User:Peter Mercator/Math snippets

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${\displaystyle \phi }$ ${\displaystyle \lambda }$ ${\displaystyle \alpha }$

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

${\displaystyle a\delta \phi }$ ${\displaystyle a}$ ${\displaystyle (a\cos \phi )\delta \lambda }$ ${\displaystyle (a\cos \phi )}$

${\displaystyle \delta x=a\delta \lambda }$   ${\displaystyle \delta y}$

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

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

${\displaystyle x=a\lambda \qquad \qquad y=a\phi ,}$

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

${\displaystyle y'(\phi )=1}$

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

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

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

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

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

Lambert ${\displaystyle x=a\lambda \qquad \qquad y=a\sin \phi }$

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

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

40,000 km

${\displaystyle 1<k<1.0004}$

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

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

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

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

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

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