05-第3章--圆锥投影

1、2021/3/10讲解：XX1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2021/3/10讲解：XX2n圆锥投影的一般公式n等角圆锥投影（Lambert）n等面积圆锥投影（Albers）n等距离圆锥投影n斜轴、横轴圆锥投影n圆锥投影的分析和应用2021/3/10讲解：XX33.1 圆锥投影的一般公式n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，就得到圆锥投影圆锥投影。2021/3/10讲解：XX4切圆锥投影、割圆锥投影n圆

2、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2021/3/10讲解：XX52021/3/10讲解：XX6AAAyxyxss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。n正轴圆锥投影的一般公式2021/3/10讲解：XX7Conic projections, in the normal polar aspect, have as distinctive featu

3、res: （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 meridian

4、s. 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

5、 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.2021/3/10讲解：XX8正轴圆锥投影的极坐标公式：)(f正轴圆锥投影的直角坐标公式：sincosyxs：纬

6、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。syAxyxs图3.1 正轴圆锥投影示意2021/3/10讲解：XX9圆锥投影经、纬线长度比、面积比和角度变形公式：)(cos( )sin()sin()cos(yxddyddxsincosyxs对上式求偏导数：一般公式：)(f2021/3/10讲解：XX10batgbabanmPrrGnMddMEm)445(2sin或沿经纬线长度比、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。0)cos()sin()()()()sin()cos()()(22222222222yyxxFyxGdd

7、ddddyxE代入高斯系数：2021/3/10讲解：XX11ddABCDABCDddSdd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ABCD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)：2021/3/10讲解：XX12batgbabaMrddnmPrrddABBAnMddADDAm)445(2sin或经纬线长度比、面积比和角度变形公式：ddABCDABCDddSdd2021/3/10讲解：XX13sincos2sin )

8、(yxbabamnPrnMddmfs正轴圆锥投影的一般公式为：对于椭球体：对于球体：sincos2sincos )(yxbabamnPRnRddmfs2021/3/10讲解：XX143.2 等角圆锥投影（Lambert）n等角圆锥投影条件： ()0mnab或或rMddrMddKdeeedKdeeeKedeNMddlncos)sin1 (coscos lncos)sin1 (cos)sin1 (lncos)sin1 ()1 (lncos22121122122122122121改写为：积分：将长度比公式代入，得：2021/3/10讲解：XX15)sin1sin1)(245()245()245( l

9、nlnln)245()245(lnln)245(ln)245(lnlncoscoscosln211121111eeeeetgtgtgUUKKUKtgtgKtgetgKded其中令esin=sin，则式中 K 为积分常数，当 = 0 时， = K，故 K 为赤道的投影半径为赤道的投影半径。e1为第一偏心率2021/3/10讲解：XX16)sin1sin1)(245( lnlnln)sin1sin1)(245(ln ln)sin1ln(2)sin1ln(2)245(ln ln)sin()sin11sin11(21)245(ln lnsin1coscosln211211111111112211111

10、eeeetgUUKKUKeetgKeeeetgKedeeetgKdeeed其中或按照以下方法进行积分2021/3/10讲解：XX17正轴等角圆锥投影的长度比、面积比和角度变形公式：0)(222rUKnmPrUKrnm2021/3/10讲解：XX180)(sin ,cos ,)sin1sin1)(245( ,22222212111rUKnmPrUKrnmyxabaeeetgUUKse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。2021/3/10讲解：XX19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：rn2rddrddrddn因为rMd

11、drMdd所以2021/3/10讲解：XX20sin)sin1 ()1 (sin)sin1 ()cossin1 (sin )sin1 (sincos)sin1 (sin )sin1 ()cossin2()sin1 (21cos)sin1 (sin )sin1 (cos()cos(2322121232212212212322122122122121212212122121221MeeaeeeaeeeaeeeeaeadddNdddr根据下式2021/3/10讲解：XX21)(sin)sin(2rMrMrrMrddnsin sin 0应为零，所以则ddn要使所以以0表示最小长度比的纬圈。2021/3

12、/10讲解：XX22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。NMrrMrdddnd)(sin(220)sin1()1(2212100000202eenNMrdnd以0代入，得:2021/3/10讲解：XX23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n0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0sin为了使通过 0 处长度比 n0 无变形，即在该纬线上保持主比例尺不变，有，10n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等角切圆锥投影。000001UrKUrKn即2021/3/10讲解：XX243.2.2

13、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n1 = n2 =1，亦即12211UrKUrK22111221lglglglgUrUrKUUrr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双标准纬线等角圆锥投影或等角割圆锥投影等角割圆锥投影（或或 Lambert投影投影）。2021/3/10讲解：XX253.3 等面积圆锥投影(Albers)根据等面积条件P = mn = 1，得：1rMddnm移项后积分得式中C为积分常数，S为椭球面上经差为为椭球面上经差为1弧度，纬差为弧度，纬差为0 到到 的梯形面积。的梯形面积。Mrdd1)

14、(2)(1)(1222SCSCMrdC或2021/3/10讲解：XX26Map in Alberss conic projection, rendered with standard parallels 60N and 30N; reference parallel 45N, central meridian 0The German Heinrich C. Albers published his equal-area conic projection in 1805. As usual, there is little distortion along the central paralle

15、l 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. I

16、n 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-st

17、andard 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 l

18、ess frequently than Alberss. In fact, it is probably the least known of Lamberts projections. 2021/3/10讲解：XX27babaPnmrnyxrsCrs2sin11,sin ,cos)(222222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：2021/3/10讲解：XX28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n2对的一阶导数，并使之等于零。22222)(2rSCrn23242sin)(22sin2)(2rSCrMrMSCMrrrddn0sin)( 2200

19、0rSC020sinn按故假设在0处有极值，必须使同样，也可证明n2对的二阶导数大于零，说明n0为最小值。化简得2021/3/10讲解：XX293.3.1 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n0 =1，且为最小。可得：0020sinsin n因为1000rn将代入，并导出0，则000ctgN所以0202SC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等面积切圆锥投影。2021/3/10讲解：XX303.3.2 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n1= n2=1 ，有 n12= n22=1 。

20、由222211)(2)(2rSCrSC两式相减，可得：)(2122221SSrr22222)(2rSCrn得：2021/3/10讲解：XX31标准纬线1和2的投影半径：2211rr可得：22212122SSC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正轴等面积割圆锥投影或Albers投影投影。2021/3/10讲解：XX323.4 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：MddMddm或1积分sCMdC或式中C为积分常数，s为由赤道至纬度为由赤道至纬度 的一段经线弧长的一段经线弧长。Equidistant conic map, standard parall

21、els 60N and Equator, central meridian 0. A full map is presented for illustration only, since this projection is seldom used for worldwide maps. 2021/3/10讲解：XX33Detail of equidistant conic map with standard parallels as chosen by Mendeleyev (90N and 55N); central meridian 100E. Only a tiny missing w

22、edge prevents it from being a full azimuthal map. Equidistant conic map, standard parallels 30N and 60S 2021/3/10讲解：XX34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：babarsCrnPmyxsCs2sin)(, 1sin,cos2021/3/10讲解：XX35依照前面方法，确定长度比最小的纬线。纬线长度比rsCrn)(计算n对的导数，并整理得：rsCrMddnsin)(2设0处有极值，则000000sincosctgNN0sin000rsC）（由sC 得2021/3

23、/10讲解：XX36将00ctgN代入长度比公式，有：0000sinrn或00sinn为证明在0处为极小，可求二阶导数，验证其是否大于零。0)sin11(cos)(022121000000202eenrMrsCdnd由此可证明n0为极小值。2021/3/10讲解：XX373.4.1 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n0 = 1。0sin000ctgNsC此投影有一条标准纬线，故又称为等距离切圆锥投影等距离切圆锥投影。00sinn由得又000ctgN因此，00000sinsrsC所以2021/3/10讲解：XX383.4.2 指定投影区域中两条纬线上无长度变形设在投影区域内

24、已选定1、2两条纬线，要求n1=n2=1，据此条件有，1)()(2211rsCrsC由此得2211211221sCrsCrrrsrsrC此投影有两条标准纬线，故又称为等距离割圆锥投影等距离割圆锥投影。12211rr或2021/3/10讲解：XX393.5 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。2021/3

25、/10讲解：XX403.6 圆锥投影的分析和应用n圆锥投影变形的分析及其应用2021/3/10讲解：XX41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n1= n2 =1，变形自1、2向中间和向外逐渐增加，而且在1 、2之间n1。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

26、的地图投影。2021/3/10讲解：XX42n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。2021/3/10讲解：XX43Relatively few projections are called conic; nevertheless, many others are ruled by conic principles, since the cone is a li

27、miting 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 r

28、elaxed 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). 2021/3/10讲解：XX44Conformal conic map with standard parallels 50N and 10S, clipped at 50S. The same

29、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 Mercators. These are in fact special cases of the conformal conic, ob

30、tained 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 mo

31、st 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 pol

32、e 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. 2021/3/10讲解：XX45Braun stereographic conic mapActually the only conic projection presented here which is defined by a s

33、imple 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. 2021/3/10讲解：XX46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