Graphics Reference
In-Depth Information
That's it! You can run the program to see how well it works. When you click
only two points, your polygon looks like a line segment. When you then subdi-
vide, the line segment appears shorter. Explain to yourself why subdividing again
doesn't make it shorter still.
A similar process can be used to take a 3D polyhedron and “cut corners” in an
attempt to smooth it out. Will it always get smoother? We'll discuss this further
in Chapter 23. In the meantime, having seen how rapidly subdivision seems to
smooth out curves, you can ask yourself, “How could I analyze whether the curve
approaches a limit, and whether such a limit is smooth at a particular point?” This
is addressed (for a somewhat different subdivision method, but the principles are
similar) in Chapter 22.
We've given you the tools to create simple applications in WPF for experiment-
ing with ideas from graphics, and worked through a subdivision application as
an example of using those tools. The exercises in this chapter will give you the
opportunity to explore the power of the 2D test bed, and to discover some inter-
esting ideas in graphics that you'll encounter again later in this topic. We strongly
advise you to complete at least a couple of these exercises so that working with
this framework will be simple for you in the future.
This test bed was built for a reason. Lots of years of working in computer
graphics have taught us another principle:
T
HE FIRST PIXEL PRINCIPLE
:
The first pixel is the hardest.
When you write a new program in graphics, the most common first result is
a completely black screen. That's almost impossible to debug, because any of
a million things could be causing the problem. Usually when you manage to get
anything
to show up on the screen, you're finished with a great deal of your debug-
ging. It's therefore proven useful to start with a program that does something like
what you're trying to do, and to gradually modify it until it
does
do what you want,
but make the modifications so that at every stage you can tell, by looking at it, that
it's doing what you expected. The test bed provides you with a starting point for
a whole host of possible programs. We hope it will save you countless hours of
debugging.
Exercise 4.1:
Modify the corner-cutting program to place the cutoff points one-
fourth and three-fourths of the way along the line; describe the results.
Exercise 4.2:
Modify the corner-cutting program to be a “dualizing” program:
It replaces a polygon with its
dual
, a polygon whose vertices are the midpoints of
the original polygon, connected in the same order, so that the dual of a square is a
diamond. Experiment: Does repeated dualizing tend to remove crossings of self-
intersecting polygons? Can you find a polygon whose successive duals are
always
self-intersecting?
