Processing math: 100%

2. 球面での回転

前項でみた球面への写像のつぎに、球面を回転させる。

等角写像後の緯度 ˆφ, 経度 ˆλ から回転後の緯度 φ′, 経度 λ′ は次のように与えられる：

φ′=sin−1[sinˆφPsinˆφ+cosˆφPcosˆφcos(ˆλ−ˆλP)]

λ′=cos−1cosˆφPsinφ′−sinˆφcosˆφPcosφ′

気象庁では別の式が使われていたが、これは atan を使うため 180 度の不定性がでて使いにくい。お勧めではない。

λ′=tan−1cosˆφsin(ˆλ−ˆλP)cosˆφPsinˆφ−sinˆφPcosˆφcos(ˆλ−ˆλP)

パラメタとして政春 (2011) に従い次を用いると、

回転楕円体上の経緯度グリッド点が投影される球面上の点の経緯度は次表で与えられる。

投影につづく

Rotation on the aposphere
lat/lon122E123E124E125E126E127E128E129E130E131E132E133E134E135E136E137E138E139E140E141E142E143E144E145E146E147E148E
46N(64.00, 83.39)(63.45, 84.35)(62.89, 85.28)(62.33, 86.20)(61.78, 87.10)(61.22, 87.98)(60.66, 88.84)(60.10, 89.69)(59.54, 90.52)(58.98, 91.34)(58.43, 92.15)(57.87, 92.94)(57.31, 93.73)(56.75, 94.50)(56.20, 95.26)(55.64, 96.02)(55.09, 96.76)(54.53, 97.50)(53.98, 98.23)(53.43, 98.95)(52.87, 99.66)(52.32, 100.37)(51.78, 101.07)(51.23, 101.77)(50.68, 102.46)(50.14, 103.15)(49.59, 103.83)
45N(63.38, 81.62)(62.83, 82.60)(62.28, 83.57)(61.72, 84.51)(61.16, 85.44)(60.61, 86.34)(60.05, 87.23)(59.49, 88.11)(58.93, 88.97)(58.37, 89.81)(57.82, 90.64)(57.26, 91.46)(56.70, 92.27)(56.14, 93.07)(55.58, 93.85)(55.03, 94.63)(54.47, 95.39)(53.91, 96.15)(53.36, 96.90)(52.80, 97.64)(52.25, 98.37)(51.70, 99.10)(51.15, 99.82)(50.60, 100.53)(50.05, 101.24)(49.50, 101.94)(48.96, 102.64)
44N(62.74, 79.92)(62.19, 80.93)(61.64, 81.92)(61.09, 82.89)(60.53, 83.84)(59.98, 84.77)(59.42, 85.69)(58.86, 86.59)(58.30, 87.47)(57.75, 88.34)(57.19, 89.19)(56.63, 90.03)(56.07, 90.86)(55.51, 91.68)(54.95, 92.48)(54.40, 93.28)(53.84, 94.06)(53.28, 94.84)(52.73, 95.61)(52.17, 96.37)(51.61, 97.12)(51.06, 97.86)(50.51, 98.60)(49.96, 99.33)(49.41, 100.06)(48.86, 100.77)(48.31, 101.49)
43N(62.08, 78.29)(61.54, 79.33)(60.99, 80.34)(60.43, 81.33)(59.88, 82.31)(59.33, 83.26)(58.77, 84.20)(58.22, 85.12)(57.66, 86.02)(57.10, 86.91)(56.54, 87.79)(55.98, 88.65)(55.43, 89.50)(54.87, 90.33)(54.31, 91.16)(53.75, 91.97)(53.19, 92.78)(52.64, 93.57)(52.08, 94.36)(51.52, 95.13)(50.97, 95.90)(50.41, 96.66)(49.86, 97.41)(49.30, 98.16)(48.75, 98.90)(48.20, 99.63)(47.65, 100.36)
42N(61.41, 76.73)(60.86, 77.79)(60.31, 78.83)(59.76, 79.84)(59.21, 80.83)(58.66, 81.81)(58.11, 82.77)(57.55, 83.71)(57.00, 84.63)(56.44, 85.54)(55.88, 86.43)(55.33, 87.31)(54.77, 88.18)(54.21, 89.03)(53.65, 89.87)(53.09, 90.71)(52.53, 91.53)(51.98, 92.34)(51.42, 93.14)(50.86, 93.93)(50.30, 94.71)(49.75, 95.49)(49.19, 96.26)(48.64, 97.02)(48.08, 97.77)(47.53, 98.52)(46.98, 99.26)
41N(60.71, 75.24)(60.17, 76.32)(59.62, 77.37)(59.08, 78.41)(58.53, 79.42)(57.98, 80.41)(57.43, 81.39)(56.88, 82.34)(56.32, 83.28)(55.77, 84.21)(55.21, 85.12)(54.65, 86.02)(54.10, 86.90)(53.54, 87.77)(52.98, 88.63)(52.42, 89.48)(51.86, 90.31)(51.30, 91.14)(50.75, 91.96)(50.19, 92.76)(49.63, 93.56)(49.07, 94.35)(48.52, 95.14)(47.96, 95.91)(47.40, 96.68)(46.85, 97.44)(46.30, 98.19)
40N(60.00, 73.82)(59.46, 74.91)(58.92, 75.98)(58.38, 77.03)(57.83, 78.06)(57.28, 79.07)(56.74, 80.06)(56.18, 81.03)(55.63, 81.99)(55.08, 82.93)(54.52, 83.86)(53.97, 84.77)(53.41, 85.67)(52.85, 86.55)(52.30, 87.43)(51.74, 88.29)(51.18, 89.14)(50.62, 89.98)(50.06, 90.81)(49.50, 91.63)(48.95, 92.44)(48.39, 93.25)(47.83, 94.04)(47.27, 94.83)(46.72, 95.61)(46.16, 96.38)(45.61, 97.15)
39N(59.27, 72.45)(58.74, 73.56)(58.20, 74.64)(57.66, 75.70)(57.12, 76.75)(56.58, 77.77)(56.03, 78.78)(55.48, 79.76)(54.93, 80.74)(54.38, 81.69)(53.82, 82.63)(53.27, 83.56)(52.71, 84.47)(52.16, 85.37)(51.60, 86.26)(51.04, 87.13)(50.48, 88.00)(49.93, 88.85)(49.37, 89.69)(48.81, 90.53)(48.25, 91.35)(47.69, 92.17)(47.13, 92.98)(46.58, 93.78)(46.02, 94.57)(45.46, 95.35)(44.91, 96.13)
38N(58.53, 71.14)(58.00, 72.26)(57.47, 73.36)(56.93, 74.43)(56.39, 75.49)(55.85, 76.53)(55.31, 77.54)(54.76, 78.54)(54.21, 79.53)(53.66, 80.50)(53.11, 81.45)(52.56, 82.39)(52.00, 83.31)(51.45, 84.23)(50.89, 85.12)(50.34, 86.01)(49.78, 86.89)(49.22, 87.75)(48.66, 88.61)(48.10, 89.46)(47.55, 90.29)(46.99, 91.12)(46.43, 91.94)(45.87, 92.75)(45.31, 93.55)(44.76, 94.35)(44.20, 95.14)
37N(57.78, 69.89)(57.26, 71.02)(56.73, 72.13)(56.19, 73.21)(55.66, 74.28)(55.12, 75.32)(54.58, 76.35)(54.03, 77.37)(53.49, 78.36)(52.94, 79.34)(52.39, 80.31)(51.84, 81.26)(51.28, 82.19)(50.73, 83.12)(50.18, 84.03)(49.62, 84.93)(49.06, 85.81)(48.51, 86.69)(47.95, 87.56)(47.39, 88.41)(46.83, 89.26)(46.27, 90.10)(45.71, 90.93)(45.16, 91.75)(44.60, 92.57)(44.04, 93.37)(43.48, 94.17)
36N(57.02, 68.69)(56.50, 69.83)(55.97, 70.94)(55.44, 72.04)(54.91, 73.11)(54.37, 74.17)(53.83, 75.21)(53.29, 76.23)(52.75, 77.23)(52.20, 78.22)(51.66, 79.20)(51.11, 80.16)(50.55, 81.11)(50.00, 82.04)(49.45, 82.96)(48.89, 83.87)(48.34, 84.77)(47.78, 85.66)(47.22, 86.53)(46.67, 87.40)(46.11, 88.26)(45.55, 89.10)(44.99, 89.94)(44.43, 90.78)(43.87, 91.60)(43.32, 92.42)(42.76, 93.23)
35N(56.25, 67.54)(55.73, 68.68)(55.20, 69.80)(54.68, 70.90)(54.15, 71.99)(53.62, 73.05)(53.08, 74.10)(52.54, 75.13)(52.00, 76.15)(51.46, 77.14)(50.91, 78.13)(50.36, 79.10)(49.81, 80.05)(49.26, 81.00)(48.71, 81.93)(48.16, 82.85)(47.60, 83.75)(47.05, 84.65)(46.49, 85.54)(45.93, 86.41)(45.38, 87.28)(44.82, 88.14)(44.26, 88.98)(43.70, 89.83)(43.14, 90.66)(42.58, 91.48)(42.03, 92.30)
34N(55.46, 66.43)(54.95, 67.58)(54.43, 68.71)(53.91, 69.82)(53.38, 70.91)(52.85, 71.98)(52.32, 73.03)(51.78, 74.07)(51.24, 75.09)(50.70, 76.10)(50.16, 77.09)(49.61, 78.07)(49.07, 79.03)(48.52, 79.98)(47.97, 80.92)(47.41, 81.85)(46.86, 82.77)(46.31, 83.67)(45.75, 84.57)(45.19, 85.45)(44.64, 86.33)(44.08, 87.19)(43.52, 88.05)(42.96, 88.90)(42.40, 89.74)(41.84, 90.57)(41.29, 91.40)
33N(54.67, 65.37)(54.16, 66.52)(53.64, 67.65)(53.12, 68.77)(52.60, 69.86)(52.07, 70.94)(51.55, 72.00)(51.01, 73.05)(50.48, 74.07)(49.94, 75.09)(49.40, 76.09)(48.85, 77.07)(48.31, 78.04)(47.76, 79.00)(47.21, 79.95)(46.66, 80.88)(46.11, 81.81)(45.56, 82.72)(45.00, 83.62)(44.45, 84.51)(43.89, 85.40)(43.33, 86.27)(42.77, 87.14)(42.22, 87.99)(41.66, 88.84)(41.10, 89.68)(40.54, 90.52)
32N(53.86, 64.34)(53.36, 65.50)(52.85, 66.64)(52.33, 67.76)(51.81, 68.86)(51.29, 69.94)(50.76, 71.00)(50.23, 72.05)(49.70, 73.09)(49.17, 74.11)(48.63, 75.11)(48.09, 76.10)(47.54, 77.08)(47.00, 78.05)(46.45, 79.00)(45.90, 79.94)(45.35, 80.87)(44.80, 81.79)(44.24, 82.70)(43.69, 83.60)(43.13, 84.49)(42.58, 85.37)(42.02, 86.24)(41.46, 87.11)(40.90, 87.96)(40.35, 88.81)(39.79, 89.65)
31N(53.05, 63.36)(52.55, 64.52)(52.04, 65.66)(51.53, 66.78)(51.02, 67.88)(50.50, 68.97)(49.98, 70.04)(49.45, 71.10)(48.92, 72.13)(48.39, 73.16)(47.85, 74.17)(47.31, 75.17)(46.77, 76.15)(46.23, 77.12)(45.68, 78.08)(45.13, 79.03)(44.58, 79.96)(44.03, 80.89)(43.48, 81.81)(42.93, 82.71)(42.37, 83.61)(41.82, 84.50)(41.26, 85.37)(40.70, 86.25)(40.15, 87.11)(39.59, 87.96)(39.03, 88.81)
30N(52.23, 62.42)(51.74, 63.57)(51.23, 64.72)(50.73, 65.84)(50.21, 66.95)(49.70, 68.04)(49.18, 69.11)(48.65, 70.17)(48.13, 71.21)(47.60, 72.24)(47.06, 73.25)(46.53, 74.26)(45.99, 75.24)(45.45, 76.22)(44.90, 77.18)(44.36, 78.14)(43.81, 79.08)(43.26, 80.01)(42.71, 80.93)(42.16, 81.84)(41.60, 82.75)(41.05, 83.64)(40.49, 84.52)(39.94, 85.40)(39.38, 86.27)(38.82, 87.13)(38.26, 87.99)
29N(51.41, 61.50)(50.91, 62.66)(50.42, 63.81)(49.91, 64.93)(49.40, 66.04)(48.89, 67.13)(48.37, 68.21)(47.85, 69.27)(47.33, 70.31)(46.80, 71.35)(46.27, 72.36)(45.74, 73.37)(45.20, 74.36)(44.66, 75.34)(44.12, 76.31)(43.58, 77.27)(43.03, 78.22)(42.48, 79.15)(41.93, 80.08)(41.38, 81.00)(40.83, 81.91)(40.28, 82.80)(39.72, 83.69)(39.17, 84.58)(38.61, 85.45)(38.05, 86.32)(37.49, 87.18)
28N(50.57, 60.62)(50.09, 61.78)(49.59, 62.93)(49.09, 64.05)(48.59, 65.16)(48.08, 66.25)(47.56, 67.33)(47.05, 68.40)(46.53, 69.44)(46.00, 70.48)(45.47, 71.50)(44.94, 72.51)(44.41, 73.51)(43.87, 74.49)(43.33, 75.46)(42.79, 76.43)(42.24, 77.38)(41.70, 78.32)(41.15, 79.25)(40.60, 80.17)(40.05, 81.08)(39.50, 81.99)(38.94, 82.88)(38.39, 83.77)(37.83, 84.65)(37.28, 85.52)(36.72, 86.39)
27N(49.74, 59.78)(49.25, 60.93)(48.76, 62.08)(48.26, 63.20)(47.76, 64.31)(47.26, 65.41)(46.75, 66.48)(46.23, 67.55)(45.71, 68.60)(45.19, 69.64)(44.67, 70.66)(44.14, 71.67)(43.61, 72.67)(43.07, 73.66)(42.53, 74.64)(41.99, 75.60)(41.45, 76.56)(40.91, 77.50)(40.36, 78.44)(39.81, 79.36)(39.26, 80.28)(38.71, 81.19)(38.16, 82.09)(37.61, 82.98)(37.05, 83.86)(36.50, 84.74)(35.94, 85.61)
26N(48.89, 58.96)(48.41, 60.11)(47.92, 61.25)(47.43, 62.38)(46.93, 63.49)(46.43, 64.58)(45.92, 65.66)(45.41, 66.73)(44.90, 67.78)(44.38, 68.82)(43.86, 69.85)(43.33, 70.86)(42.80, 71.86)(42.27, 72.85)(41.73, 73.83)(41.20, 74.80)(40.66, 75.76)(40.11, 76.71)(39.57, 77.65)(39.02, 78.58)(38.47, 79.50)(37.92, 80.41)(37.37, 81.31)(36.82, 82.21)(36.26, 83.09)(35.71, 83.98)(35.15, 84.85)
25N(48.04, 58.16)(47.56, 59.32)(47.08, 60.46)(46.59, 61.58)(46.10, 62.69)(45.60, 63.78)(45.10, 64.87)(44.59, 65.93)(44.08, 66.98)(43.56, 68.02)(43.04, 69.05)(42.52, 70.07)(41.99, 71.07)(41.46, 72.06)(40.93, 73.05)(40.39, 74.02)(39.85, 74.98)(39.31, 75.93)(38.77, 76.87)(38.22, 77.80)(37.68, 78.73)(37.13, 79.64)(36.58, 80.55)(36.03, 81.45)(35.47, 82.34)(34.92, 83.23)(34.36, 84.10)
24N(47.18, 57.40)(46.71, 58.55)(46.23, 59.69)(45.75, 60.81)(45.26, 61.92)(44.76, 63.01)(44.26, 64.09)(43.76, 65.16)(43.25, 66.21)(42.73, 67.25)(42.22, 68.28)(41.70, 69.30)(41.17, 70.30)(40.64, 71.30)(40.11, 72.28)(39.58, 73.25)(39.04, 74.22)(38.51, 75.17)(37.96, 76.11)(37.42, 77.05)(36.88, 77.98)(36.33, 78.89)(35.78, 79.80)(35.23, 80.71)(34.68, 81.60)(34.12, 82.49)(33.57, 83.37)
23N(46.32, 56.66)(45.85, 57.81)(45.38, 58.94)(44.90, 60.06)(44.41, 61.17)(43.92, 62.26)(43.42, 63.34)(42.92, 64.40)(42.41, 65.46)(41.90, 66.50)(41.39, 67.53)(40.87, 68.54)(40.35, 69.55)(39.83, 70.55)(39.30, 71.53)(38.77, 72.51)(38.23, 73.47)(37.70, 74.43)(37.16, 75.37)(36.61, 76.31)(36.07, 77.24)(35.52, 78.16)(34.98, 79.07)(34.43, 79.98)(33.88, 80.88)(33.32, 81.77)(32.77, 82.65)
22N(45.46, 55.94)(44.99, 57.08)(44.52, 58.22)(44.04, 59.33)(43.56, 60.44)(43.07, 61.53)(42.58, 62.61)(42.08, 63.67)(41.58, 64.72)(41.07, 65.76)(40.56, 66.79)(40.04, 67.81)(39.52, 68.82)(39.00, 69.81)(38.48, 70.80)(37.95, 71.78)(37.41, 72.74)(36.88, 73.70)(36.34, 74.65)(35.80, 75.59)(35.26, 76.52)(34.72, 77.44)(34.17, 78.36)(33.62, 79.27)(33.07, 80.17)(32.52, 81.06)(31.97, 81.95)