等角投影面上等角航线的曲率分析

等角航线又名恒向线,在椭球面上它是与经线方向保持定角α的航向线。如图1所示,设Sl为等角投影面上等角航线的表象,该等角航线的航向角为α,坐标方位角为Φ。P点的子午线收敛角为γ,根据微分几何学[1]关于曲率的定义,可以写出等角投影面上的经、纬线和斜航线的曲率公式,令P点图1等角投影面上的等角航线的子午圈曲率半径为M和卵西圈曲率半径为N,投影长度比为m,则有：或者下面分析常用等角投影面上的等角航线的曲率情况：（1）墨卡托投影面上（基准纬度中0）代人（1）、（2）、（3）式得到：凡一K,＝KI—0说明等角航线被表象为…     

契比雪夫投影的研究

根据等角投影理论,推导出了契比雪夫投影公式的具体形式,并对契比雪夫投影在制作中国全图的应用和变形与等角方位投影、等角圆锥投影进行了比较分析.结果表明,契比雪夫投影要优于等角方位投影和等角圆锥投影.     

契比雪夫投影的研究

根据等角投影理论,推导出了契比雪夫投影公式的具体形式,并对契比雪夫投影在制作中国全图的应用和变形与等角方位投影、等角圆锥投影进行了比较分析.结果表明,契比雪夫投影要优于等角方位投影和等角圆锥投影.     

等角投影理论和方法综述

等角投影与其它性质投影比较之，其研究尤为深刻，应用也最为广泛。本文对等角投影的历史发展作了简单回顾，重点对等角投影的数学基础、等角投影的一般公式、等角投影变形量度、具有极值特性的等角投影和探求等角投影的方法进行了综述。最后提出需要继续研究的若干问题作为等角投影研究展望。     

常用等角投影及其解析变换的复变函数表示

借助复变函数理论讨论了常用等角投影及其解析变换的复变函数表示;给出了高斯投影、墨卡托投影和等角圆锥投影正反解的复变函数表示模型;在此基础上系统地推导出了高斯投影、墨卡托投影和等角圆锥投影间解析变换的复变函数表达式.这些复数变换公式是含参考椭球第一偏心率的符号形式,可解决不同参考椭球下的变换问题.与传统的实数变换公式相比,其结构更为简单、理论更为严密.     

等角投影理论和方法综述

等角投影比其它性质投影之研究更为深刻，应用也最为广泛。本文对等角投影的历史发展作了简单回顾，重点对等角投影的数学基础、等角投影的一般公式、等角投影变形量度［８］、具有极值特性的等角投影和探求等角投影的方法等进行了综述。最后提出需要继续研究的若干问题作为等角投影研究展望。     

关于投影平面上等长性质的研究

投影平面上的等长性质跟等角性质、等积性质一样，都是说明地球椭球面描写到地图平面上后的投影不变性的。所不同的是前者说明原面上某一方向的曲线投影后的长度不变性，后者说明原面上的微分圆投影后的相似性和等积性。本文研究投影平面上的等长性质。     

拉格朗日投影与常用等角投影间解析变换的复变函数表示

借助复变函数理论讨论了拉格朗日投影与常用等角投影间的解析变换问题,导出了拉格朗日投影正反解的复变函数表达式,在此基础上系统地建立了该投影与高斯投影、墨卡托投影和等角圆锥投影间解析变换的复变函数表示模型。这些复数变换公式是含参考椭球第一偏心率的符号形式,可解决不同参考椭球下的投影变换问题,与传统的实数变换公式相比,其结构更为简单,理论更为严密,便于实际使用。     

等角投影变换的正形变换多项式及其应用的研究

本文系统地讨论了等角投影变换的正形变换多项式的理论、方法及其在计算机制图和地理信息系统建设中的若干应用问题,并同时介绍了《等角投影变换BASIC、FORTRAN程序软件包》的有关内容。     

等角投影有限元变换法研究

本文论证了任意两等角投影间的变换，可以归结为求拉普拉斯方程狄利克莱问题的解。并指出，一些著名等角投影变换的解析表达式，实际是狄利克莱问题在某种特殊条件下的精确解。但是，因为在大多数情形下求精确解是困难的，为此本文引进了求解狄利克莱问题的有限元法，并给出了应用有限元法进行等角投影变换的计算方法。最后通过实例评价了这一方法的精度。     

The Hammer projection of the ellipsoid of revolution (azimuthal, transverse, rescaled equiareal)

The classical Hammer projection of the sphere, which is azimuthal transverse rescaled equiareal, is generalized to the ellipsoid of revolution. Its first constituent, the azimuthal transverse equiareal projection of the biaxial ellipsoid, is derived giving equations for an equiareal transverse azimuthal projection. The second constituent, the equiareal mapping of the biaxial ellipsoid with respect to a transverse frame of reference and a change of scale, is then considered; results give collections of the general mapping equations generating the ellipsoidal Hammer projection, which lead to a world map. Received: 14 May 1996 / Accepted: 21 February 1997     

Equal-Area Projections of the Triaxial Ellipsoid: First Time Derivation and Implementation of Cylindrical and Azimuthal Projections for Small Solar System Bodies

Many small solar system bodies such as asteroids or small satellites have irregular shapes, often approximated by the reference surface of a triaxial ellipsoid. Map projections for the triaxial ellipsoid are needed to present the incoming data in the form of maps. In this paper the formulae of equal-area cylindrical and azimuthal projections of the triaxial ellipsoid were derived and practically implemented for the first time using as an example the asteroid 253 Mathilde. This paper is the final in a series of papers devoted to all main classes of projections of the triaxial ellipsoid. Before this, the authors obtained equidistant along meridians projection and Jacobi conformal projection for the triaxial ellipsoid.     

等角投影数值变换的正型多项式方法及其应用分析

数值变换是地图投影变换中常用的方法,鉴于二元n次多项式直接求解法针对性差、计算效率低、稳定性差等缺点,针对等角投影采用正型多项式法进行数值变换,本文着重介绍正型多项式法及其应用。     

等分分层组合投影研究

针对中国地理格网(1°、10°等多级格网系统)的分割方法,设计了一种适合该格网系统的新型地图投影——分层组合投影。从微分几何的观点出发,把地球椭球按等纬度分割成若干层圆台,分别建立每个圆台的投影模型,即可得到一种地图投影。这种投影还可根据格网间隔的不同进行细分,从而发展成为一种适合多分辨率格网模型的动态地图投影。通过对该投影进行变形计算表明,该投影可以保持等角,而且面积和长度变形都很小,特别是在高纬度地区,与Mercator投影相比变形明显减小。     

菜单式通用高斯投影计算程序(CASIOfx-4500P)

本文简化了子午线弧长公式 ,并将之用于各种椭球上的通用高斯投影计算 ,编写CASIOfx -4 5 0 0P计算器的菜单式通用高斯投影计算程序 ,以便一般测绘单位推广应用 1 980年国家大地坐标系。     

等角圆锥投影基准纬度非迭代算法

为简化传统正轴等角圆锥投影求解基准纬度时繁琐的迭代算法,引入平均纬度和平均纬差的概念,借助计算机代数系统Mathematica,在平均纬差处级数展开,导出了基于球体模型的正轴等角圆锥投影求解基准纬度的非迭代算法。以全国和不同纬差的省区为例,将其与传统椭球迭代算法进行对比分析。结果表明,推导的基于球体模型的非迭代公式计算基准纬度B0、B1、B2的相对误差最大值为2.011%,长度变形的相对误差小于1×10-6,基本可满足全国以及各省区地图制图的精度要求,从而验证了所研究算法的精确性与实用性。     

极区不分带高斯投影的正反解表达式

针对传统高斯投影公式在极区难以应用的问题,通过引入等角余纬度及等量纬度的表达式,推导出严密的复数等角余纬度公式,进而得到严密的极区高斯投影正解表达式;借助符号迭代法及指数函数与三角函数间的关系式,推导出对应的极区高斯投影反解表达式;基于极区高斯投影正解表达式,推导出可用于极区的长度比、子午线收敛角公式;最后,以CGCS2000椭球为例,与实数型幂级数高斯投影公式计算的结果进行对比,验证了本文推导公式的正确性。由于本文推导公式不受带宽限制,且可用于整个极区的表示,对于编制极区地图及极区导航具有重要的参考价值。     

大椭圆线椭球高斯投影在高速铁路工程中的应用

为了减少东西走向长线工程中的控制点数据在进行高斯投影转换时需要频繁换带的问题,提出了一种方法,即以工程线路中心线或其附近的大椭圆线为中央子午线,建立大椭圆线椭球,根据非线性规划最优理论求出大椭圆线椭球参数,并对其进行椭球变换,然后以此新椭球为基础进行高斯投影。     

ON THE LAMBERT CONFORMAL PROJECTION AS APPLIED IN CHINA

Abstract

There are no proper projections for use in geodetic work in a country which has great extensions both in latitude and longitude. For, if a single projection of any kind be applied in such a case, the linear and angular distortions would be so great at the boundary that it is very difficult or even impossible to apply the corrections to them. In order to render it possible for any projection to be applied, the area in question should be divided either into strips bounded by meridians or into zones bounded by parallels. In the former case the Transverse Mercator or Gauss’ projection may be used, while in the latter, the Lambert conformal projection is the most suitable. China is such a country as that mentioned above. It covers an area extending from 16°N. to 53°N. in latitude and of no less than sixty-five degrees in longitude. The problem of choosing a projection for geodetic work depends only on how the area is to be divided. It has been decided by the Central Land Survey of China to adopt the Lambert conformal projection as the basis for the co-ordinate system, and, in order to meet the requirements of geodetic work, the whole country is subdivided into eleven zones bounded by parallels including a spacing of 3½ degrees in latitude-difference. To each of these zones is applied a Lambert projection, properly chosen so as to fit it best. The two standard parallels of the projection are situated at one-seventh of the latitude-difference of the zone from the top and bottom. Thus, the spacing between the standard parallels is 2½ degrees. This gives a maximum value of the scale factor of less than one part in four thousand, thus reducing the distortions of any kind to a reasonable amount. The area between these parallels belongs to the zone proper, while those outside are the overlapping regions with the adjacent ones. All the zones can be extended indefinitely both eastwards and westwards to include the boundaries of the country.     

Generalization of the Lambert–Lagrange projection

The Lagrange projection represents conformally the terrestrial globe within a circle. This is achieved by compressing the latitude and longitude and by applying the new coordinates into the equatorial stereographic projection. The same concept can be generalized to any conformal projection, although the application of this technique to other analytical functions is less known. In this work, the general Lambert–Lagrange projection formula is proposed and the application of the modified coordinates is discussed on projections: stereographic, conformal conic and Gauss–Schreiber. In general, the results are merely a curiosity, except for the case of Gauss–Schreiber, where the use of coordinates with altered scale can be applied in the optimization of conformal projections.