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].