%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}
\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}
}%
}
