Graphics Reference
In-Depth Information
at 1, the shape of the graph of z(u) can be realized by a parabola as shown in the
figure. The function z(u) achieves its maximum value at u = 0.5. (By uniqueness, since
a parabola is able to satisfy the given data, it
must
be the curve. Actually solving for
z(u) would give us
2
()
=-
(
)
zu
5052 5
u
-
.
+
.
,
but we do not need this precise formula.) It follows that as u moves from 0 to 1, x(u)
increases steadily from 1 to 6 in a uniform way and the function z(u) starts at 1 and
increases until u = 0.5, then it decreases back to 1. This leads to the indicated sketch
of the projection of p(u) to the x-z plane.
Curve (c).
See Figure 11.7(c). The graph of x(u) needs to have the shape shown since
its slope is 20 at both 0 and 1. It is a cubic. The only fact that we need to believe that
requires perhaps a little extra experience with functions is that the local maximum
and minimum occur at some values a and b, respectively, where 0 < a < 0.5 < b < 1.
The function z(u) has slope 40 at 0 and -40 at 1. Therefore, since this can again be
realized by a parabola which takes on its maximum value at u = 0.5, it is that parabola.
Its actual formula happens to be
2
()
=-
(
)
zu
20
u
-
0 5
.
+
6
,
but this is again not important for what we are doing. All that we need to
know is that as u moves from 0 to a, the x-coordinate of p(u) is increasing and
so is the z-coordinate. As we move from a to 0.5, x is decreasing, but z is still
increasing. Moving from 0.5 to b, both x and z are decreasing. Finally, as u moves
from b to 1, x is increasing, but z is decreasing. The reader should check that the
x- and z-coordinates of the self-intersecting loop shown on the right in Figure
11.7(c) behave in the same way as one moves from the left to the right endpoint of
that curve.
Curve (d).
See Figure 11.7(d). The graph of x(u) is again a straight line and the
shape of the graph of z(u) is forced by its slope of 10 at both ends to be the cubic
as shown. The rest of the argument is, like for curve (c), based on an analysis of
the regions where x and z are increasing and decreasing. This finishes Example
11.3.1.
Example 11.3.1 and others such as Exercise 11.3.2 should begin to give the reader
a feel for how changing
p
0
and
p
1
affects a curve.
One other useful matrix form is the one for a cubic curve that interpolates four
points
p
0
,
p
1
,
p
2
, and
p
3
. Although equation (11.8) already described a general solu-
tion for this problem, it is worthwhile to state the special uniform case explicitly. That
is the case where the u
i
are chosen to be the numbers 0, 1/3, 2/3, and 1, in other words,
p
i
= p(i/3). Let
