Antipodal point

Japanese: 対心点 - たいしんてん（英語表記）antipodal point

Two points that are the ends of a diameter in a sphere, that is, two points on a sphere that are symmetric with respect to the center of the sphere, are called antipodes or antipodal points of each other. If the antipodal point of P is represented as P ^* , then for the spherical triangle ABC, the spherical triangle A ^* B ^* C ^* is called an antipodal triangle (Figure). An antipodal triangle is congruent to the original spherical triangle in the reverse direction. The following theorem is called Borsuk's antipodal point theorem. 〈If f and g are two continuous functions defined on the sphere S , then there exists a point P on S such that f (P) = f (P ^* ) and g (P) = g (P ^* ).〉

Source: Heibonsha World Encyclopedia, 2nd Edition Information

Japanese:

球において一つの直径の両端となっている2点，すなわち球の中心に関して対称な球面上の2点を互いに他の対心点，または対蹠（たいしよ）点という。Pの対心点をP^*で表すとき，球面三角形ABCに対し，球面三角形A^*B^*C^*を対心三角形という(図)。対心三角形はもとの球面三角形と逆向きに合同である。次の定理をボルスクBorsukの対心点定理という。〈f,gを球面S上で定義された二つの連続関数とするとき，f(P)＝f(P^*)，g(P)＝g(P^*)を満たす点PがS上に必ず存在する〉。

出典　株式会社平凡社世界大百科事典第２版について　情報

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