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Diffstat (limited to 'info/mathtrip/src/geom.tex')
-rw-r--r-- | info/mathtrip/src/geom.tex | 84 |
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} + }% +} |