\chapter{Solid geometry} \label{chap:solidgeom} 

Solid geometry or \nuvo{stereometry} \pron{£bster£i£h£om£itr£i} deals with the objects of the space such as points, 
lines, planes and so on. 
   
\section{Definitions and first properties}   
   
\subsection{Points, lines and planes}   
   
\begin{prop}   
  Two points determine a line. 
\end{prop}\index{point}\index{line}   
   
The line passing through the points $A$ and $B$ is denoted by $(AB)$. 
   
\begin{prop}   
  Three non-collinear points determine a plane. 
\end{prop}\index{plane}   
   
If $A$, $B$ and $C$ are three non-collinear points, the plane they determine is denoted by $(ABC)$. $A$ is in the plane 
$(ABC)$. One can write: $A \in (ABC)$. 
   
Points which are in the same plane are said to be \nuvo{coplanar} \pron{k£w£hpl£qn£e£Z}. 
   
\begin{prop}   
  If $A$ and $B$ are in the plane $\EuScript P$ then all the points of the line $(AB)$ are in the plane $\EuScript P$. 
  The line $(AB)$ is said to be in the plane $\EuScript P$. 
\end{prop}   
   
Instead of `the line \Li is in the plane $\EuScript P$', one can say `the line \Li is \nuvo{included in} the plane 
$\EuScript P$'. 
   
One writes: $ \Li \subset \EuScript P$ to denote that the line \Li is in the plane \Pl.\index{$\subset$} 
   
\begin{prop}   
  Two intersecting lines determine a plane. So two intersecting lines are coplanar. 
   
  One line and one point not on the line determine a plane. 
\end{prop}   
   
\subsection{Parallelism}   
   
One uses the sign \vide{} to denote an empty set, for example $\Li \cap \Lip = \vide$ means that lines \Li and \Lip have 
no common point.\index{$\vide$} 
   
\begin{Def}   
  Two lines are \nuvo{parallel} if and only if they are coplanar and do not intersect. 
\end{Def}   
   
Two parallel lines which are not identical have no common point. If two parallel lines have a common point then they are 
identical.  So if \Li and $\Li'$ are parallel or $\Li = \Li'$ or $\Li \cap \Li' = \vide$.  In the space two lines can 
have no common point without being parallel. That is the case when they are not coplanar then they are said to be 
\nuvo{skew} \pron{skj£U}. 
   
The following figure is a \nuvo{cabinet drawing} \pron{£hk£ab£in£it £hdr£O£i£n} of the cube $ABCDHEFG$. 

\begin{fgre}
  \IG{solge01}
\end{fgre}
   
\begin{rem}    
  $(AB)$ and $(CG)$ do not meet and are not parallel. They are not coplanar. $(AC)$ and $(BD)$ are coplanar but not 
  parallel. 
\end{rem}   
   
\emph{Pictorial} \pron{p£ik£ht£Or£Ql} \emph{drawing} shows a `picture' of an object in three dimensions, as if one were
looking at the object itself. One form of pictorial drawing is \emph{oblique} \pron{£e£hbl£Ik} \emph{drawing}.
\index{oblique drawing} In oblique drawing, a front view of the object is drawn. From the front view, lines at any angle
are drawn. This angle is usually at 45 degrees, although other angles are used. Cabinet drawing is the most common form
of oblique drawing. In this form of oblique drawing, the sloping lines are at 45 degrees and the lengths along the
sloping lines are drawn to half
scale.\index{drawing!oblique}\index{drawing!pictorial}\index{drawing!cabinet}\index{pictorial}
   
   
\begin{prop}   
  Two parallel lines without a common point determine a plane. 
\end{prop}   
   
\begin{Def}   
  Two planes are \nuvo{parallel} if and only if they are identical or if they have no common point. 
\end{Def}   
   
On the figure \ref{chap:solidgeom}--1, planes $(ABC)$ and $(EFG)$ are parallel. 
   
\begin{prop}   
  If two planes are not parallel they intersect in a line. 
\end{prop}   
   
In other words, if \Pl and $\Pl'$ are not parallel then $\Pl \cap \Pl'$ is a line. 
   
On the figure \ref{chap:solidgeom}--1, planes $(ABC)$ and $(FEA)$ intersect in the line $(AB)$. 
   
\begin{prop}   
  Let planes \Pl and $\EuScript Q$ be parallel. If a plane $\EuScript R$ intersects \Pl then it intersects $\EuScript 
  Q$.  Let then $d$ be $\Pl \cap \EuScript R$ and §d be $\EuScript Q \cap \EuScript R$, $d$ and §d are parallel. 
\end{prop}   
   
\begin{Def}   
  The line \Li is \nuvo{parallel} to the plane \Pl if \Li is in \Pl or if \Li and \Pl have no common point. 
\end{Def}   
   
Beware! On the figure \ref{chap:solidgeom}--1, lines $(EF)$ and $(EG)$ are both parallel to the plane $(ABC)$ \emph{but} 
they \emph{do} intersect. Parallelism is a relation which is innocuous when dealing with lines only or with planes only 
but which is rather dangerous when dealing with lines and planes at the same time. 
   
   
\begin{fgre}
  \IG{solge02a}\hspace{2cm}\IG{solge02b}
\end{fgre}
   
We can now state these usefull propositions: 
\begin{prop}   
  If the line $§d$ is parallel to the line $d$ and if the line $d$ is in the plane \Pl then the line §d is parallel to 
  the plane \Pl. (See figure \ref{chap:solidgeom}--2 left) 
\end{prop}   
   
\begin{prop}   
  If the line §d is parallel to the plane \Pl and to the plane $\EuScript Q$ and if the two planes intersect in the line 
  $d$ then the line §d is parallel to the line $d$.  (See figure \ref{chap:solidgeom}--2 right). 
\end{prop}   
   
\subsection{Orthogonality}   
\begin{Def}   
  Two lines of space are said to be \nuvo{perpendicular} \pron{£bp£Epen£hd£ikj£ul£e£Z} iff they are coplanar and 
  perpendicular in their common plane. 
\end{Def}   
   
\begin{Def}   
  Two lines $d$ and §d are said to be \nuvo{orthogonal} \pron{£O£h£f£og(£e)n(£e)l} iff there exists a line which is 
  parallel to $d$ and perpendicular to §d. 
\end{Def}   
   
So orthogonal lines are not necessarily coplanar and do not always intersect. For example, $(AB)$ and $(AD)$ are 
perpendicular whereas $(AB)$ and $(DH)$ are orthogonal and do not intersect. (See figure 1) 
   
If the lines are not coplanar some strange events can happen: $(AB)$ is perpendicular to $(AD)$ which is perpendicular 
to $(DH)$ but $(AB)$ and $(DH)$ are \emph{not} parallel. The theorem which states that \emph{if two lines are perpendicular to 
a third line then they are parallel}, is true only if the three lines are coplanar. 
   
\begin{Def}   
  The line $d$ is said to be perpendicular to the plane \Pl iff $d$ is orthogonal to two intersecting lines of \Pl. 
\end{Def}   
   
\begin{prop}   
  If the line $d$ is perpendicular to the plane \Pl then $d$ intersects \Pl moreover $d$ is orthogonal to each line of 
  \Pl. 
\end{prop}   
   
\begin{prop}   
\begin{itemize}   
\item If two lines are perpendicular to the same plane then they are parallel. 
\item If a line is perpendicular to two planes then the planes are parallel. 
\end{itemize}   
\end{prop}   
   
\begin{Def}   
  Let $A$ and $B$ be two distinct points. The plane \Pl is said to be the \emph{perpendicular bisector} 
  \pron{b£p£hsekt£e£Z} of $[AB]$ iff \Pl passes through the midpoint of $[AB]$ and $(AB)$ is perpendicular to \Pl. 
\end{Def}\index{perpendicular!bisector}\index{bisector!perpendicular}   
   
\begin{prop}   
  The perpendicular bisector of $[AB]$ is the set of all the points which are equidistant \pron{£b£Ikw£i£hd£ist£ent} 
  from $A$ and $B$. 
\end{prop}   
   
\begin{Def}   
  The plane \Q is said to be perpendicular to the plane \Pl iff \Q contains a line which is perpendicular to \Pl. 
\end{Def}   

\vspace*{\stretch{1}}   

$$ \includegraphics[keepaspectratio=true, totalheight=8cm]{figures/Vignette10}$$

\vspace*{\stretch{1}}
\endinput 

%%% Local Variables:    
%%% mode: latex   
%%% TeX-master: "../2dcours"   
%%% End:    
% correction 2001-01-18 : 
% correction pour v1.2 2002-03-11
% correction pour v1.3 2003-06-15
% correction pour v1.4 2003-07-04

