05 第3章圆锥投影

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,1,第三章 圆锥投影（,Conical Projection,）,蒲英霞,南京大学地理与海洋科学学院,2011,年,10,月,11,日,2,圆锥投影的一般公式,等角圆锥投影（,Lambert,）,等面积圆锥投影（,Albers,）,等距离圆锥投影,斜轴、横轴圆锥投影,圆锥投影的分析和应用,3,3.1,圆锥投影的一般公式,圆锥投影的概念,设想用一个圆锥套在地球椭球体上，然后把地球椭球面上的经纬线网按照一定条件投影到圆锥面上，最后沿着一条母线（经线）将圆锥面切开而展成平面，就得到,圆锥投影,。,4,切圆锥投影、割圆锥投影,圆锥投影的分类,按圆锥面与地球椭球体之间的关系,等角投影、等面积投影和任意投影,按圆锥面与地球椭球体所处的不同位置,正轴圆锥投影、横轴圆锥投影、斜轴圆锥投影,按变形性质,5,6,A,A,A,y,x,y,x,s,s,图,3.1,正轴圆锥投影示意,a.,投影面同原面的关系,b.,投影面展开后的情况,纬线：投影为同心圆圆弧；,经线：投影为相交于一点的直线束，且夹角与经差（,）成正比；,经纬线：正交。,正轴圆锥投影的一般公式,7,Conic projections, in the normal polar,aspect, have as distinctive features:,（,1,）,meridians are straight equidistant lines, converging at a point which may or not be a pole. Compared with the sphere, angular distance between meridians is always reduced by a fixed factor, the,cone constant.,（,2,）,parallels are arcs of circle, concentric in the point of convergence of meridians. As a consequence, parallels cross all meridians at right angles. Distortion is constant along each parallel.,Due to simple construction and inherent distortion pattern, conic projections have been widely employed in regional or national maps of temperate zones (while,azimuthal,and,cylindrical,maps were favored for polar and tropical zones, respectively), especially for areas bounded by two not too-distant meridians, like Russia or the conterminous United States. On the other hand, conic projections are seldom appropriate for world maps.,8,正轴圆锥投影的极坐标公式：,正轴圆锥投影的直角坐标公式：,：纬线投影半径；,f,：取决于投影性质；,：投影常数；,：,经差。,s,：纬线,s,的投影半径，在一定投影中是常数。,s,y,A,x,y,x,s,图,3.1,正轴圆锥投影示意,9,圆锥投影经、纬线长度比、面积比和角度变形公式：,对上式求偏导数：,一般公式：,10,沿经纬线长度比、面积比和最大角度变形公式：,m,之所以取负号在于,和,的起算方向相反。,代入高斯系数：,11,A,B,C,D,A,B,C,D,S,图,3-2,圆锥投影中两个面的微分线段,由图,3-2,，设平面梯形,A,B,C,D,是地球面上微分梯形,ABCD,的投影，,d,是两经线的微小夹角,d,的投影，,d,是椭球体面上纬度的微小变化,(,d,),而产生投影后纬线半径的微小增量。,正轴圆锥投影经、纬线长度比、面积比和角度变形公式推导,(,第二种方法,),：,12,经纬线长度比、面积比和角度变形公式：,A,B,C,D,A,B,C,D,S,13,正轴圆锥投影的一般公式为：,对于椭球体：,对于球体：,14,3.2,等角圆锥投影（,Lambert,）,等角圆锥投影条件：,改写为：,积分：,将长度比公式代入，得：,15,令,esin,=sin,，则,式中,K,为积分常数，当,= 0,时，,=,K,，故,K,为赤道的投影半径,。,e,1,为第一偏心率,16,或按照以下方法进行积分,17,正轴等角圆锥投影的长度比、面积比和角度变形公式：,18,正轴等角圆锥投影的一般公式为：,在上式中，尚有常数,和,K,还需要进一步确定。,19,首先研究本投影中长度比,(,n),的变化情况。,为此先求,n,对,的一阶导数：,因为,所以,20,根据下式,21,要使,所以,以,0,表示最小长度比的纬圈。,22,为了证实,0,纬圈的长度比为最小，应证明,n,对,的二阶导数大于零。,以,0,代入，得,:,23,由于制图区域中只有一条纬线无长度变形，表明,0,处长度比,n,0,为最小，其余纬线上的长度比皆大于,1,，即,3.2.1,指定投影区域中一条纬线无长度变形,为了使通过,0,处长度比,n,0,无变形，即在该纬线上保持主比例尺不变，有，,该投影在制图区域内具有一条标准纬线，称为,等角切圆锥投影,。,即,24,3.2.2,指定投影区域中两条纬线无长度变形,在投影区域内，确定纬度为,1,和,2,的两条纬线，并要求其长度比都等于,1,。条件为,n,1,=,n,2,=1,，亦即,得,本投影具有两条标准纬线，称为,双标准纬线等角圆锥投影,或,等角割圆锥投影,（,或,Lambert,投影,）。,25,3.3,等面积圆锥投影,(Albers),根据等面积条件,P =,mn,= 1,，得：,移项后,积分得,式中,C,为积分常数，,S,为椭球面上经差为,1,弧度，纬差为,0,到,的梯形面积。,26,Map in Alberss conic projection, rendered with standard parallels 60N and 30N; reference parallel 45N, central meridian 0,The German Heinrich C. Albers published his equal-area conic projection in 1805. As usual, there is little distortion along the central parallel and none on the standard ones. The standard parallels may lie on different hemispheres, but if equidistant from the Equator, the projection degenerates into an,equal-area cylindrical,.,This projection was commonly applied to official American maps after usage of the,polyconic,projection declined.,In a particular case of Alberss conic projection, either 90N or 90S is chosen as a standard parallel, and therefore meridians converge at a pole. Published by Lambert in 1772, this projection preserves areas, thus parallels are farther apart near the vertex, getting closer together towards the non-standard pole. When 0 is chosen as the other standard parallel, the result is a cone constant of 1/2 and a semicircular map. Lambert himself chose a constant of 7/8 for his map of Europe: the resulting standard parallel, roughly 4835N, lies between Paris and Munich.,This projection was employed much less frequently than Alberss. In fact, it is probably the least known of Lamberts projections.,27,本投影中仍有两个常数,和,C,待确定。,正轴等面积投影的一般公式为：,28,依照前面所使用的方法，先确定长度比最小的纬线。为此先求,n,2,对,的一阶导数，并使之等于零。,按,故,假设在,0,处有极值，必须使,同样，也可证明,n,2,对,的,二阶导数大于零，说明,n,0,为最小值。,化简得,29,3.3.1,指定投影区域中一条纬线无长度变形且长度比为最小,根据投影条件，指定无长度变形所在纬线的纬度为,0,，其上,n,0,=1,，且为最小。,可得：,因为,将,代入，并导出,0,，则,所以,本投影只有一条无变形的纬线即单标准纬线，故又称为,等面积切圆锥投影,。,30,3.3.2,指定投影区域中两条纬线无长度变形,根据指定条件：,1,和,2,上,n,1,=,n,2,=1,，有,n,1,2,=,n,2,2,=1,。由,两式相减，可得：,得：,31,标准纬线,1,和,2,的投影半径：,可得：,本投影在两条纬线上无长度变形，称为,正轴等面积割圆锥投影或,Albers,投影,。,32,3.4,等距离圆锥投影,通常保持经线投影后无长度变形，即,m,= 1,。有：,积分,式中,C,为积分常数，,s,为由赤道至纬度,的一段经线弧长,。,Equidistant conic map, standard parallels 60N and Equator, central meridian 0. A full map is presented for illustration only, since this projection is seldom used for worldwide maps.,33,Detail of equidistant conic map with standard parallels as chosen by Mendeleyev (90N and 55N); central meridian 100E. Only a tiny missing wedge prevents it from being a full azimuthal map.,Equidistant conic map, standard parallels 30N and 60S,34,本投影有两个常数,和,C,待定。,正轴等距离投影的一般公式为：,35,依照前面方法，确定长度比最小的纬线。,纬线长度比,计算,n,对,的导数，并整理得：,设,0,处有极值，则,由,得,36,将,代入长度比公式，有：,或,为证明在,0,处为极小，可求二阶导数，验证其是否大于零。,由此可证明,n,0,为极小值。,37,3.4.1,指定投影区域中某纬线上长度比等于,1,且为最小,根据条件，有,n,0,=,1,。,此投影有一条标准纬线，故又称为,等距离切圆锥投影,。,由,得,又,因此，,所以,38,3.4.2,指定投影区域中两条纬线上无长度变形,设在投影区域内已选定,1,、,2,两条纬线，要求,n,1,=,n,2,=1,，据此条件有，,由此得,此投影有两条标准纬线，故又称为,等距离割圆锥投影,。,或,39,3.5,斜轴、横轴圆锥投影,当投影区域不是沿纬线延伸时，适宜采用斜轴或横轴圆锥投影。,在斜轴或横轴圆锥投影中，等高圈投影为一组同心圆弧（相当于正轴投影的纬线），垂直圈投影为过圆心的一组射线，且两直线间的夹角与相应的两垂直圈之间的夹角成正比（相当于正轴投影的经线）。经纬线投影为复杂的曲线，只是通过新极点的经线投影为直线，且成为其它经线的对称轴。,40,3.6,圆锥投影的分析和应用,圆锥投影变形的分析及其应用,41,正轴圆锥投影的变形仅与纬度有关，而与经度无关。同一条纬线上变形相等。,单标准纬线圆锥投影在纬线,0,上,n,=1,，其余均大于,1,；双标准纬线圆锥投影中，纬线,1,、,2,的长度比,n,1,=,n,2,=1,，变形自,1,、,2,向中间和向外逐渐增加，而且在,1,、,2,之间,n,1,。,任何圆锥投影的变形，自标准纬线起向高纬度增长快，向低纬度增长慢。沿经线长度比，则根据投影的变形性质而不同。,在同一投影区域内，割圆锥投影中变形增长的绝对值比切圆锥投影要小些。因此，前者比后者优越，在实际应用中也较广泛。,圆锥投影最适宜用作沿纬线延伸的中纬度地区的地图投影。,42,圆锥投影标准纬线的选择,制图区域南北纬差不大，只有,34,度，就可以采用单标准纬线。单标准纬线的选择非常简单，只要由制图区域南北边纬线的纬度取中数并凑整为整度或半度就可以。,制图区域跨纬差稍大一些，一般多采用双标准纬线。双标准纬线的选择通常有两种情况：预先选定和由所指定的条件决定。,43,Relatively few projections are called conic; nevertheless, many others are ruled by conic principles, since the cone is a limiting case of both the circle (a cone with no height, and cone constant 1) and the cylinder (a cone with vertex at infinity, with standard parallels symmetrical north and south of the Equator). There is only one type of,equal-area,conic projection, and only one is,conformal,.,Conic constraints are relaxed by,pseudoconic,(with curved meridians) and,polyconic,(with nonconcentric parallels) projections. Conic and coniclike are among the oldest projections, being the base for Ptolemys maps (ca. 100).,44,Conformal conic map with standard parallels 50N and 10S, clipped at 50S.,The same paper (1772) with Lamberts equal-area conic projection included his,conformal,conic design: Lambert explicitly investigated a conic approach as intermediary between the then known conformal projections,azimuthal stereographic,and,Mercator,s. These are in fact special cases of the conformal conic, obtained respectively when one pole is the single standard parallel and when the standard parallels are symmetrically spaced above and below the Equator.,This projection remained essentially ignored until World War I, when it was employed by the French military. Since then, it has become one of the most widely used projections for large-scale mapping, second only to Mercators.,Like in all conformal projections, scale distortion is greatly exaggerated in the borders of a worldwide map, although less than in Mercators. Meridians converge at the pole nearest the standard parallels; the opposite pole lies at infinity and can not be shown. Scale distortion is constant along each parallel. Meridian scale is less than true between the standard parallels, and greater outside them.,45,Braun stereographic conic map,Actually the only conic projection presented here which is defined by a simple geometric construction, the stereographic projection created by C.Braun (1867) encloses the globe in a cone aligned with the north-south axis, 1.5 times as tall as the globe and tangent at the 30N parallel.The projection center is the South pole and the resulting map fits a perfect semicircle.,46,为什么说正轴圆锥投影变形仅与纬度有关，而与经差无关？,正轴圆锥投影沿经线长度比,m,中的负号是怎样得出的？,请写出球体（半径为,R,）在正轴圆锥投影中的长度比、面积比、角度变形公式。,等角圆锥投影纬线半径,的推导过程中，出现的常数,K,有什么含义？,绘出等角、等面积、等距离圆锥投影在两条标准纬线的情况下，沿经线的变形椭圆形状（注明,m,、,n,、,P,大于、小于或等于,1,的情况）。,试证明，在圆锥投影中变形自标准纬线向北增长要快于向南增长。,思考题,47,上机实习作业,经度：,纬度：,经纬线网密度：,标准纬线：,中央经线：,主比例尺：,1,：,1000,万,地球椭球：克拉索夫斯基椭球,1,、正轴等角割圆锥投影经纬网的绘制（以教材第,105,页为例,),48,序号,大写,小写,英文注音,国际音标注音,中文注音,1,alpha,a:lf,阿尔法,2,beta,bet,贝塔,3,gamma,ga:m,伽马,4,delta,delt,德尔塔,5,epsilon,epsilon,伊普西龙,6,zeta,zat,截塔,7,eta,eit,艾塔,8,thet,it,西塔,9,iot,aiot,约塔,10,kappa,kap,卡帕,11,lambda,lambd,兰布达,12,mu,mju,缪,13,nu,nju,纽,14,xi,ksi,克西,15,omicron,omikron,奥密克戎,16,pi,pai,派,17,rho,rou,肉,18,sigma,sigma,西格马,19,tau,tau,套,20,upsilon,jupsilon,宇普西龙,21,phi,fai,佛爱,22,chi,phai,西,23,psi,psai,普西,24,omega,omiga,欧米伽,49,50,试证明，在圆锥投影中变形自标准纬线向北增长要快于向南增长。,两边,取对数,在,0,处展开级数，有,证明（以等角圆锥投影为例）：,由纬线长度变形公式，有,51,式中,0,为最小长度比纬线的纬度。,而,令,在,=,0,处为零,故,52,假设地球为球体，则有：,利用变形近似式,ln,n,有,53,在切圆锥投影中，设标准纬线为,0,，标准纬线以北、以南纬差为的两纬线纬度分别为,0,+,，,0,-,。它们的纬线长度比为,n,1,n,2,。则有：,有,1,2,。故向高纬度变形快于低纬度。,其他情况类似。,