%This macro provides the text for the 'geometry' part in
%the third column of the page 5
%
%The macro has one parameter
\newcommand\TFiveGeometry[1]{%
   \parbox[t]{#1}{%
      \TFiveGeomFontSize
      \DisplaySpace{\TFiveDisplaySpace}{\TFiveDisplayShortSpace}
      %Since the column is narrow, ragged right looks better
      \raggedright

      \TFiveTitle{Projective coordinates:}
       The triples $(x,y,z)$,
      not all $x$, $y$ and $z$ zero.
      \begin{DisplayFormulae}{1}{0pt}{4ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
        \Fm{\MathRemark[\relax]{\forall c \neq 0}}
        \Fm{(x,y,z) = (cx,cy,cz)}.
      \end{DisplayFormulae}

      \begin{tabular}{ll}
      Cartesian & Projective \\\hline
      $(x,y)$ &$(x,y,1)$ \\
      $y = mx + b$ &$(m,-1,b)$ \\
      $x = c$ &$(1,0,-c)$ \\
      \end{tabular}

      \TFiveTitle{Distance formula, $L_p$ and $L_\infty$ metric:}
      \AdjustSpace{1.5ex plus .2ex minus .5ex}
      \begin{DisplayFormulae}{1}{0pt}{4ex plus 1ex minus .5ex}{\BigChar}{\StyleWithoutNumber}
         \Fm{\sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}},
         \Fm{\big[ \vert x_1 - x_0 \vert^p + \vert y_1 - y_0 \vert^p \big]^{1/p}},
         \begin{multline*}
                \lim_{p \to \infty} \big[ \vert x_1 - x_0 \vert^p + \\[-3ex plus .5ex minus .5ex]
                                          \vert y_1 - y_0 \vert^p \big]^{1/p}
             \end{multline*}.
      \end{DisplayFormulae}

      \TFiveTitle{Area of triangle $(x_0,y_0)$, $(x_1,y_1)$ and $(x_2,y_2)$:}
      \AdjustSpace{1.5ex plus .2ex minus .5ex}
      \begin{DisplayFormulae}{1}{0pt}{4ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
         \Fm{\tfrac{1}{2}\abs\left\vert 
                                \begin{matrix}
                                   x_1-x_0 &y_1-y_0  \\
                                   x_2-x_0 &y_2-y_0  \\
                                \end{matrix}%
                             \right\vert
            }
      \end{DisplayFormulae}

      \TFiveTitle{Angle formed by three points:}
      \input{angle.tex}
      \centerline{\usebox\AngleBox}
      \begin{DisplayFormulae}{1}{0pt}{4ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
           \Fm{\cos \theta = \frac{(x_1, y_1) \cdot (x_2, y_2)}{\ell_1 \ell_2}}
      \end{DisplayFormulae}

      \TFiveTitle{Line through two points $(x_0,y_0)$ and $(x_1,y_1)$:}
      \begin{DisplayFormulae}{1}{0pt}{4ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
      \Fm{\left\vert 
                  \begin{matrix}
                      x &y & 1\\
                      x_0 &y_0 & 1\\
                      x_1 &y_1 & 1\\
                  \end{matrix}%
          \right\vert = 0
         }
      \end{DisplayFormulae}

      \TFiveTitle{Area of circle, volume of sphere:}
      \begin{DisplayFormulae}{1}{0pt}{4ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
           \Fm{A= \pi r^2} 
           \Fm{V= \tfrac{4}{3} \pi r^3}
      \end{DisplayFormulae}

      \TFiveTitle{Area and volume of a circumscribed cylinder to a sphere:}
      \begin{DisplayFormulae}{1}{0pt}{4ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
           \Fm{A_{cyl}= \frac{3}{2}A_{sph}}, 
           \Fm{V_{cyl}= \frac{3}{2}V_{sph}}
      \end{DisplayFormulae}
      \begin{flushright}
            Archimedes
      \end{flushright}
   }%
}