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墨卡托投影、地理坐标系、地面分辨率、地图比例尺

GIS理论(墨卡托投影、地理坐标系、地面分辨率、地图比例尺、Bing Maps Tile System)

  墨卡托投影(Mercator Projection),又名“等角正轴圆柱投影”,荷兰地图学家墨卡托(Mercator)在1569年拟定,假设地球被围在一个中空的圆柱里,其赤道与圆柱相接触,然后再假想地

球中心有一盏灯,把球面上的图形投影到圆柱体上,再把圆柱体展开,这就是一幅标准纬线为零度(即赤道)的“墨卡托投影”绘制出的世界地图。

一、墨卡托投影坐标系(Mercator Projection

  墨卡托投影以整个世界范围,赤道作为标准纬线,本初子午线作为中央经线,两者交点为坐标原点,向东向北为正,向西向南为负。南北极在地图的正下、上方,而东西方向处于地图的正右、左。

  由于Mercator Projection在两极附近是趋于无限值得,因此它并没完整展现了整个世界,地图上最高纬度是85.05度。为了简化计算,我们采用球形映射,而不是椭球体形状。虽然采用Mercator Projection只是为了方便展示地图,需要知道的是,这种映射会给Y轴方向带来0.33%的误差。

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earthRadius =6378137

20037508.3427892 = earthRadius * (math.pi - 0)

85.05112877980659 = (math.atan(math.exp(aa / earthRadius))-math.pi/4)*2 * 180 / math.pi

image = 512 * 512

groundResolution(1 level)  = (20037508.3427892 * 2) / 512 = 78271.516964

screendpi = 96

mapScale = groundResolution * 96 / 0.0254 = 295829355.455

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  由于赤道半径为6378137米,则赤道周长为2*PI*r = 20037508.3427892,因此X轴的取值范围:[-20037508.3427892,20037508.3427892]。当纬度φ接近两极,即90°时,Y值趋向于无穷。因此通常把Y轴的取值范围也限定在[-20037508.3427892,20037508.3427892]之间。因此在墨卡托投影坐标系(米)下的坐标范围是:最小为(-20037508.3427892, -20037508.3427892 )到最大 坐标为(20037508.3427892, 20037508.3427892)。

二、地理坐标系(Geographical coordinates

  地理经度的取值范围是[-180,180],纬度不可能到达90°,通过纬度取值范围为[20037508.3427892,20037508.3427892]反计算可得到纬度值为85.05112877980659。因此纬度取值范围是[-85.05112877980659,85.05112877980659]。因此,地理坐标系(经纬度)对应的范围是:最小地理坐标(-180,-85.05112877980659),最大地理坐标(180, 85.05112877980659)。

三、地面分辨率(Ground Resolution

  地面分辨率是以一个像素(pixel)代表的地面尺寸(米)。以微软Bing Maps为例,当Level为1时,图片大小为512*512(4个Tile),那么赤道空间分辨率为:赤道周长/512。其他纬度的空间分辨率则为 纬度圈长度/512,极端的北极则为0。Level为2时,赤道的空间分辨率为 赤道周长/1024,其他纬度为 纬度圈长度1024。很明显,Ground Resolution取决于两个参数,缩放级别Level和纬度latitude ,Level决定像素的多少,latitude决定地面距离的长短。

地面分辨率的公式为,单位:米/像素:

ground resolution = (cos(latitude * pi/180) * 2 * pi * 6378137 meters) / (256 * 2level pixels)

  最低地图放大级别(1级),地图是512 x 512像素。每下一个放大级别,地图的高度和宽度分别乘于2:2级是1024 x 1024像素,3级是2048 x 2048像素,4级是4096 x 4096像素,等等。通常而言,地图的宽度和高度可以由以下式子计算得到:map width = map height = 256 * 2^level pixels

四、地图比例尺(Map Scale

  地图比例尺是指测量相同目标时,地图上距离与实际距离的比例。通过地图分辨率在计算可知由Level可得到图片的像素大小,那么需要把其转换为以米为单位的距离,涉及到DPI(dot per inch),暂时可理解为类似的PPI(pixel per inch),即每英寸代表多少个像素。256 * 2level / DPI 即得到相应的英寸inch,再把英寸inch除以0.0254转换为米。实地距离仍旧是:cos(latitude * pi/180) * 2 * pi * 6378137 meters; 因此比例尺的公式为:

map scale = 256 * 2level / screen dpi / 0.0254 / (cos(latitude * pi/180) * 2 * pi * 6378137)

  比例尺= 1 : (cos(latitude * pi/180) * 2 * pi * 6378137 * screen dpi) / (256 * 2level * 0.0254)

  地面分辨率和地图比例尺之间的关系:

map scale = 1 : ground resolution * screen dpi / 0.0254 meters/inch

缩放级别

地图宽度、高度(像素)

地面分辨率(米/像素)

地图比例尺(以96dpi为例)

1

512

78,271.5170

1 : 295,829,355.45

2

1,024

39,135.7585

1 : 147,914,677.73

3

2,048

19,567.8792

1 : 73,957,338.86

4

4,096

9,783.9396

1 : 36,978,669.43

5

8,192

4,891.9698

1 : 18,489,334.72

6

16,384

2,445.9849

1 : 9,244,667.36

7

32,768

1,222.9925

1 : 4,622,333.68

8

65,536

611.4962

1 : 2,311,166.84

9

131,072

305.7481

1 : 1,155,583.42

10

262,144

152.8741

1 : 577,791.71

11

524,288

76.4370

1 : 288,895.85

12

1,048,576

38.2185

1 : 144,447.93

13

2,097,152

19.1093

1 : 72,223.96

14

4,194,304

9.5546

1 : 36,111.98

15

8,388,608

4.7773

1 : 18,055.99

16

16,777,216

2.3887

1 : 9,028.00

17

33,554,432

1.1943

1 : 4,514.00

18

67,108,864

0.5972

1 : 2,257.00

19

134,217,728

0.2986

1 : 1,128.50

20

268,435,456

0.1493

1 : 564.25

21

536,870,912

0.0746

1 : 282.12

22

1,073,741,824

0.0373

1 : 141.06

23

2,147,483,648

0.0187

1 : 70.53

五、Bing Maps像素坐标系和地图图片编码

  为了优化地图系统性能,提高地图下载和显示速度,所有地图都被分割成256 x 256像素大小的正方形小块。由于在每个放大级别下的像素数量都不一样,因此地图图片(Tile)的数量也不一样。每个tile都有一个XY坐标值,从左上角的(0, 0)至右下角的(2^level–1, 2^level–1)。例如在3级放大级别下,所有tile的坐标值范围为(0, 0)至(7, 7),如下图:

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  已知一个像素的XY坐标值时,我们很容易得到这个像素所在的Tile的XY坐标值:

tileX = floor(pixelX / 256) tileY = floor(pixelY / 256)

  为了简化索引和存储地图图片,每个tile的二维XY值被转换成一维字串,即四叉树键值(quardtree key,简称quadkey)。每个quadkey独立对应某个放大级别下的一个tile,并且它可以被用作数据库中B-tree索引值。为了将坐标值转换成quadkey,需要将Y和X坐标二进制值交错组合,并转换成4进制值及对应的字符串。例如,假设在放大级别为3时,tile的XY坐标值为(3,5),quadkey计算如下:

tileX = 3 = 011(二进制)

tileY = 5 = 101(二进制)

quadkey = 100111(二进制) = 213(四进制) = “213”

Quadkey还有其他一些有意思的特性。第一,quadkey的长度等于该tile所对应的放大级别;第二,每个tile的quadkey的前几位和其父tile(上一放大级别所对应的tile)的quadkey相同,下图中,tile 2是tile 20至23的父tile,tile 13是tile 130至133的父级:

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  最后,quadkey提供的一维索引值通常显示了两个tile在XY坐标系中的相似性。换句话说,两个相邻的tile对应的quadkey非常接近。这对于优化数据库的性能非常重要,因为相邻的tile通常被同时请求显示,因此可以将这些tile存放在相同的磁盘区域中,以减少磁盘的读取次数。

  下面是微软Bing Maps的TileSystem相关算法:

 

using System;using System.Text;namespace Microsoft.MapPoint{    static class TileSystem    {        private const double EarthRadius = 6378137;        private const double MinLatitude = -85.05112878;        private const double MaxLatitude = 85.05112878;        private const double MinLongitude = -180;        private const double MaxLongitude = 180;        /// <summary>        /// Clips a number to the specified minimum and maximum values.        /// </summary>        /// <param name="n">The number to clip.</param>        /// <param name="minValue">Minimum allowable value.</param>        /// <param name="maxValue">Maximum allowable value.</param>        /// <returns>The clipped value.</returns>        private static double Clip(double n, double minValue, double maxValue)        {            return Math.Min(Math.Max(n, minValue), maxValue);        }        /// <summary>        ///Determines the map width and height (in pixels) at a specified level        /// of detail.        /// </summary>        /// <param name="levelOfDetail">Level of detail, from 1 (lowest detail)        /// to 23 (highest detail).</param>        /// <returns>The map width and height in pixels.</returns>        public static uint MapSize(intlevelOfDetail)        {            return (uint) 256 << levelOfDetail;        }        /// <summary>        ///Determines the ground resolution (in meters per pixel) at a specified        /// latitude and level of detail.        /// </summary>        /// <param name="latitude">Latitude (in degrees) at which to measure the        /// ground resolution.</param>        /// <param name="levelOfDetail">Level of detail, from 1 (lowest detail)        /// to 23 (highest detail).</param>        /// <returns>The ground resolution, in meters per pixel.</returns>        public static double GroundResolution(double latitude, int levelOfDetail)        {            latitude = Clip(latitude, MinLatitude, MaxLatitude);            return Math.Cos(latitude * Math.PI / 180) * 2 * Math.PI * EarthRadius / MapSize(levelOfDetail);        }        /// <summary>        ///Determines the map scale at a specified latitude, level of detail,        /// and screen resolution.        /// </summary>        /// <param name="latitude">Latitude (in degrees) at which to measure the        /// map scale.</param>        /// <param name="levelOfDetail">Level of detail, from 1 (lowest detail)        /// to 23 (highest detail).</param>        /// <param name="screenDpi">Resolution of the screen, in dots per inch.</param>        /// <returns>The map scale, expressed as the denominator N of the ratio 1 : N.</returns>        public static double MapScale(double latitude, int levelOfDetail, intscreenDpi)        {            return GroundResolution(latitude, levelOfDetail) * screenDpi / 0.0254;        }        /// <summary>        /// Converts a point from latitude/longitude WGS-84 coordinates (in degrees)        /// into pixel XY coordinates at a specified level of detail.        /// </summary>        /// <param name="latitude">Latitude of the point, in degrees.</param>        /// <param name="longitude">Longitude of the point, in degrees.</param>        /// <param name="levelOfDetail">Level of detail, from 1 (lowest detail)        /// to 23 (highest detail).</param>        /// <param name="pixelX">Output parameter receiving the X coordinate in pixels.</param>        /// <param name="pixelY">Output parameter receiving the Y coordinate in pixels.</param>        public static void LatLongToPixelXY(double latitude, double longitude, intlevelOfDetail, out int pixelX, out int pixelY)        {            latitude = Clip(latitude, MinLatitude, MaxLatitude);            longitude = Clip(longitude, MinLongitude, MaxLongitude);            double x = (longitude + 180) / 360;             double sinLatitude = Math.Sin(latitude * Math.PI / 180);            double y = 0.5 - Math.Log((1 + sinLatitude) / (1 - sinLatitude)) / (4 * Math.PI);            uint mapSize = MapSize(levelOfDetail);            pixelX = (int) Clip(x * mapSize + 0.5, 0, mapSize - 1);            pixelY = (int) Clip(y * mapSize + 0.5, 0, mapSize - 1);        }        /// <summary>        /// Converts a pixel from pixel XY coordinates at a specified level of detail        /// into latitude/longitude WGS-84 coordinates (in degrees).        /// </summary>        /// <param name="pixelX">X coordinate of the point, in pixels.</param>        /// <param name="pixelY">Y coordinates of the point, in pixels.</param>        /// <param name="levelOfDetail">Level of detail, from 1 (lowest detail)        /// to 23 (highest detail).</param>        /// <param name="latitude">Output parameter receiving the latitude in degrees.</param>        /// <param name="longitude">Output parameter receiving the longitude in degrees.</param>        public static void PixelXYToLatLong(int pixelX, int pixelY, intlevelOfDetail, out double latitude, out double longitude)        {            double mapSize = MapSize(levelOfDetail);            double x = (Clip(pixelX, 0, mapSize - 1) / mapSize) - 0.5;            double y = 0.5 - (Clip(pixelY, 0, mapSize - 1) / mapSize);            latitude = 90 - 360 * Math.Atan(Math.Exp(-y * 2 * Math.PI)) / Math.PI;            longitude = 360 * x;        }        /// <summary>        /// Converts pixel XY coordinates into tile XY coordinates of the tile containing        /// the specified pixel.        /// </summary>        /// <param name="pixelX">Pixel X coordinate.</param>        /// <param name="pixelY">Pixel Y coordinate.</param>        /// <param name="tileX">Output parameter receiving the tile X coordinate.</param>        /// <param name="tileY">Output parameter receiving the tile Y coordinate.</param>        public static void PixelXYToTileXY(int pixelX, int pixelY, out int tileX, out int tileY)        {            tileX = pixelX / 256;            tileY = pixelY / 256;        }        /// <summary>        /// Converts tile XY coordinates into pixel XY coordinates of the upper-left pixel        /// of the specified tile.        /// </summary>        /// <param name="tileX">Tile X coordinate.</param>        /// <param name="tileY">Tile Y coordinate.</param>        /// <param name="pixelX">Output parameter receiving the pixel X coordinate.</param>        /// <param name="pixelY">Output parameter receiving the pixel Y coordinate.</param>        public static void TileXYToPixelXY(int tileX, int tileY, out int pixelX, out int pixelY)        {            pixelX = tileX * 256;            pixelY = tileY * 256;        }        /// <summary>        /// Converts tile XY coordinates into a QuadKey at a specified level of detail.        /// </summary>        /// <param name="tileX">Tile X coordinate.</param>        /// <param name="tileY">Tile Y coordinate.</param>        /// <param name="levelOfDetail">Level of detail, from 1 (lowest detail)        /// to 23 (highest detail).</param>        /// <returns>A string containing the QuadKey.</returns>        public static string TileXYToQuadKey(int tileX, int tileY, intlevelOfDetail)        {            StringBuilder quadKey = newStringBuilder();            for (int i = levelOfDetail; i > 0; i--)            {                char digit = 0;                int mask = 1 << (i - 1);                if ((tileX & mask) != 0)                {                    digit++;                }                if ((tileY & mask) != 0)                {                    digit++;                    digit++;                }                quadKey.Append(digit);            }            return quadKey.ToString();        }        /// <summary>        /// Converts a QuadKey into tile XY coordinates.        /// </summary>        /// <param name="quadKey">QuadKey of the tile.</param>        /// <param name="tileX">Output parameter receiving the tile X coordinate.</param>        /// <param name="tileY">Output parameter receiving the tile Y coordinate.</param>        /// <param name="levelOfDetail">Output parameter receiving the level of detail.</param>        public static void QuadKeyToTileXY(string quadKey, out int tileX, out int tileY, out intlevelOfDetail)        {            tileX = tileY = 0;            levelOfDetail = quadKey.Length;            for (int i = levelOfDetail; i > 0; i--)            {                int mask = 1 << (i - 1);                switch (quadKey[levelOfDetail - i])                {                    case 0:                        break;                    case 1:                        tileX |= mask;                        break;                    case 2:                        tileY |= mask;                        break;                    case 3:                        tileX |= mask;                        tileY |= mask;                        break;                    default:                        throw new ArgumentException("Invalid QuadKey digit sequence.");                }            }        }    }}

 

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当我们在用arcgis server 构建切片时,我们会发现在缓存生成的conf.xml中有这样的片段:

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在上述片段中<LODInfo>代表了每一级切片的信息,<LevelID>代表切片的级数。

在这里,<Scale>代表比例尺。比例尺是表示图上距离比实地距离缩小的程度,也叫缩尺。公式为:比例尺=图上距离/实地距离。用数字的比例式或分数式表示比例尺的大小。例如地图上1厘米代表实地距离500千米,可写成:1∶50,000,000或写成:1/50,000,000。

  <Resolution>,代表分辨率。Resolution 的实际含义代表当前地图范围内,1像素代表多少地图单位(X地图单位/像素),地图单位取决于数据本身的空间参考。

当我们在进行Web API的开发时,经常会碰到根据Resolution来缩放地图的情况。但是实际需求中我们更需要根据Scale来缩放,因此就涉及到Scale和Resolution的转换。

Resolution和Scale的转换算法:

Resolution跟dpi有关,跟地图的单位有关。(dpi代表每英寸的像素数)

  Resolution和Scale的转换算法

举例:

案例一:如果地图的坐标单位是米, dpi为96

           1英寸= 2.54厘米;

           1英寸=96像素;

最终换算的单位是米;

如果当前地图比例尺为1: 125000000,则代表图上1米实地125000000米;

米和像素间的换算公式:

           1英寸=0.0254米=96像素

           1像素=0.0254/96 米

则根据1:125000000比例尺,图上1像素代表实地距离是125000000*0.0254/96 = 33072.9166666667米。我们这个换算结果和切片的结果略微有0.07米的误差。这个误差产生的原因是英寸换算厘米的参数决定的,server使用的换算参数1英寸约等于0.0254000508米。

案例二:如果地理坐标系是wgs84,地图的单位是度,dpi为96

           Server中度和米之间的换算参数:

             1度约等于 111194.872221777米

接下来就需要进行度和像素间的换算:

当比例尺为1:64000000米时,相当于1像素 = 64000000*0.0254000508/96 = 16933.3672米

再将米转换为度 16933.3672/111194.872221777 = 0.1522855043731385度

因此当地图单位为度时,近似计算在1:64000000 对应的Resolution为0.1522855043731385度

验证结果:

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double resolution = scale * 0.0254000508/96/111194.872221777;

墨卡托投影、地理坐标系、地面分辨率、地图比例尺