Minimum distortion pointed-polar projections for world maps by applying graticule transformation

Figures & data

Table 1. Mean overall distortion values $E_K$ of some often used projections (pp: pointed-polar; fp: flat-polar; mixed: partly pointed-polar, partly flat-polar).

Projection	$E_k$	Pole	Projection	$E_k$	Pole
Web Mercator	0.69104	fp	Extended Apianus II	0.46485	pp
Mercator–Sanson	0.66474	pp	Eckert V	0.42009	fp
van der Grinten	0.57682	pp	Baranyi IV	0.40674	pp
conformal cylindrical $({\varphi }_s=\pm {42}^{\circ })$	0.54896	fp	Eckert III	0.40345	fp
Mollweide	0.53375	pp	Robinson	0.39287	fp
Aitoff	0.52187	pp	Kavrayskiy VII	0.36930	fp
Equirectangular	0.48864	fp	Winkel Tripel $({\varphi }_s={50}^{\circ }{28}^{\prime })$	0.36699	fp
Ortelius	0.47146	mixed	Pseudocylindrical with minimized distortion	0.35184	pp

Figure 1. The isolines of the maximum angular deformation $2\omega \left({}^{\circ }\right)$ and the area scale p for the minimum distortion pseudocylindrical projection (a).

Figure 2. The isolines of the maximum angular deformation $2\omega \left({}^{\circ }\right)$ and the area scale p for the minimum distortion pseudocylindrical projection (b).

Table 2. Coefficients $c_i$ and mean overall distortion values of the studied projections.

proj.	$c_1$	$c_2$	$c_3$	$c_4$	$c_5$	$c_6$	$c_7$	$E_k$
(a)	0.73044	4.20041	$2.00000$	0.00471	–	–	–	0.36756
(b)	0.75762	$2.00000$	4.63375	0.00264	–	–	–	0.35222
(c)	0.76158	1.67084	5.17538	0.00272	–	–	–	0.35184
(d)	0.71416	3.79209	$2.00000$	0.00902	0.87550	0.01004	0.00273	0.34558
(e)	0.74532	$2.00000$	4.04753	0.00730	0.93884	0.00271	0.00450	0.31959
(f)	0.77172	$2.00000$	3.26655	0.00649	0.88525	0.00950	0.00305	0.32531

Figure 3. The isolines of the maximum angular deformation $2\omega \left({}^{\circ }\right)$ and the area scale p for the minimum distortion non-conical projection (d).

Figure 4. The isolines of the maximum angular deformation $2\omega {(}^{\circ })$ and the area scale p for the minimum distortion non-conical projection (e).

Figure 5. The isolines of the maximum angular deformation $2\omega {(}^{\circ })$ and the area scale p for the non-conical projection (f) derived from the projection (e) with reduced maximal outline curvature.