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在常見的讲述球面三角学的书籍里,一般都用投影法或平面三角方法导出余弦定理、正弦定理等球面三角基本公式。华罗庚先生在他所著“高等数学引論”第一卷第一分册中,則应用了矢量分析方法。这里,給出另一种較为簡洁的方法,即应用矩陣运算来推导球面三角基本公式。在未导出这些公式之前,先簡单介紹一下“旋轉矩陣”的概念和基水性貭。考虑空間直角坐标系的旋轉变換。設原坐标系(O;x,y,z)繞x軸沿正方向(逆时針)旋轉角α后变換为新坐标系(ο;ξ,η,ζ)(如图1),則由解析几何学便知,新旧坐标系之間有如下关系:
In the common books dealing with spherical trigonometry, spherical sphere triangle basic formulae such as cosine theorem and sine theorem are generally derived by projection method or plane triangle method. Mr. Hua Luogeng applied the vector analysis method in the first volume of his book “Introduction to Advanced Mathematics”. Here, another relatively simple method is given, that is, applying a matrix operation to derive a spherical triangle basic formula. Before exporting these formulas, let’s briefly introduce the concept of “rotation matrix” and basic water-repellency. Consider the rotation transformation of a rectangular coordinate system. Let the original coordinate system (O;x,y,z) be rotated in the positive direction (counterclockwise) around the x-axis and then transformed into a new coordinate system (o;ξ,η,ζ) (see Figure 1). According to analytical geometry, there are the following relationships between the old and new coordinate systems: