User:Peter Mercator/Math snippets

../Sandbox

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


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

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

$$\delta x=a\delta\lambda$$ $$\delta y$$
 * 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].$$