summaryrefslogtreecommitdiff
path: root/info/mathtrip/src/geom.tex
diff options
context:
space:
mode:
Diffstat (limited to 'info/mathtrip/src/geom.tex')
-rw-r--r--info/mathtrip/src/geom.tex84
1 files changed, 84 insertions, 0 deletions
diff --git a/info/mathtrip/src/geom.tex b/info/mathtrip/src/geom.tex
new file mode 100644
index 0000000000..26c8578176
--- /dev/null
+++ b/info/mathtrip/src/geom.tex
@@ -0,0 +1,84 @@
+%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}
+ }%
+}