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。
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(pixelper 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:

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:

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:

1 using System;
2  using System.Text;
3
4 namespace Microsoft.MapPoint
5 {
6 static class TileSystem
7 {
8 private const double EarthRadius = 6378137;
9 private const double MinLatitude = -85.05112878;
10 private const double MaxLatitude = 85.05112878;
11 private const double MinLongitude = -180;
12 private const double MaxLongitude = 180;
13
14
15 /// <summary>
16 /// Clips a number to the specified minimum and maximum values.
17 /// </summary>
18 /// <param name="n">The number to clip.</param>
19 /// <param name="minValue">Minimum allowable value.</param>
20 /// <param name="maxValue">Maximum allowable value.</param>
21 /// <returns>The clipped value.</returns>
22 private static double Clip(double n, double minValue, double maxValue)
23 {
24 return Math.Min(Math.Max(n, minValue), maxValue);
25 }
26
27
28
29 /// <summary>
30 /// Determines the map width and height (in pixels) at a specified level
31 /// of detail.
32 /// </summary>
33 /// <param name="levelOfDetail">Level of detail, from 1 (lowest detail)
34 /// to 23 (highest detail).</param>
35 /// <returns>The map width and height in pixels.</returns>
36 public static uint MapSize(int levelOfDetail)
37 {
38 return (uint) 256 << levelOfDetail;
39 }
40
41
42
43 /// <summary>
44 /// Determines the ground resolution (in meters per pixel) at a specified
45 /// latitude and level of detail.
46 /// </summary>
47 /// <param name="latitude">Latitude (in degrees) at which to measure the
48 /// ground resolution.</param>
49 /// <param name="levelOfDetail">Level of detail, from 1 (lowest detail)
50 /// to 23 (highest detail).</param>
51 /// <returns>The ground resolution, in meters per pixel.</returns>
52 public static double GroundResolution(double latitude, int levelOfDetail)
53 {
54 latitude = Clip(latitude, MinLatitude, MaxLatitude);
55 return Math.Cos(latitude * Math.PI / 180) * 2 * Math.PI * EarthRadius / MapSize(levelOfDetail);
56 }
57
58
59
60 /// <summary>
61 /// Determines the map scale at a specified latitude, level of detail,
62 /// and screen resolution.
63 /// </summary>
64 /// <param name="latitude">Latitude (in degrees) at which to measure the
65 /// map scale.</param>
66 /// <param name="levelOfDetail">Level of detail, from 1 (lowest detail)
67 /// to 23 (highest detail).</param>
68 /// <param name="screenDpi">Resolution of the screen, in dots per inch.</param>
69 /// <returns>The map scale, expressed as the denominator N of the ratio 1 : N.</returns>
70 public static double MapScale(double latitude, int levelOfDetail, int screenDpi)
71 {
72 return GroundResolution(latitude, levelOfDetail) * screenDpi / 0.0254;
73 }
74
75
76
77 /// <summary>
78 /// Converts a point from latitude/longitude WGS-84 coordinates (in degrees)
79 /// into pixel XY coordinates at a specified level of detail.
80 /// </summary>
81 /// <param name="latitude">Latitude of the point, in degrees.</param>
82 /// <param name="longitude">Longitude of the point, in degrees.</param>
83 /// <param name="levelOfDetail">Level of detail, from 1 (lowest detail)
84 /// to 23 (highest detail).</param>
85 /// <param name="pixelX">Output parameter receiving the X coordinate in pixels.</param>
86 /// <param name="pixelY">Output parameter receiving the Y coordinate in pixels.</param>
87 public static void LatLongToPixelXY(double latitude, double longitude, int levelOfDetail, out int pixelX, out int pixelY)
88 {
89 latitude = Clip(latitude, MinLatitude, MaxLatitude);
90 longitude = Clip(longitude, MinLongitude, MaxLongitude);
91
92 double x = (longitude + 180) / 360;
93 double sinLatitude = Math.Sin(latitude * Math.PI / 180);
94 double y = 0.5 - Math.Log((1 + sinLatitude) / (1 - sinLatitude)) / (4 * Math.PI);
95
96 uint mapSize = MapSize(levelOfDetail);
97 pixelX = (int) Clip(x * mapSize + 0.5, 0, mapSize - 1);
98 pixelY = (int) Clip(y * mapSize + 0.5, 0, mapSize - 1);
99 }
100
101
102
103 /// <summary>
104 /// Converts a pixel from pixel XY coordinates at a specified level of detail
105 /// into latitude/longitude WGS-84 coordinates (in degrees).
106 /// </summary>
107 /// <param name="pixelX">X coordinate of the point, in pixels.</param>
108 /// <param name="pixelY">Y coordinates of the point, in pixels.</param>
109 /// <param name="levelOfDetail">Level of detail, from 1 (lowest detail)
110 /// to 23 (highest detail).</param>
111 /// <param name="latitude">Output parameter receiving the latitude in degrees.</param>
112 /// <param name="longitude">Output parameter receiving the longitude in degrees.</param>
113 public static void PixelXYToLatLong(int pixelX, int pixelY, int levelOfDetail, out double latitude, out double longitude)
114 {
115 double mapSize = MapSize(levelOfDetail);
116 double x = (Clip(pixelX, 0, mapSize - 1) / mapSize) - 0.5;
117 double y = 0.5 - (Clip(pixelY, 0, mapSize - 1) / mapSize);
118
119 latitude = 90 - 360 * Math.Atan(Math.Exp(-y * 2 * Math.PI)) / Math.PI;
120 longitude = 360 * x;
121 }
122
123
124
125 /// <summary>
126 /// Converts pixel XY coordinates into tile XY coordinates of the tile containing
127 /// the specified pixel.
128 /// </summary>
129 /// <param name="pixelX">Pixel X coordinate.</param>
130 /// <param name="pixelY">Pixel Y coordinate.</param>
131 /// <param name="tileX">Output parameter receiving the tile X coordinate.</param>
132 /// <param name="tileY">Output parameter receiving the tile Y coordinate.</param>
133 public static void PixelXYToTileXY(int pixelX, int pixelY, out int tileX, out int tileY)
134 {
135 tileX = pixelX / 256;
136 tileY = pixelY / 256;
137 }
138
139
140
141 /// <summary>
142 /// Converts tile XY coordinates into pixel XY coordinates of the upper-left pixel
143 /// of the specified tile.
144 /// </summary>
145 /// <param name="tileX">Tile X coordinate.</param>
146 /// <param name="tileY">Tile Y coordinate.</param>
147 /// <param name="pixelX">Output parameter receiving the pixel X coordinate.</param>
148 /// <param name="pixelY">Output parameter receiving the pixel Y coordinate.</param>
149 public static void TileXYToPixelXY(int tileX, int tileY, out int pixelX, out int pixelY)
150 {
151 pixelX = tileX * 256;
152 pixelY = tileY * 256;
153 }
154
155
156
157 /// <summary>
158 /// Converts tile XY coordinates into a QuadKey at a specified level of detail.
159 /// </summary>
160 /// <param name="tileX">Tile X coordinate.</param>
161 /// <param name="tileY">Tile Y coordinate.</param>
162 /// <param name="levelOfDetail">Level of detail, from 1 (lowest detail)
163 /// to 23 (highest detail).</param>
164 /// <returns>A string containing the QuadKey.</returns>
165 public static string TileXYToQuadKey(int tileX, int tileY, int levelOfDetail)
166 {
167 StringBuilder quadKey = new StringBuilder();
168 for (int i = levelOfDetail; i > 0; i--)
169 {
170 char digit = '0';
171 int mask = 1 << (i - 1);
172 if ((tileX & mask) != 0)
173 {
174 digit++;
175 }
176 if ((tileY & mask) != 0)
177 {
178 digit++;
179 digit++;
180 }
181 quadKey.Append(digit);
182 }
183 return quadKey.ToString();
184 }
185
186
187
188 /// <summary>
189 /// Converts a QuadKey into tile XY coordinates.
190 /// </summary>
191 /// <param name="quadKey">QuadKey of the tile.</param>
192 /// <param name="tileX">Output parameter receiving the tile X coordinate.</param>
193 /// <param name="tileY">Output parameter receiving the tile Y coordinate.</param>
194 /// <param name="levelOfDetail">Output parameter receiving the level of detail.</param>
195 public static void QuadKeyToTileXY(string quadKey, out int tileX, out int tileY, out int levelOfDetail)
196 {
197 tileX = tileY = 0;
198 levelOfDetail = quadKey.Length;
199 for (int i = levelOfDetail; i > 0; i--)
200 {
201 int mask = 1 << (i - 1);
202 switch (quadKey[levelOfDetail - i])
203 {
204 case '0':
205 break;
206
207 case '1':
208 tileX |= mask;
209 break;
210
211 case '2':
212 tileY |= mask;
213 break;
214
215 case '3':
216 tileX |= mask;
217 tileY |= mask;
218 break;
219
220 default:
221 throw new ArgumentException("Invalid QuadKey digit sequence.");
222 }
223 }
224 }
225 }
226 }

 

posted on 2010-11-10 14:35  liongis  阅读(2355)  评论(0编辑  收藏  举报