%%%% arc_1.txt
%%%% Created by Laurence D. Finston (LDF) Thu Mar 1 11:51:50 CET 2007
%% * (1) Copyright and License.
%%%% This file is part of GNU 3DLDF, a package for three-dimensional drawing.
%%%% Copyright (C) 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014,
%%%% 2015, 2016, 2017, 2018 The Free Software Foundation
%%%% GNU 3DLDF is free software; you can redistribute it and/or modify
%%%% it under the terms of the GNU General Public License as published by
%%%% the Free Software Foundation; either version 3 of the License, or
%%%% (at your option) any later version.
%%%% GNU 3DLDF is distributed in the hope that it will be useful,
%%%% but WITHOUT ANY WARRANTY; without even the implied warranty of
%%%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
%%%% GNU General Public License for more details.
%%%% You should have received a copy of the GNU General Public License
%%%% along with GNU 3DLDF; if not, write to the Free Software
%%%% Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
%%%% GNU 3DLDF is a GNU package.
%%%% It is part of the GNU Project of the
%%%% Free Software Foundation
%%%% and is published under the GNU General Public License.
%%%% See the website http://www.gnu.org
%%%% for more information.
%%%% GNU 3DLDF is available for downloading from
%%%% http://www.gnu.org/software/3dldf/LDF.html.
%%%% Please send bug reports to Laurence.Finston@gmx.de
%%%% The mailing list help-3dldf@gnu.org is available for people to
%%%% ask other users for help.
%%%% The mailing list info-3dldf@gnu.org is for the maintainer of
%%%% GNU 3DLDF to send announcements to users.
%%%% To subscribe to these mailing lists, send an
%%%% email with ``subscribe '' as the subject.
%%%% The author can be contacted at:
%%%% Laurence D. Finston
%%%% c/o The Free Software Foundation, Inc.
%%%% 51 Franklin St, Fifth Floor
%%%% Boston, MA 02110-1301
%%%% USA
%%%% Laurence.Finston@gmx.de
%% * (1)
%% Run the following commands:
%% 3dldf arc_1.ldf
%% mpost arc_1.mp
%% tex arc_1.txt
%% dvips -o arc_1.ps arc_1.dvi
\input epsf
\nopagenumbers
\pageno=1
%% Uncomment for DIN A3 portrait.
%% \special{papersize=297mm, 420mm} %% DIN A3 Portrait
%% \vsize=420mm
%% \hsize=297mm
%% Uncomment for A3 landscape.
%\special{papersize=420mm, 297mm} %% DIN A3 Landscape
%\vsize=297mm
%\hsize=420mm
%% Uncomment for A4 landscape.
%\special{papersize=297mm, 210mm}
%\hsize=297mm
%\vsize=210mm
%\advance\voffset by -1in
%\advance\hoffset by -1in
\parindent=0pt
\def\epsfsize#1#2{#1}
\font\small=cmr8
%% *** (3)
\iftrue % \iffalse
\pageno=-1
%\pageno=0
\vbox to \vsize{%
\vskip.5cm
\centerline{Arc 1}
\centerline{Laurence D. Finston}
\vskip2cm
\iftrue % \iffalse
{\small
\hsize=.75\hsize
\hskip1cm
\vbox{\vskip2\baselineskip
This document is part of GNU 3DLDF, a package for three-dimensional
drawing.
\vskip\baselineskip
Copyright {\copyright} 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014,
%%%% 2015, 2016, 2017, 2018 The Free Software Foundation
\vskip\baselineskip
GNU 3DLDF is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 3 of the License, or
(at your option) any later version.
\vskip\baselineskip
GNU 3DLDF is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
\vskip\baselineskip
You should have received a copy of the GNU General Public License
along with GNU 3DLDF; if not, write to the Free Software
Foundation, Inc.,\hfil\break
51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA}}
\fi
\vss}
\vfil\eject
\fi
%% *** (3)
\pageno=1
\vbox to \vsize{%
Given a circular arc $q$, find the radius $r$ and the angle
$\theta$
subtended by $q$ at the center $M$ of circle $k$.
\vskip\baselineskip
Solution:
\vskip\baselineskip
Let $q$ be the arc of a circle $k$ with unknown center $M$ and
unknown radius $r$.
Let $p_0$ and $p_1$ be the end points of $q$.
Let $p_2$ be the point halfway between $p_0$ and $p_1$.
Let $p_3$ be the intersection of $q$ and the perpendicular
to the line $p_1p_0$ through $p_2$.
Let $p_4$ be the point halfway between $p_0$ and $p_3$.
Let $p_5$ be the intersection of $q$ and the perpendicular
to $p_0p_3$ through $p_4$.
The lines $p_3p_2$ and $p_5p_4$ intersect at $M$.
Let $a$ be the distance from $p_3$ to $p_2$, i.e.,
the magnitude of $p_3 - p_2$,
$b$ the distance from $p_0$ to $p_2$, and
$c$ the distance from $p_0$ to $p_3$.
Let $\alpha$ be $\angle p_3p_0p_2$ and $\beta$ be
$\angle p_0p_3p_2$. Then $\alpha = \sin (a/c)$ and $\beta = \sin (b/c)$.
Let $\gamma$ be $\angle p_4Mp_3$. Since $\angle p_3p_4M$
is a right angle, $\gamma = \alpha$. Since the arc
$p_0p_5$ is a quarter of the arc $p_0p_1$, the angle $\theta$
subtended by $q$ at $M$ is $4\gamma$.
%
\vskip.5cm
\line{\hskip1cm\epsffile{arc_1.1}\hss}
\vss
}
%% **** (4) End here.
\bye