User:Peter Mercator/Math snippets

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$\phi$ $\lambda$ $\alpha$

$k(\lambda ,\,\phi ,\,\alpha )=\lim _{Q\to P}{\frac {P'Q'}{PQ}},$

$a\delta \phi$ $a$ $(a\cos \phi )\delta \lambda$ $(a\cos \phi )$

$\delta x=a\delta \lambda$   $\delta y$

horizontal scale factor   $\quad k\;=\;{\frac {\delta x}{a\cos \phi \,\delta \lambda \,}}=\,\sec \phi \qquad \qquad {}$

vertical scale factor     $\quad h\;=\;{\frac {\delta y}{a\delta \phi \,}}={\frac {y'(\phi )}{a}}$

$x=a\lambda \qquad \qquad y=a\phi ,$

$\pi /180$) $[{-}\pi ,\pi ]$ $\phi$ $[{-}\pi /2,\pi /2]$.

$y'(\phi )=1$

horizontal scale, $\quad k\;=\;{\frac {\delta x}{a\cos \phi \,\delta \lambda \,}}=\,\sec \phi \qquad \qquad {}$       vertical scale $\quad h\;=\;{\frac {\delta y}{a\delta \phi \,}}=\,1$

$y$-direction $x$-direction $2\pi a\cos \phi$$\sec \phi$ $2\pi a$

$x=a\lambda \qquad \qquad y=a\ln \left(\tan \left({\frac {\pi }{4}}+{\frac {\phi }{2}}\right)\right)$

horizontal scale,  $\quad k\;=\;{\frac {\delta x}{a\cos \phi \,\delta \lambda \,}}=\,\sec \phi \qquad \qquad {}$

vertical scale     $\quad h\;=\;{\frac {\delta y}{a\delta \phi \,}}=\,\sec \phi$

Lambert $x=a\lambda \qquad \qquad y=a\sin \phi$

horizontal scale,  $\quad k\;=\;{\frac {\delta x}{a\cos \phi \,\delta \lambda \,}}=\,\sec \phi \qquad \qquad {}$

vertical scale     $\quad h\;=\;{\frac {\delta y}{a\delta \phi \,}}=\,\cos \phi$

40,000 km

$1<k<1.0004$

$x=0.9996a\lambda \qquad \qquad y=0.9996a\ln \left(\tan \left({\frac {\pi }{4}}+{\frac {\phi }{2}}\right)\right).$

${\text{(a)}}\quad \tan \alpha ={\frac {a\cos \phi \,\delta \lambda }{a\,\delta \phi }},$

${\text{(b)}}\quad \tan \beta ={\frac {\delta x}{\delta y}}={\frac {a\delta \lambda }{\delta y}},$

${\text{(c)}}\quad \tan \beta ={\frac {a\sec \phi }{y'(\phi )}}\tan \alpha .\,$

$\mu _{\alpha }=\lim _{Q\to P}{\frac {P'Q'}{PQ}}=\lim _{Q\to P}{\frac {\sqrt {\delta x^{2}+\delta y^{2}}}{\sqrt {a^{2}\,\delta \phi ^{2}+a^{2}\cos ^{2}\!\phi \,\delta \lambda ^{2}}}}.$

$\mu _{\alpha }(\phi )=\sec \phi \left[{\frac {\sin \alpha }{\sin \beta }}\right].$