Comparing finite and infinitesimal map distortion measures

Figures & data

Table 1. Correlation matrices for scale-dependent and scale-independent distortions.

	$E_K$	$E_p$	$E_l$	$E_{\mathrm{\Delta }d}$	$E_{\mathrm{\Delta }p}$
$E_K$	1	0.7537	0.9694	0.9601	0.9042
$E_p$	0.7537	1	0.8834	0.9017	0.9294
$E_l$	0.9694	0.8834	1	0.997	0.9612
$E_{\mathrm{\Delta }d}$	0.9601	0.9017	0.997	1	0.9711
$E_{\mathrm{\Delta }p}$	0.9042	0.9294	0.9612	0.9711	1
	$E_a$	$E_K^{\prime }$	$E_p^{\prime }$	$E_l^{\prime }$	$E_f$	$E_s$	$E_G$	$E_{\mathrm{\Delta }\delta }$	$E_{\mathrm{\Delta }d}^{\prime }$	$E_{\mathrm{\Delta }p}^{\prime }$
$E_a$	1	0.6296	−0.5918	0.262	0.5895	0.3986	0.5749	0.5867	0.219	0.0566
$E_K^{\prime }$	0.6296	1	0.1615	0.8867	0.5673	0.5131	0.6183	0.9397	0.8672	0.6367
$E_p^{\prime }$	−0.5918	0.1615	1	0.5871	−0.1476	0.0176	−0.0872	0.2258	0.6125	0.6557
$E_l^{\prime }$	0.262	0.8867	0.5871	1	0.4068	0.4645	0.4929	0.8783	0.9938	0.8612
$E_f$	0.5895	0.5673	−0.1476	0.4068	1	0.575	0.8688	0.6633	0.3881	−0.0583
$E_s$	0.3986	0.5131	0.0176	0.4645	0.575	1	0.8997	0.5684	0.4496	0.0494
$E_G$	0.5749	0.6183	−0.0872	0.4929	0.8688	0.8997	1	0.7046	0.4716	−0.019
$E_{\mathrm{\Delta }\delta }$	0.5867	0.9397	0.2258	0.8783	0.6633	0.5684	0.7046	1	0.8662	0.5448
$E_{\mathrm{\Delta }d}^{\prime }$	0.219	0.8672	0.6125	0.9938	0.3881	0.4496	0.4716	0.8662	1	0.8609
$E_{\mathrm{\Delta }p}^{\prime }$	0.0566	0.6367	0.6557	0.8612	−0.0583	0.0494	−0.019	0.5448	0.8609	1

Figure 1. Relations between various criteria.

Figure 2. The best pseudoazimuthal projections for the Atlantic Ocean according to different criteria.

Figure 3. The best pseudoazimuthal projection for the Atlantic Ocean according to the average of flexion and skewness.

Table

Projection	$E_K$	$E_p$	$E_a$	$E_l$	$E_K^{\prime }$	$E_p^{\prime }$	$E_l^{\prime }$	$E_{\mathrm{\Delta }d}$	$E_{\mathrm{\Delta }p}$	$E_{\mathrm{\Delta }\delta }$	$E_{\mathrm{\Delta }d}^{\prime }$	$E_{\mathrm{\Delta }p}^{\prime }$	$E_f$	$E_s$	$E_G$
Lambert Azim. (Transverse)	1.0073	0	1.4245	0.5419	1.0073	0	0.506	0.4614	1.1688	0.2631	0.4232	1.1661	1.2157	1.4055	1.314
Postel (Transverse)	0.8698	0.8698	0.8698	0.6548	0.7477	0.6013	0.5201	0.5785	1.2859	0.2146	0.4396	1.1913	1.294	1.3256	1.3099
Lambert cylindrical	0.691	0	0.9773	0.3636	0.691	0	0.35	0.3315	0.79	0.1976	0.3029	0.7894	0.9052	1.7152	1.3714
Behrmann	0.59	0	0.8344	0.3095	0.59	0	0.3009	0.2698	0.79	0.1668	0.252	0.7894	0.9863	1.6501	1.3594
Gall–Peters	0.5535	0	0.7827	0.286	0.5535	0	0.2779	0.2446	0.79	0.172	0.233	0.7894	1.1019	1.5565	1.3485
Plate carrée	0.4886	0.4886	0.4886	0.3508	0.4413	0.3882	0.3033	0.3376	0.883	0.1622	0.2772	0.8543	0.911	1.386	1.1728
Equrectangular	0.4172	0.4172	0.4172	0.2976	0.403	0.3882	0.2814	0.2775	0.858	0.1495	0.2567	0.8543	0.9547	1.3156	1.1494
Mercator	0.691	0.9773	0	0.4886	0.549	0.7763	0.3882	0.4389	–	0.1801	0.331	–	1.0353	1.0353	1.0353
Perspectíve cylindrical	1.0926	1.4659	0.4886	0.8159	0.893	1.1645	0.6647	0.7384	–	0.3493	0.5841	–	2.1662	2.1651	2.1657
Braun	0.5208	0.667	0.3123	0.379	0.4291	0.5204	0.3083	0.3595	0.9615	0.1547	0.279	0.8993	0.9463	1.2157	1.0893
Gall	0.4154	0.5277	0.258	0.2942	0.4107	0.5204	0.292	0.2664	0.9143	0.148	0.2634	0.8993	1.0105	1.1012	1.0568
Miller	0.5534	0.7474	0.2322	0.3993	0.4452	0.5852	0.3198	0.3736	1.0106	0.1555	0.2853	0.9322	0.9599	1.1302	1.0485
Mercator–Sanson	0.6647	0	0.9401	0.34	0.6647	0	0.3247	0.2839	0.7645	0.2074	0.2734	0.7638	1.8029	1.4543	1.6379
Sanson–Wagner	0.4321	0.3222	0.5192	0.2778	0.4117	0.2635	0.2479	0.2406	0.8027	0.1384	0.2084	0.7927	1.1339	1.0548	1.0951
Kavrayskiy VI	0.4884	0	0.6908	0.2484	0.4884	0	0.242	0.2044	0.77	0.1512	0.1994	0.7692	1.4102	1.3877	1.399
Eckert V	0.4201	0.2683	0.5301	0.2476	0.4168	0.2577	0.2476	0.2088	0.8041	0.14	0.2086	0.793	1.1436	1.0529	1.0992
Eckert VI	0.494	0	0.6986	0.2511	0.494	0.0001	0.2445	0.2072	0.7703	0.1535	0.2022	0.7695	1.3972	1.3791	1.3882
Apian II	0.4607	0.2766	0.5899	0.2882	0.4446	0.2178	0.2589	0.242	0.7663	0.1465	0.2124	0.759	1.2601	1.033	1.1522
Apian–Wagner	0.4282	0.3543	0.491	0.2853	0.4011	0.2841	0.2514	0.2492	0.8053	0.135	0.2108	0.7913	1.0983	1.0682	1.0833
Mollweide	0.5337	0	0.7547	0.2722	0.5337	0	0.2641	0.2215	0.7505	0.1645	0.2162	0.7494	1.6612	1.3959	1.5343
Mollweide–Wagner	0.476	0	0.6732	0.2424	0.476	0	0.2366	0.1971	0.759	0.1449	0.1922	0.7582	1.4829	1.4405	1.4619
Eckert III	0.4035	0.3312	0.4646	0.261	0.3973	0.3161	0.2599	0.2217	0.8176	0.1352	0.2207	0.8003	1.0055	1.1204	1.0645
Eckert IV	0.4698	0.0001	0.6644	0.2405	0.4698	0	0.2351	0.1909	0.7512	0.1374	0.1852	0.7505	1.452	1.4709	1.4615
Kavrayskiy VII	0.3693	0.2921	0.433	0.2386	0.3662	0.2841	0.2317	0.2001	0.7913	0.1239	0.1941	0.7913	1.1045	0.9873	1.0476
Baranyi IV	0.4066	0.2916	0.4957	0.2574	0.4032	0.2819	0.2487	0.2146	0.7746	0.1365	0.2079	0.7745	1.2824	1.049	1.1715
Robinson	0.3929	0.2003	0.5182	0.2333	0.3922	0.1976	0.2328	0.1939	0.7751	0.1284	0.1936	0.7698	1.2262	1.1837	1.2051
Loximuthal	0.4348	0.3301	0.5189	0.2844	0.4114	0.2631	0.252	0.2436	0.7785	0.1376	0.2085	0.7678	1.1217	1.001	1.0631
Werner	0.7793	0	1.1021	0.4006	0.7793	0	0.3768	0.3488	0.9102	0.2486	0.332	0.9096	1.5927	1.1801	1.4017
Bonne	0.7386	0	1.0445	0.3794	0.7386	0	0.3591	0.3287	0.8666	0.2351	0.3144	0.866	1.6533	1.2587	1.4693
American polyconic	0.7682	0.7469	0.7888	0.551	0.6558	0.4878	0.4206	0.4918	1.1119	0.2065	0.3643	1.0047	1.1235	0.645	0.9161
War Office	0.7738	0.6285	0.8958	0.5118	0.7398	0.5404	0.4441	0.4375	1.0333	0.2328	0.3708	0.9913	1.3636	1.0065	1.1984
Equal-area polyconic	0.7618	0	1.0773	0.3898	0.7618	0	0.3666	0.3498	0.8943	0.2543	0.3343	0.8935	1.2371	0.6512	0.9886
Lagrange	0.8416	1.1903	0	0.5951	0.5042	0.7131	0.3565	0.5771	1.4039	0.1526	0.3167	0.9787	0.791	0.791	0.791
Van der Grinten I	0.5767	0.7972	0.172	0.4083	0.4413	0.5999	0.3096	0.3657	1.0191	0.1422	0.2624	0.9098	0.9114	0.7167	0.8198
Ginzburg V	0.3738	0.2036	0.4878	0.2166	0.3725	0.1988	0.2105	0.1852	0.7876	0.1285	0.1819	0.7876	1.1546	0.8616	1.0187
Aitoff	0.5219	0.2383	0.6985	0.3097	0.5022	0.1284	0.2662	0.2658	0.7911	0.1655	0.2284	0.7778	1.2824	0.8808	1.1001
Aitoff–Wagner	0.4184	0.341	0.4835	0.2736	0.3866	0.2553	0.2342	0.24	0.8143	0.1326	0.1993	0.7962	1.0806	0.9267	1.0066
Hammer	0.5792	0	0.8191	0.2946	0.5792	0	0.284	0.2505	0.7833	0.1885	0.2442	0.7823	1.4	1.0187	1.2243
Hammer–Wagner	0.4592	0	0.6494	0.2327	0.4592	0	0.2273	0.1994	0.7764	0.1511	0.1958	0.776	1.306	1.2428	1.2748
Eckert-Greifendorff	0.6119	0	0.8653	0.3129	0.6119	0	0.3009	0.2602	0.765	0.1912	0.2516	0.7641	1.7211	1.4246	1.5798
Winkel III	0.367	0.2027	0.4779	0.2112	0.367	0.2024	0.2085	0.1841	0.774	0.1299	0.1834	0.7727	1.1852	0.8412	1.0277
Lee elliptical	1.7291	2.4453	0	1.2227	0.6167	0.8721	0.4361	1.2147	2.5811	0.1563	0.4043	1.0817	0.9847	1.3208	1.1649
Eisenlohr	1.681	2.3772	0	1.1886	0.6945	0.9822	0.4911	1.1781	2.5184	0.1672	0.4593	1.1562	0.7059	0.7059	0.7059
Goode	0.3482	0	0.4924	0.1763	0.3482	0	0.1739	0.1303	0.6108	0.0982	0.128	0.6103	1.0356	0.7268	0.8946
Boggs	0.3226	0	0.4562	0.1634	0.3226	0	0.1615	0.1245	0.6172	0.0948	0.1233	0.6169	1.0394	0.6884	0.8815
Berghaus	0.3733	0.2505	0.4646	0.2079	0.3631	0.2187	0.1886	0.1532	0.9429	0.0902	0.1317	0.8863	0.9081	0.85	0.8795
William–Olsson	0.3491	0	0.4937	0.1753	0.3491	0	0.1728	0.126	0.7993	0.0967	0.1238	0.795	0.794	0.3588	0.6161
Cahill gnomonic	0.388	0.5205	0.1735	0.2729	0.2449	0.2997	0.166	0.2536	0.9159	0.0806	0.1108	0.7379	0	0.8442	0.5969
Cahill conformal	0.2108	0.2981	0	0.149	0.1608	0.2275	0.1137	0.1353	0.8048	0.0644	0.0895	0.7514	0.5587	0.5587	0.5587
Waterman	0.1577	0.1691	0.1455	0.0962	0.1141	0.0697	0.061	0.0879	0.7258	0.0309	0.042	0.7256	0.6392	0.5438	0.5934

Table

Figure	$r_1$	$r_3$	$r_5$	$w_{11}$	$w_{12}$	$w_{13}$	$w_{21}$	$w_{22}$	$w_{31}$	${\lambda }_P$	${\lambda }_M$
2(a)	0.9871	−0.0030	−0.0016	−0.0717	0.0246	−0.0037	−0.0408	−0.0082	0.0365	−31.1451${}^{\circ }$	19.7945${}^{\circ }$
2(b)	0.9905	0.0065	−0.0059	−0.1245	0.0358	−0.0076	0.0073	−0.0054	0.0179	−27.8526${}^{\circ }$	22.1870${}^{\circ }$
2(c)	0.9973	0.0036	−0.0049	−0.1994	0.0476	−0.0099	0.1211	−0.0107	−0.0274	−26.3221${}^{\circ }$	24.2428${}^{\circ }$
2(d)	0.9940	0.0015	−0.0026	−0.1891	0.0450	−0.0112	0.1023	−0.0069	−0.0198	−25.5266${}^{\circ }$	23.2368${}^{\circ }$
3	1	0.0287	−0.0024	−0.1817	0.0310	−0.0076	0.0727	0.0015	−0.0144	−28.5086${}^{\circ }$	24.3099${}^{\circ }$