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  • 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)转载

    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:

    复制代码
    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(int levelOfDetail)
    {
    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, int screenDpi) { 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, int levelOfDetail, 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, int levelOfDetail, 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, int levelOfDetail) { StringBuilder quadKey = new StringBuilder(); 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 int levelOfDetail) { 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."); } } } }}
    复制代码
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  • 原文地址:https://www.cnblogs.com/chao3/p/4524124.html
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