Graphics Reference
In-Depth Information
Figure 9.14: A screen capture for the texture-mapping program of Exercise 9.6, showing
the texture on the left, a large triangle, and the locations of the triangle's vertices in texture
coordinates atop the texture. The resultant texturing on the triangle's interior is shown on
the right.
spaced points, but I want them to include all the original points!” That's gener-
ally not possible, but we can come close. Suppose that the shortest edge of the
original polygon has length
s
. Show that you can place approximately
L
/
s
points
Q
0
,
Q
1
,
,
P
n
, with the
property that the ratio of the greatest gap between adjacent points and the smallest
gap is no more than 2.
(c) Suppose you let yourself place
CL
...
on the original polyline, including all the points
P
0
,
...
s
points with the same constraints as in
the previous part, for some
C
greater than one. Estimate the max-min gap ratio in
terms of
C
.
Exercise 9.8:
Consider the interval
[
p
,
q
]
where
p
/
=
q
. If we define
α
(
x
)=
x
−
p
(
x
)=
x
−
q
p
p
and
β
q
, then
α
and
β
are called the
barycentric coordinates of
x
.
q
−
−
(a) Show that for
x
∈
[
p
,
q
]
, both
α
(
x
)
and
β
(
x
)
are between 0 and 1.
(b) Show that
α
(
x
)+
β
(
x
)=
1.
can be defined on the rest of the real line, and these defi-
nitions depend on
p
and
q
; if we call them
(c) Clearly
α
and
β
α
pq
and
β
pq
, then we can, for another
