椭球体的重要参数和公式

基准和椭球体

地球表面非常的不规则,如果按照大地水准面来定义地球表面,别说是球体了,估计连一个椭球体都得不到。假想一个扁率极小的椭圆,绕大地球体短轴旋转所形成的规则椭球体称之为地球椭球体。地球椭球体表面是一个规则的数学表面,可以用数学公式表达,所以在测量和制图中就用它替代地球的自然表面。椭球体是通过长半轴和短半轴(或长半轴和扁率)来定义或控制。椭球通过基准来定义,一般包括中心和方向。大地基准是大地测量计算时的参考依据和尺度。大地基准面(Geodetic Datum),由椭球体本身及椭球体和地表上一组点视为原点间之关系来定义。一般认为此关系能以6个量来定义,通常是大地纬度、大地经度、原点高度、原点垂线偏差之两分量及原点至某点的大地方位角。这些概念并不是特别容易理清,暂时不理会。

国际上使用最广泛的World Geodetic System 1984 (WGS84),是一种地心坐标系,坐标原点为地球质心。包括了一个基准和一个坐标系统。至于为什么总是用年份来标识基准,主要因为大陆漂移等原因,用来定义基准的那些点位,总是在变化的。所以通常使用定义基准的年份来命名,所以在84之前,还有WGS72,WGS66和WGS60。GPS的广播星历是以WGS84坐标系为依据的,而据说当年定义这个基准的时候,有人忘记加入欧洲的控制点,所以欧洲又有了一套ETRS89(…懵逼)。

在我国,北京54采用的是克拉索夫斯基椭球,在该坐标系内进行了许多地区的局部平差,其成果得到了广泛的应用。1978年4月在西安召开全国天文大地网平差会议,确定重新定位,即建立西安80坐标系,此后西安80坐标系取代北京54坐标系成为新的国家坐标系。西安80采用的是1975国际椭球。

常用椭球

坐标系 椭球 长半轴 短半轴 扁率
WGS84大地坐标系 WGS84地球椭球 6378137 6356752.314 1/298.257223563
北京54坐标系 克拉索夫斯基椭球 6738245 6356863 1/298.3
西安80坐标系 IAG75地球椭球 6378140 6356755 1/298.257
国家2000大地坐标系 CGCS2000地球椭球 6378137 6356752.31414 1/298.257222101

重要公式

椭球体(或称旋转椭球体)有两个半径,即长半轴(a)和短半轴(b)。椭球体可以由a和b定义,也可以由a和扁率定义,扁率f(Flattening)的定义为:
\[f={{a-b}\over a}\]

在坐标系中除了长半轴、短半轴和扁率这三个直观数字外,偏心率的平方,也可以用来描述椭球体形状,加上其余的一些参数,汇总如下:

参数 公式
第一偏心率 \(e_1=\sqrt{a^2-b^2\over a^2} \)
第二偏心率 \(e_2=\sqrt{a^2-b^2\over b^2} \)
第一辅助参数 \(W=\sqrt {1-e^2 \sin^2B}\)
卯酉圈曲率半径 \(N ={ a \over W } = {a \over \sqrt {1-e^2 \sin^2B}}\)
子午线曲率半径 \(M = {a(1-e^2) \over W^3} = { a(1-e^2) \over (1-e^2 \sin^2B)^{3 \over 2} }\)

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