Hi Denis,
At the bottom-most rendering level monotonic curves can be cool to deal
with, but I'm dubious that they help with widening. For one things, I
think you need more breaks than they would give you and also they might
sometimes break a curve when it doesn't need it.
One way in which they may not break enough is that I think that
inflections also need to be broken in order to find a parallel curve
(though I suppose a very tiny inflection might still be approximated by
a parallel curve easily) and inflections can happen at any angle without
going horizontal or vertical.
Secondly, although a circle tends to be represented by quadrant sections
which are monotonic, a circle rotated by 45 degrees would have
horizontal and vertical sections in the middle of each quadrant and you
would split those, but really they can be made parallel regardless of
angle so these would be unnecessary splits.
My belief is that lengths and angles of the control polygon help
determine if it is well-behaved and can be made parallel simply by
offsetting. Some formula involving those values would likely be happy
with circle sections regardless of their angle of rotation. I believe
the Apache Harmony version of BasicStroke used those criteria...
...jim
On 8/25/2010 2:36 PM, Denis Lila wrote:
Hello Jim.
I think a more dynamic approach that looked at how "regular" the curve
was would do better. Regular is hard to define, but for instance a
bezier section of a circle could have parallel curves computed very
easily without having to flatten or subdivide further. Curves with
inflections probably require subdividing to get an accurate parallel
curve.
I'm not sure if you read it, but after the email with the webrev link
I sent another describing a different method of widening: split the
curve at every t where dx/dt == 0 and dy/dt == 0. This guarantees that
there will be no more than 5 curves per side, and since each curve will
be monotonic in both x and y the curve that interpolates its parallel
should do a pretty good job.
This is far better than interpolating at regular t intervals, but I'm
trying to find a better way. I don't like this because the split
depend not only on the curve itself, but also on the axes. The axes are
arbitrary, so this is not good. For example a curve like this
|
\_ would get widened by 1 curve per side (which is good and optimal), but
if we rotate this curve by, say, 30 degrees it would be widened by 2 curves
per side.
It also doesn't handle cusps and locations of high curvature very well (although
I think the latter is a numerical issue that could be made better by using
doubles).
Regards,
Denis.
----- "Jim Graham"<james.gra...@oracle.com> wrote:
Hi Denis,
On 8/23/2010 4:18 PM, Denis Lila wrote:
To widen cubic curves, I use a cubic spline with a fixed number
of curves for
each curve to be widened. This was meant to be temporary, until I
could find a
better algorithm for determining the number of curves in the spline,
but I
discovered today that that won't do it.
For example, the curve p.moveTo(0,0),p.curveTo(84.0, 62.0,
32.0, 34.0, 28.0, 5.0)
looks bad all the way up to ~200 curves. Obviously, this is
unacceptable.
It would be great if anyone has any better ideas for how to go about
this.
To me it seems like the problem is that in the webrev I chop up the
curve to be
interpolated at equal intervals of the parameter.
Perhaps looking at the rate of change of the slope (2nd and/or 3rd
derivatives) would help to figure out when a given section of curve
can
be approximated with a parallel version?
I believe that the BasicStroke class of Apache Harmony returns widened
curves, but when I tested it it produced a lot more curves than Ductus
(still, probably a lot less than the lines that would be produced by
flattening) and it had some numerical problems. In the end I decided
to
leave our Ductus stuff in there until someone could come up with a
more
reliable Open Source replacement, but hopefully that code is close
enough to be massaged into working order.
You can search the internet for "parallel curves" and find lots of
literature on the subject. I briefly looked through the web sites,
but
didn't have enough time to remember enough calculus and trigonometry
to
garner a solution out of it all with the time that I had. Maybe
you'll
have better luck following the algorithms... ;-)
As far as flattening at the lowest level when doing scanline
conversion,
I like the idea of using forward differencing as it can create an
algorithm that doesn't require all of the intermediate storage that a
subdividing flattener requires. One reference that describes the
technique is on Dr. Dobbs site, though I imagine there are many
others:
http://www.drdobbs.com/184403417;jsessionid=O5N5MDJRDMIKHQE1GHOSKH4ATMY32JVN
You can also look at the code in
src/share/native/sun/java2d/pipe/ProcessPath.c for some examples of
forward differencing in use (and that code also has similar techniques
to what you did to first chop the path into vertical pieces). BUT
(*Nota Bene*), I must warn you that the geometry of the path is
somewhat
perturbed in that code since it tries to combine "path normalization"
and rasterization into a single process. It's not rendering pure
geometry, it is rendering tweaked geometry which tries to make non-AA
circles look round and other such aesthetics-targeted impurities.
While
the code does have good examples of how to set up and evaluate forward
differencing equations, don't copy too many of the details or you
might
end up copying some of the tweaks and the results will look strange
under AA. The Dr. Dobbs article should be your numerical reference
and
that reference code a practical, but incompatible, example...
...jim
