地图投影计算机代数分析研究进展

地图投影是现代地图学的重要组成部分,涉及大量的椭圆函数幂级数展开、隐函数复合函数微分、椭圆积分、复变函数运算等一系列烦琐的数学分析过程,人工推导不但费时费力,而且容易出错,有时由于难以忍受的复杂性等各种原因,甚至根本无法实现。本文主要从椭球各纬度间正反解符号表达式、不同变形性质地图投影间的直接变换、高斯投影的复变函数表示、斜轴墨卡托投影数学分析、极区海图投影及变换等5个方面,论述了地图投影计算机代数分析取得的研究进展,讨论了该领域有待进一步解决的主要问题,对推动地图投影学的发展具有积极意义。     

Enhancing adaptive composite map projections: Wagner transformation between the Lambert azimuthal and the transverse cylindrical equal-area projections

The adaptive composite map projection technique changes the projection to minimize distortion for the geographic area shown on a map. This article improves the transition between the Lambert azimuthal projection and the transverse equal-area cylindrical projection that are used by adaptive composite projections for portrait-format maps. Originally, a transverse Albers conic projection was suggested for transforming between these two projections, resulting in graticules that are not symmetric relative to the central meridian. We propose the alternative transverse Wagner transformation between the two projections and provide equations and parameters for the transition. The suggested technique results in a graticule that is symmetric relative to the central meridian, and a map transformation that is visually continuous with changing map scale.     

关系数据库的符号化预处理

从实践的角度分析当前3种主要的符号化控制方式,提出基于关系数据库的符号化预处理方式,实践证明,该方式既可以很好地解决符号化的准确性问题,还能保证地图的显示速度。     

THE NOMENCLATURE OF MAP PROJECTIONS

Abstract

Map Projections.—A matter that should have been mentioned in the original article under this title (E.S.R., vii, 51, 190) is the definition of a map projection. In the list of carefully worded “Definitions of Terms used in Surveying and Mapping” prepared by the American Society of Photogrammetry (Photogrammetrie Engineering, vol. 8,1942, pp. 247–283), a map projection is defined as “a systematic drawing of lines on a plane surface to represent the parallels of latitude and the meridians of longitude of the earth or a section of the earth”, and most other published works in which a definition appears employ a somewhat similar wording. This, however, is an unnecessary limitation of the term. Many projections are (and all projections can be) plotted from rectangular grid co-ordinates, and meridians and parallels need not be drawn at all; but a map is still on a projection even when a graticule is not shown. Objection could be raised also to the limitation to “plane surface”, since we may speak of the projection of the spheroid upon a sphere, or of the sphere upon a hemisphere. Hence, it is suggested that “any systematic method of representing the whole or a part of the curved surface of the Earth upon another (usually plane) surface” is an adequate definition of a map projection.     

地图投影反解变换的一种新方法

通常地图投影反解变换有2种方法,即多项式拟合法和投影方程解析法.多项式法利用已知控制点的坐标对应关系,通过最小二乘法拟合求解地图投影反解变换的多项式函数,其优点是反解模型与地图投影无关,算法具有通用性,缺点是反算精度较低.解析法根据地图投影正算公式,在一定条件下通过解方程求得地图投影反解变换解析式,其优点是反解变换精度高,缺点是解法复杂.本文利用计算数学方法,根据地图投影变换的基本数学原理,提出了一种新的地图投影反解变换方法,双向迭代逼近法(BDIRA).具有反解变换精度高、收敛速度快、算法通用和GIS软件编程实现方便等特点.     

基于ArcGIS Engine地图符号化效率的几点思考

地图数据符号化是地图可视化的重要手段,是GIS实现过程中计算机时间最长的工作之一.因此,快捷、实时实现GIS系统地图数据符号化是衡量和评价GIS系统性能的重要指标.本文从应用出发.介绍了地图数据符号化的基本过程,提出了提高地图数据符号化效率的若干方法,分析了符号化方法的特点,从而为相关应用系统的开发提供了有价值的技术经验,提高了系统的开发效率.     

地图投影解析变换的数值实现方法

系统阐述了地图投影、地图投影变换,以地图投影及其变换为基础,采用Visual C++6.0作为软件平台,开发了GIS的地图投影变换软件。本系统分为解析正算和解析反算两部分,其中解析反算先用数值变换方法求得相应变换系数,然后再按照解析变换的方法实现相应功能,这也是本文的主要思想。同时本文以公式较为简单的墨卡托投影和公式较为复杂的圆锥投影为例说明了以上过程。该系统不仅可以单独使用,也可以与其他GIS软件进行接口,增强GIS的地图投影变换功能。     

An iterative approach for inverse transformation of map projections

ABSTRACT

Map projections are given by forward transformation equations. Inverse transformation is derived from forward transformation analytically or numerically. In this paper, a numerical approach for inverse transformation of map projections is proposed, which is based on numerical differentiation and Newton–Raphson root finding method. This approach can facilitate the program developments for map projections when inverse transformation is needed. Numerical differentiation is tested with three map projections. It is seen that seven-digit precision or more can be reached. Boundary conditions and initial guess problem in inverse transformation are discussed. In terms of initial guess, map projections are divided into three categories, and appropriate initial guess values for cylindrical, pseudocylindrical, azimuthal, and conical projections in normal aspect are suggested. Newton–Raphson method with numerical differentiation is tested with 20 different map projections by using test data sets. The results show that the proposed approach is applicable if appropriate initial guess is available.     

海岸带数据集成中的空间坐标转换方法研究

就海岸带空间数据集成中所涉及的多种坐标系统一、多种地图投影转换等问题作了详细分析。并提出了有效的解决方案。     

地图投影变形球面大圆弧的度量指标

地图投影是地图学的重要研究内容。任何地图投影都不可避免地存在变形问题。针对地图投影的变形,本文提出球面大圆弧和互补比率均值相结合的地图投影面积变形与形状变形指标。通过算例验证和相关性分析,大圆指标一方面简化了小圆指标(即互补比率均值)的计算过程,并能与小圆指标的结果保持一致;另一方面,大圆指标与微分指标之间也具有较高的一致性(形状变形指标的皮尔森积矩相关系数大于0.988)。由于大圆指标不依赖于微分计算,且计算简捷,因此大圆指标更具通用性。本文进一步采用回归分析对大圆指标进行分析,结果表明,大圆指标与微分指标具有较好的线性关系(线性回归的平均误差小于1.10‰)。为了降低采样点数量和解决采样点不统一问题,本文还提出了基于随机采样的指标计算方法,并对随机方法进行了验证和分析。依据大圆指标与微分指标的一致性和线性关系,可以认为使用大圆指标能够有效地评估地图投影的变形情况。     

中国海岸带专用地图投影设计

地图投影是各类海岸带信息的空间定位框架要素之一。通过对目前我国海岸带常用的各种地图投影现状分析,设计并实现全国海岸带专用投影———斜轴等角圆锥投影,该投影不仅使得整个中国海岸带实现无缝的一体化表达,而且使变形达到最小。文中详细阐述该投影的设计思想和设计原理,并将结果与其他各种海岸带投影进行详细的分析比较。     

基于COM的地图投影选择专家系统实现

现有的GIS软件中通常提供一定数量的地图投影方式供用户选择,地图投影选择是一个复杂的系统问题,需要考虑多方面的因素,对投影的各种参数进行评估,没有现成的公式、方案或规范可以遵循。本文通过分析地图投影选择问题的信息层次,论述了专家系统解决地图投影选择问题的可行性,讨论了详细的系统建立方法,并通过COM技术实现了地图投影选择的专家系统,并将系统与专业的GIS软件集成,为地图投影选择问题提供了一条解决思路。     

浅析中国古地图符号表示方法

地图内容是通过地图符号来表达的,随着地图所承载的内容不断丰富以及对地图精度的要求不断提高,地图符号也在不断地发展演变。基于此,通过研究从战国到清朝的有代表性的中国古地图,围绕中国古地图符号象形这一最大的特点,总结了中国古地图符号的山水画法、抽象画法和符号化3种表示方法,阐述了中国古地图符号表达方式的变化。     

Shapes on a plane: evaluating the impact of projection distortion on spatial binning

ABSTRACT

One method for working with large, dense sets of spatial point data is to aggregate the measure of the data into polygonal containers, such as political boundaries, or into regular spatial bins such as triangles, squares, or hexagons. When mapping these aggregations, the map projection must inevitably distort relationships. This distortion can impact the reader’s ability to compare count and density measures across the map. Spatial binning, particularly via hexagons, is becoming a popular technique for displaying aggregate measures of point data sets. Increasingly, we see questionable use of the technique without attendant discussion of its hazards. In this work, we discuss when and why spatial binning works and how mapmakers can better understand the limitations caused by distortion from projecting to the plane. We introduce equations for evaluating distortion’s impact on one common projection (Web Mercator) and discuss how the methods used generalize to other projections. While we focus on hexagonal binning, these same considerations affect spatial bins of any shape, and more generally, any analysis of geographic data performed in planar space.     

基于Super Map的国界网络电子地图设计与实现

国界网络电子地图是我国边界地区信息可视化的窗口,要求国界显示必须位置精确,与其他要素关系明了,符号表达符合标准。本文介绍了国界的基本类型,总结了国界表示的基本要求,从国界网络电子地图的需求和特点出发,针对国界网络电子地图的功能、图层、符号、色彩与注记进行了设计,并基于SuperMap软件制作了多比例尺国界网络电子地图,总结了一套实用的国界符号化方法。其中,国界的表示达到了公开出版的要求,为网络电子地图中国界的标准化、规范化显示提供了参考。     

Mixed cylindric map projections of the ellipsoid of revolution

The mixed spherical map projections of equiareal, cylindric type are based upon the Lambert projection and the sinusoidal Sanson–Flamsteed projection. These cylindric and pseudo-cylindric map projections of the sphere are generalized to the ellipsoid of revolution (biaxial ellipsoid). They are used in consequence by two lemmas to generate a horizontal and a vertical weighted mean of equiareal cylindric map projections of the ellipsoid of revolution. Its left–right deformation analysis via further results leads to the left–right principal stretches/eigenvalues and left–right eigenvectors/eigenspace, as well as the maximal left–right angular distortion for these new mixed cylindric map projections of ellipsoidal type. Detailed illustrations document the cartographic synergy of mixed cylindric map projections. Received: 23 April 1996 / Accepted: 19 April 1997     

基于Web的地图符号设计与实现

地图符号化是当前数字地图研究的重要内容之一。着重论述了Web环境下地图符号的构成特征、地图符号的组织及符号配置,以及符号库的基本传输方法,最后给出具体实例,说明了该实践的有效性与可行性。     

New Equal-Area Map Projections For Noncircular Regions

A series of new equal-area map projections has been devised. Called Oblated Equal-Area, its lines of constant distortion follow approximately oval or rectangular paths instead of the circles of the Lambert Azimuthal Equal-Area projection or the straight lines of the Cylindrical Equal-Area projection. The projection series permits design of equal-area maps of oblong regions with less overall distortion of shape and scale than equal-area maps on other projections.     

An adaptable equal-area pseudoconic map projection

Equivalence (the equal-area property of a map projection) is important to some categories of maps. However, unlike conformal projections, completely general techniques do not exist for creating new, computationally reasonable equal-area projections. The literature describes many specific equal-area projections and a few equal-area projections that are more or less configurable, but flexibility is still sparse. This work describes a new, highly configurable equal-area projection system consisting of arcs of concentric circles, placing it in the pseudoconic class. The system uses a novel technique to hybridize the Bonne pseudoconic projection and the Albers conic projection, subsuming many existing projections as degenerate cases. With the resulting system and the technique used to develop it, map projection designers will have greater choice in tailoring the projection to the need. The system may be particularly suited to maps that dynamically adapt to changing scale and region of interest, such as required for online maps.     

Direct georeferencing of airborne LiDAR data in national coordinates

The topographic mapping products of airborne light detection and ranging (LiDAR) are usually required in the national coordinates (i.e., using the national datum and a conformal map projection). Since the spatial scale of the national datum is usually slightly different from the World Geodetic System 1984 (WGS 84) datum, and the map projection frame is not Cartesian, the georeferencing process in the national coordinates is inevitably affected by various geometric distortions. In this paper, all the major direct georeferencing distortion factors in the national coordinates, including one 3D scale distortion (the datum scale factor distortion), one height distortion (the earth curvature distortion), two length distortions (the horizontal-to-geodesic length distortion and the geodesic-to-projected length distortion), and three angle distortions (the skew-normal distortion, the normal-section-to-geodesic distortion, and the arc-to-chord distortion) are identified and demonstrated in detail; and high-precision map projection correction formulas are provided for the direct georeferencing of the airborne LiDAR data. Given the high computational complexity of the high-precision map projection correction approach, some more approximate correction formulas are also derived for the practical calculations. The simulated experiments show that the magnitude of the datum scale distortion can reach several centimeters to decimeters for the low (e.g., 500 m) and high (e.g., 8000 m) flying heights, and therefore it always needs to be corrected. Our proposed practical map projection correction approach has better accuracy than Legat’s approach,¹ but it needs 25% more computational cost. As the correction accuracy of Legat’s approach can meet the requirements of airborne LiDAR data with low and medium flight height (up to 3000 m above ground), our practical correction approach is more suitable to the high-altitude aerial imagery. The residuals of our proposed high-precision map projection correction approach are trivial even for the high flight height of 8000 m. It can be used for the theoretical applications such as the accurate evaluation of different GPS/INS attitude transformation methods to the national coordinates.