%%%% 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 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
%%%% Kreuzbergring 41
%%%% D-37075 Goettingen
%%%% Germany
%%%% Laurence.Finston@gmx.de
%% * (1)
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\centerline{Sundial 1}
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\centerline{Laurence D. Finston}
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\centerline{Last updated: October 2, 2007}
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This document is part of GNU 3DLDF, a package for three-dimensional
drawing.
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Copyright {\copyright} 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014 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.
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GNU 3DLDF is distributed in the hope that it will be useful,
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GNU General Public License for more details.
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51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA}}
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See following page for explanation.
\vskip4cm
\line{Copyright {\copyright} 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014 The Free Software Foundation %
\hskip 1cm Author: Laurence D. Finston\hss}
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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 The Free Software Foundation %
\hskip 1cm Author: Laurence D. Finston\hss}
\endgroup
\eject
%% **** (4) Figure 2
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\line{\hskip 1cm\epsffile{sundl_01.2}\hss}
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\line{{\Largebx Parallel projection onto plane of equatorial dial.\hfil}}
\vskip 20cm
\line{Copyright {\copyright} 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014 The Free Software Foundation %
\hskip 1cm Author: Laurence D. Finston\hss}\vss}
\vfil\eject
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\line{\hskip 1cm\epsffile{sundl_01.3}\hss}
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\line{{\Largebx Parallel projection onto the skew plane r3.\hfil}}
\vskip 22cm
\line{Copyright {\copyright} 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014 The Free Software Foundation %
\hskip 1cm Author: Laurence D. Finston\hss}\vss}
\vfil\eject
%% **** (4) End here.
\bye
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