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RE: MaxBend in ParametricPlot?
 To: mathgroup at smc.vnet.net
 Subject: [mg12961] RE: [mg12924] MaxBend in ParametricPlot?
 From: Ersek_Ted%PAX1A at mr.nawcad.navy.mil
 Date: Sun, 28 Jun 1998 02:51:50 0400
 Sender: ownerwrimathgroup at wolfram.com
Matthias Weber wrote:


I am having trouble with the results I get by ParametricPlot. The
documentation is sufficiently incomplete to prevent me from calling
this a bug, but at least I would like to report the _behaviour_ I am
unhappily confronted with:

When I draw a round circle with ParametricPlot, I would expect to get
the sample points more or less equidistributed, but the resulting
distribution is rather strange. At the end is some code illustrating
what I will explain now.

The online doc says that Plot and ParametricPlot use an adaptive
algorithm so that the kink angle between successive segments does not
exceed MaxBend, which is '10' by default. (10 degrees, I suppose?).
The minimal number of sample points is given by PlotPoints, which
initially is 25.

If I set MaxBend>1000 (degrees??), I get a list of 39 points. When
setting PlotDivision>1 (which, as I understand it, should prevent any
further subdivision), I get 40 points. All this lacks an explanation
to me.

The kink angle between consectutive segments on the drawn unit circle
with initial settings varies (roughly) between 0.4 and 8.0 degrees,
points with small kink angle accumulate near the points
{0,0},{0,1},{1,0},{0,1}.

For the application I have in mind I want to extract the data computed
by ParametricPlot and to use them, but I was hoping that
ParametricPlot gives 'more adaptive' results. Is there a parameter
setting which gives (near) equidistribution for a circle and is
nevertheless adaptive in general? Is there
a desription of the algorithm behind ParametricPlot available? Has
somebody written his/her own version already? 

_The Mathematica Graphics Guidebook_ by C. Smith & N. Blachman discusses
(in great detail) the sampling algorithm Plot and ParametricPlot used
in Version 2.2. I suspect the algorithm hasn't changed for version
3.0.
This aspect of the sampling algorithm is discussed on pages 8993 of the
above book.
In this discussion the authors explain that "bend angle" is computed by
normalizing the distances between adjacent samples so the three points
used to form an angle fit into a box with length of one on each side.
They say if A is a vector from the middle sample to the left sample, and
B is a vector from the middle sample to the right sample. Then
Cos[BendAngle]=Dot[A,B]/(Magnitute[A] Magnitude[B])
This results in excessive sampling where the plot is vertical or
horizontal. However it also has an advantage in that you can zoom in on
a graphic (without taking more samples) and the curve still looks
smooth.
Here is an example:
In[1]:=
cir=ParametricPlot[{Cos[t],Sin[t]},{t,0,2 Pi},AspectRatio>Automatic];
In[2]:=
Show[cir, PlotRange>{{0.03,0.03},{0.9993,1.}}, AspectRatio>1 /4,
ImageSize>{250,250}, Axes>None];
I don't know how to use the options to get the results you are looking
for.
Ted Ersek
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