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 $varOmega=varOmega_{varphi}timesvarOmega_{lambda}$ 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 ``goodu0027u0027 data; their parameters can be easily adjusted. Each graticule tile borders generated over $varOmega_{k}^{g}=varOmega_{k,varphi}^{g}timesvarOmega_{k,lambda}^{g}$ 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 $Deltavarphi,Deltalambda$. Finally, several tests of the proposed algorithms are involved.