%%%% 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, 2019, 2020, 2021, 2022, 2023 The Free Software Foundation, Inc. %%%% 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, 2019, 2020, 2021, 2022, 2023 The Free Software Foundation, Inc. \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