%%%% sundl_01.txt %%%% Created by Laurence D. Finston (LDF) Thu Sep 27 18:02:55 CEST 2007 %% $Id: sundl_01.txt,v 1.8 2007/10/02 17:22:09 lfinsto1 Exp lfinsto1 $ %% * (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 %%%% Kreuzbergring 41 %%%% D-37075 Goettingen %%%% Germany %%%% Laurence.Finston@gmx.de %% * (1) \input epsf \nopagenumbers \font\large=cmr12 \font\Large=cmr17 \font\huge=cmr17 scaled \magstep2 \font\largebx=cmbx12 \font\Largebx=cmbx17 \font\hugebx=cmbx17 scaled \magstep2 \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 %% \advance\voffset by 1cm %% \advance\hoffset by .25cm \parindent=0pt \def\epsfsize#1#2{#1} \font\small=cmr8 %% *** (3) \iftrue % \iffalse \pageno=-1 %\pageno=0 \vbox to \vsize{% \vskip.5cm \centerline{Sundial 1} \vskip\baselineskip \centerline{Laurence D. Finston} \vskip\baselineskip \centerline{Last updated: October 2, 2007} \vskip2cm %\line{\hskip.5cm\epsffile{sundl_01.0}} \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) \advance\voffset by -1in \advance\hoffset by -1in \advance\voffset by 1cm \advance\hoffset by 1cm \pageno=1 \dimen0=\vsize \pageno=1 \vbox to \vsize{% \vskip1cm \line{\hskip1cm\epsffile{sundl_01.1}\hss} \vskip1cm \line{{\Largebx Perspective projection.\hfil}} \vskip1cm See following page for explanation. \vskip4cm !\line{Copyright {\copyright} 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021, 2022, 2023 The Free Software Foundation, Inc. % \hskip 1cm Author: Laurence D. Finston\hss} \vss } \vfil\eject \begingroup \parskip=.5\baselineskip \advance\hsize by -2cm Let $g_0$ and $g_1$ be points on a line passing through the origin such that the line $g_0g_1$ lies in the x-y plane and its angle to the x-z plane is $51^\circ 32'$ (the latitude of G{\"o}ttingen, Germany). $g_0g_1$ represents the gnomon. Let $c_0$ be a circle with its center at the origin and lying in a plane perpendicular to $g_0g_1$. Let $r_0$ be the square enclosing $c_0$ and $r_1$ be a larger square in the same plane as $r_0$ and $c_0$, whose center is also at the origin and whose sides are parallel to those of $r_0$. Let $r_4$ be a rectangle perpendicular to $r_1$ such that the vertices $q_0$ and $q_1$ of $r_1$ are the midpoints of the sides $q_4q_5$ and $q_6q_7$ of $r_4$. Let $r_2$ be the rectangle ${q_4}{q_6}{q_9}{q_8}$ such that the vectors $q_8 - q_4$ and $q_9 - q_6$ are vertical, i.e., their y-components are non-zero and their x and z components are 0. Let $q_{13}$ be the intersection point of the line $q_0q_1$ with the x-y plane. The line through the origin and $q_{13}$ is the intersection of the x-y plane with the plane of $c_0$ and represents the projection of the gnomon $g_0g_1$ onto the plane of $c_0$ at noon. (The section of this line within the circumference of $c_0$ is drawn in blue.) The point $q_{10}$ is the intersection of the gnomon $g_0g_1$ with the plane of $r_2$ and the line $q_{10}q_{11}$ is the intersection of the x-y plane with the plane of $r_2$. It represents the projection of the gnomon $g_0g_1$ onto the plane of $r_2$ at noon. Let point $p_{75}$ be the point on the circumference of $c_0$ such that the angle between the line from the origin to $p_{75}$ and the line from the origin through $q_{13}$ is $15^\circ$ and the z-coordinate of $p_{75}$ is positive (in a left-handed coordinate system). (The point is to the {\it right\/} of the label. This point is also labelled ``\uppercase\expandafter{\romannumeral 13} $(75^\circ)$''.) The line from the origin to $p_{75}$ thus represents the projection of the gnomon $g_0g_1$ onto the plane of $c_0$ at 1:00 PM. The origin and the points $q_{10}$ and $p_{75}$ determine the plane $w_0$. The point $q_{14}$ is an intersection point of $w_0$ with the rectangle $r_1$ and the point $q_{16}$ is an intersection point of $w_0$ with the rectangle $r_2$. The line $q_{10}q_{16}$ thus represents the projection of the gnomon onto the plane of $r_2$ at 1.00 PM. The same principle would apply to any ``hour lines'' or other lines representing time divisions on $c_0$, which represents the dial of an equatorial sundial: The intersection of the plane $w_n$ through the origin, a point on the line representing the time division, and a point on the gnomon not in the plane of $c_0$ and the plane of $r_2$ will be a line representing the same time division on the plane of $r_2$. The set of these lines on the plane of $r_2$ would constitute the dial of a vertical sundial. They would radiate from $q_{10}$. In addition, the intersection of a plane $w_n$ representing a time division on $c_0$ with any other plane $v$ will also represent the corresponding time division on a dial lying in $v$. The rectangle $r_3$ was found by rotating $r_2$ about the axis $q_4q_8$ by $5^\circ$ (counterclockwise as seen when looking downward from $q_8$ onto $q_4$). The point $q_{17} = q_{23}$ was found by taking the point $q_6$ and performing the same rotation on it. $r_3$ was then rotated about the axis $q_4q_{17}$ by $5^\circ$ (counterclockwise as seen when looking from $q_4$ onto $q_17$). The point $q_{18}$ is the intersection of the gnomon $g_0g_1$ with the plane of $r_3$. The line $q_{18}q_{22}$ is the intersection of the plane $w_0$ with the plane of $r_3$. It thus represents the projection of the gnomon onto the plane of $r_3$ at 1.00 PM. \par \vskip 27cm !\line{Copyright {\copyright} 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021, 2022, 2023 The Free Software Foundation, Inc. % \hskip 1cm Author: Laurence D. Finston\hss} \endgroup \eject %% **** (4) Figure 2 \vbox to \vsize{% \vskip.5cm \line{\hskip 1cm\epsffile{sundl_01.2}\hss} \vskip2cm \line{{\Largebx Parallel projection onto plane of equatorial dial.\hfil}} \vskip 20cm !\line{Copyright {\copyright} 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021, 2022, 2023 The Free Software Foundation, Inc. % \hskip 1cm Author: Laurence D. Finston\hss}\vss} \vfil\eject \vbox to \vsize{% \vskip.5cm \line{\hskip 1cm\epsffile{sundl_01.3}\hss} \vskip2cm \line{{\Largebx Parallel projection onto the skew plane r3.\hfil}} \vskip 22cm !!!!!!!\line{Copyright {\copyright} 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021, 2022, 2023 The Free Software Foundation, Inc. % \hskip 1cm Author: Laurence D. Finston\hss}\vss} \vfil\eject %% **** (4) End here. \bye %% * (1) Local variables for Emacs. %% Local Variables: %% mode:plain-TeX %% outline-regexp:"%% [*\f]+" %% End: