Circles are not very circular

[Ken Starks]

Yesterday I downloaded a picture of a 360 degree protractor from the
examples gallery.
http://www.texample.net/tikz/examples/degree-wheel/

The outline of the protractor is a circle of radius 2.0cm, and
tick-marks at
degree intervals run from 1.8 cm to 2.0 cm

*** However ***
Some of the tick marks do not reach the circle.
You can see this by magnifying the edge in Adobe Viewer.
The stroke-width is large, so the gap indicates a significant error.
(By the way, I can't see a gap in the PStricks version)

The tick marks are OK at 0, 90, 180, 270 degrees, and at
their worst at 45, 135, 225, 315 degrees.


I presume that the Circle is implemented as a four-curve
cubic Bezier, and suspect that the length of the
control vector is too big.  The normal value given in most software
is  4/3 of  (sqrt(2) * the radius.

See for example:  http://www.tinaja.com/glib/bezcirc.pdf

This value forces the Bezier to be exact at eight points
(0, 45, 90, ... , 315 degrees ), with small errors (always too large)
in between.
If this accuracy is unacceptable, you could always use
more cubic-bezier-segments; you could get eight segments from
a look-up table of 5 values and `four rules' arithmetic, so
you don't need to rely on a TeX-based arithmetic engine for
Trigonometry or square Roots.


I have modified the code (see below), replacing the circle command with
a low-level Bezier version, and the gaps have indeed disappeared.
But I left the ordinary circle for the `shading' so you can see the
difference.

I would appreciate comments---and if necessary action---by someone
on the programming team.

Bye for now,
Ken.

% #################### code follows ##################

% A simple compass
% Author: Dario Orescanin
% modified: Ken Starks
% to reduce errors (Magnify at 45, 135, 225, 315 degrees, near edge of
protractor)

\documentclass{minimal}
\usepackage{tikz}
\usepackage{verbatim}

\begin{comment}
:Title: Degree wheel
:Tags: Foreach

A degree wheel inspired by `an example`_ on the `PSTricks website`_.

.. _an example: http://tug.org/PSTricks/main.cgi?file=examples#compass
.. _pstricks website: http://tug.org/PSTricks/main.cgi

:Author: Dario Orescanin

\end{comment}

\begin{document}

\begin{centering}

% Define a few constants for easy configuration
\def\radius{2cm}
\def\onedegrad{1.8cm}
\def\fivedegrad{1.75cm}
\def\tendegrad{1.7cm}
\def\labelrad{1.6cm}


\begin{tikzpicture}[scale=4]

 % adding a subtle gray tone to add a bit of "personality"
 \shade[shading=radial, inner color=white, outer color=gray!15] (0,0)
circle (\radius);

 %  START OF REPLACEMENT (I've left the shading in an ordinary circle,
so you can
 % see the difference.

 % \draw (0,0) circle (\radius);

 \def\cveclen{0.55228475*\radius}
 %  (4.0 * (sqrt(2)-1)/3.0) * \radius
 % for more about accuracy, see:
 % http://www.tinaja.com/glib/bezcirc.pdf

 \pgfpathmoveto{\pgfpoint{\radius}{0cm}};
 \pgfpathcurveto
 
{\pgfpoint{\radius}{\cveclen}}{\pgfpoint{\cveclen}{\radius}}{\pgfpoint{0cm}{\radius}};


 \pgfpathcurveto  
{\pgfpoint{-\cveclen}{\radius}}{\pgfpoint{-\radius}{\cveclen}}{\pgfpoint{-\radius}{0cm}};


 \pgfpathcurveto
 
{\pgfpoint{-\radius}{-\cveclen}}{\pgfpoint{-\cveclen}{-\radius}}{\pgfpoint{0cm}{-\radius}};


 \pgfpathcurveto
 
{\pgfpoint{\cveclen}{-\radius}}{\pgfpoint{\radius}{-\cveclen}}{\pgfpoint{\radius}{0cm}};

 \pgfclosepath;
 \pgfusepath{stroke};



 \draw[fill=black] (0,0) circle (.02mm);
 \node[draw, circle, inner sep=.2mm] (a) at (0,0) {};

 % helper lines
 \foreach \x in {0, 45, ..., 360} \draw[very thin, gray!40] (a) --
(\x:\radius);

 % main lines
 \foreach \x in {0,...,359} \draw (\x:\onedegrad) -- (\x:\radius);

 % labels and longer lines at every 10 degrees
 \foreach \x in {0,10,...,350}
 {
   \node[scale=1.4, rotate=\x*-1] at (360-\x+90:\labelrad) {\x};
   \draw (\x:\tendegrad) -- (\x:\radius);
 };

 % lines at every 5 degrees
 \foreach \x in {0,5,...,355}  \draw (\x:\fivedegrad) -- (\x:\radius);

\end{tikzpicture}
\end{centering}


\end{document}



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[Mark Wibrow-2]

Hi,

Your suggestion of 0.55228475 is what I understand to give the
*maximum* error for cubic bezier approximation of circular arcs. The
minimum error (again as I understand it - I am not a maths or a CAD
person) is actually provided by 0.55191496.

However, having just looked at \pgfpathellipse I see that the "magic
number" used is actually 0.555, which is neither of these two. I don't
why this number was chosen, but there may have been a good reason.

I suggest submitting this either as a bug or a feature request.

In the mean time replacing 0.555 in \pgfpathellipse with numbers of
your choice may provide preferable results (and will effect all paths
such as circle and ellipse in TikZ). \pgfpathellipse is defined in
\pgfcorepathconstruct.code.tex

Regards

Mark

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[Till Tantau-3]

Dear Ken, dear Mark,


the number 0.555 used in pgf is actually a bit embarrassing... I  
choose the value 0.555 about a zillions ago (this was already in pgf  
when pgf.sty was still a single file (!) ). It was found by trial and  
error and looked "good enough" and no one ever complained. Naturally,  
I should have looked up the right number.

The value 0.5522847 gives much better results, indeed.

Mark, 0.55191496. does not seem to give as good results as 0.5522847.  
Also, in the calculation from the link Ken send I think it is proved  
that 4/3(sqrt(2)-1) should be optimal. Where does 0.55191496 come from?

Anyway, I fixed this in the CVS, which now uses 0.5522...

Many thanks for pointing this out.


While we are at it, do you have a good formula (implementable in  
TeX...) for computing the lengths of the support vectors for arcs? For  
instance, what is the correct length of the support vectors for an arc  
of 80 degrees? or 30.234 degrees? Currently, some more trial and error  
numbers are used and the results are, well, not-so-good...


Best regards,
Till


Am 09.10.2008 um 20:06 schrieb Mark Wibrow:

> Hi,
>
> Your suggestion of 0.55228475 is what I understand to give the
> *maximum* error for cubic bezier approximation of circular arcs. The
> minimum error (again as I understand it - I am not a maths or a CAD
> person) is actually provided by 0.55191496.
>
> However, having just looked at \pgfpathellipse I see that the "magic
> number" used is actually 0.555, which is neither of these two. I don't
> why this number was chosen, but there may have been a good reason.
>
> I suggest submitting this either as a bug or a feature request.
>
> In the mean time replacing 0.555 in \pgfpathellipse with numbers of
> your choice may provide preferable results (and will effect all paths
> such as circle and ellipse in TikZ). \pgfpathellipse is defined in
> \pgfcorepathconstruct.code.tex
>
> Regards
>
> Mark
>
> -------------------------------------------------------------------------
> This SF.Net email is sponsored by the Moblin Your Move Developer's  
> challenge
> Build the coolest Linux based applications with Moblin SDK & win  
> great prizes
> Grand prize is a trip for two to an Open Source event anywhere in  
> the world
> http://moblin-contest.org/redirect.php?banner_id=100&url=/
> _______________________________________________
> pgf-users mailing list
> pgf-users@...
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>

--
Prof. Dr. Till Tantau
www.tcs.uni-luebeck.de/mitarbeiter/tantau


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