Plotting the map projection graticule involving discontinuities based on combined sampling

This article presents  new algorithm for interval plotting the projection graticule on the interval $\varOmega=\varOmega_{\varphi}\times\varOmega_{\lambda}$ based on the combined sampling technique. The proposed method synthesizes uniform and adaptive sampling approaches and treats discontinuities of the coordinate functions $F,G$. A full set of the projection constant values represented by the projection pole $K=[\varphi_{k},\lambda_{k}]$, two standard parallels $\varphi_{1},\varphi_{2}$ and the central meridian shift $\lambda_{0}^{\prime}$ are supported. In accordance with the discontinuity direction it utilizes a subdivision of the given latitude/longitude intervals $\varOmega_{\varphi}=[\underline{\varphi},\overline{\varphi}]$, $\varOmega_{\lambda}=[\underline{\lambda},\overline{\lambda}]$ to the set of disjoint subintervals $\varOmega_{k,\varphi}^{g},$$\varOmega_{k,\lambda}^{g}$ forming tiles without internal singularities, containing only "good" data; their parameters can be easily adjusted. Each graticule tile borders generated over $\varOmega_{k}^{g}=\varOmega_{k,\varphi}^{g}\times\varOmega_{k,\lambda}^{g}$ run along singularities. For combined sampling with the given threshold $\overline{\alpha}$ between adjacent segments of the polygonal approximation the recursive approach has been used; meridian/parallel offsets are $\Delta\varphi,\Delta\lambda$. Finally, several tests of the proposed algorithms are involved