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User:Peter Mercator/Sandbox
![{\displaystyle k(\lambda ,\,\phi ,\,\alpha )=\lim _{Q\to P}{\frac {P'Q'}{PQ}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc1ca4b598f63faec0dfc5479708d3a7cce171bc)
- horizontal scale factor
![{\displaystyle \quad k\;=\;{\frac {\delta x}{a\cos \phi \,\delta \lambda \,}}=\,\sec \phi \qquad \qquad {}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da3ebb318898941c86b8d39a30aec87d9c501591)
- vertical scale factor
![{\displaystyle \quad h\;=\;{\frac {\delta y}{a\delta \phi \,}}={\frac {y'(\phi )}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a7287258fa1e803052fdb7ff3f1e1b120f4fc0)
![{\displaystyle x=a\lambda \qquad \qquad y=a\phi ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3691ae5b7eefe49298bc9adab81f1487510d4976)
)
.
- horizontal scale,
vertical scale ![{\displaystyle \quad h\;=\;{\frac {\delta y}{a\delta \phi \,}}=\,1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e40263dd0b3c8b73f3a0856c0d7e28d98dc08744)
-direction
-direction
![{\displaystyle 2\pi a\cos \phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/278ac0a4a8e9a033096a49587f66065a854748e9)
![{\displaystyle x=a\lambda \qquad \qquad y=a\ln \left(\tan \left({\frac {\pi }{4}}+{\frac {\phi }{2}}\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f737f08fa2007485660c3917c1d2e369bc73e183)
- horizontal scale,
![{\displaystyle \quad k\;=\;{\frac {\delta x}{a\cos \phi \,\delta \lambda \,}}=\,\sec \phi \qquad \qquad {}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da3ebb318898941c86b8d39a30aec87d9c501591)
- vertical scale
![{\displaystyle \quad h\;=\;{\frac {\delta y}{a\delta \phi \,}}=\,\sec \phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/89ca27c893b2d7db255d178bdfb3a5b930967819)
Lambert
- horizontal scale,
![{\displaystyle \quad k\;=\;{\frac {\delta x}{a\cos \phi \,\delta \lambda \,}}=\,\sec \phi \qquad \qquad {}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da3ebb318898941c86b8d39a30aec87d9c501591)
- vertical scale
![{\displaystyle \quad h\;=\;{\frac {\delta y}{a\delta \phi \,}}=\,\cos \phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4c423a99a4e0e8b1e4a4331832b80a323afee2)
40,000 km
![{\displaystyle {\text{(a)}}\quad \tan \alpha ={\frac {a\cos \phi \,\delta \lambda }{a\,\delta \phi }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/917e0c7bd3b6bf24a454b5a27ec1c8f7612c3c06)
![{\displaystyle {\text{(b)}}\quad \tan \beta ={\frac {\delta x}{\delta y}}={\frac {a\delta \lambda }{\delta y}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e583a164914d1e9c5c559f527e2fe41467497f23)
![{\displaystyle {\text{(c)}}\quad \tan \beta ={\frac {a\sec \phi }{y'(\phi )}}\tan \alpha .\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b92a79f8914c22fbe985e982041688812e6367b)
![{\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}}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56fa6254d564549ab4227ab20be68d09404a3d14)
![{\displaystyle \mu _{\alpha }(\phi )=\sec \phi \left[{\frac {\sin \alpha }{\sin \beta }}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/407e6b8233b4f72d35d205332238ace25fe88ed9)