This paper proposes 3 series of map projections for world map with interruption over ±180° of longitude. Root-mean-square error of area and angle, which is L2 of Kamiya （2012）, or simply noted L in this paper, is minimized in the proposed projections. The objective function L depends on one parameter wS（0 ≤ wS ≤ 1）, which is a weight for area error.
Proposed projections are polynomials of longitude and latitude under the condition that poles shall be a point. If 360° of angle at pole corresponds to θ° on a map, angle defect of pole is defined as 360−θ. This paper proposes 3 series of global optimal projections： with 0° of angle defect of pole, named whole pole； without constraint condition for angle defect of pole, named free pole；with 180° of angle defect of pole, named half pole. Order of the polynomials is 13 for whole pole and 11 for others.
Proposed projection by wS＝0.S are named “Global optimal projection SW” for whole pole, i.e. “Global optimal projection 5W” for wS＝0.5, “Global optimal projection SF” for free pole, and “Global optimal projection SH” for half pole.
Value of L, weighted root-mean-square-error, is much smaller in proposed projections than in conventional projections except almost conformal condition or almost equal area condition. Therefore, the global optimal projections are excellent projection as mean of minimizing both area and angle error.
Though value of L is slightly larger in whole pole than in free/half pole, distortion around the poles is smaller in whole pole than in free/half pole. If distortions of Arctic/Antarctic area （for example 80° or higher latitude） are important, whole pole shall be selected. Otherwise, free pole or half pole shall be selected.
Global optimal projections reduce near origin region almost isotropically under the conformal-like condition. They also stretch the region in the northsouth direction almost equivalently under equal-arealike condition. Parallel lines are concave to the pole, but less concave or sometimes convex to the pole near central meridian.
Aspect ratio of map shall be around 1.5 for minimizing distortion, though many map projections adopt 2.0. Necking at poles like Eisenlohr projection is important to decrease angular distortion.
Technical information about global optimal projections will be disclosed at http://www.gsi.go.jp/cais/geoinfo-index.html.
