Lambert Conformal Conic Projection

$$x = \rho\sin[n(\lambda-\lambda_0)]$$ (1)

$$y = \rho_0-\rho\cos[n(\lambda-\lambda_0)],$$ (2)

where

$$\rho = F\cot^n({\textstyle{1\over 4}}\pi+{\textstyle{1\over 2}}\phi)$$ (3)

$$\rho_0 = F\cot^n({\textstyle{1\over 4}}\pi+{\textstyle{1\over 2}}\phi_0)$$ (4)

$$F = {\cos\phi_1\tan^n({\textstyle{1\over 4}}\pi+{\textstyle{1\over 2}}\phi_1)\over n}$$ (5)

$$n = {\ln(\cos\phi_1\sec\phi_2)\over\ln[\tan({\textstyle{1\over 4}}\pi... ...\over 2}}\phi_2)\cot({\textstyle{1\over 4}}\pi+{\textstyle{1\over 2}}\phi_1)]}.$$ (6)

The inverse Formulas are

$$\phi = 2\tan^{-1}\left[{\left({F\over\rho}\right)^{1/n}}\right]-{\textstyle{1\over 2}}\pi$$ (7)

$$\lambda = \lambda_0+{\theta\over n},$$ (8)

where

$$\rho = \mathop{\rm sgn}\nolimits (n)\sqrt{x^2+(\rho_0-y)^2}$$ (9)

$$\theta = \tan^{-1}\left({x\over\rho_0-y}\right).$$ (10)

References

Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 104-110, 1987.