Plotting the map projection graticule involving discontinuities based on combined sampling

Tomáš Bayer

Abstract


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.

Keywords


digital cartography; mathematical cartography; adaptive sampling; graticule; meridians; parallels; recursive approach; map projection; great circle; discontinuity; visualization; sphere;

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