一个球极投影不等式的解析证明

The Analytic Proof of the Inequality for Stereographic Projection

设π:S2→R^2为球极投影,若记X∈S2的球极投影像为x,则对S2上x3坐标不超过-cos2θ(0θπ/2)的任意两点X,Y,2cos2θ｜x-y｜d(X,Y)2｜x-y｜其中等号成立当且仅当X=Y.

Let π:S2→2 be stereographic projection,if writing x is the image of the stereographic projection of X∈S2,then for any X,Y∈S2 whose x3 coordinates -cos 2θ(0θπ/2) 2cos2θ|x-y|d(X,Y)2|x-y| where equalities hold if and only if X=Y.