the course CSCI 6821 taught by Professor Alxe during the Spring '10 term at Georgia Southwestern.
 Spring '10
 Alxe
 Computer Graphics

Click to edit the document details