I just received an update from the EPSG mailing list:

New to Version 6.15 are (among other things): Added spherical Mercator coordinate operation method and associated CRS as seen in popular web mapping and visualisation applications.

It looks like they FINALLY added the spherical Mercator / Web Mercator projection used in Virtual Earth and Google Maps.

This is a big surprise. EPSG’s earlier statement whether to include it was this:

"We have reviewed the coordinate reference system used by Microsoft, Google, etc. and believe that it is technically flawed. We will not devalue the EPSG dataset by including such inappropriate geodesy and cartography.”

Guess they changed their mind, or did they just devalue their dataset? Then again, judging from the remarks EPSG put in there, their arrogance still shines through. There´s absolute nothing wrong with using a sphere instead of a flattened sphere. Sure it's not as accurate as for instance WGS84, but then again WGS84 is not accurate either - no ellipsoid is. But we know the exact differences between the two, and as always you will need to take these things into account so I don´t see the real issue. Viisually the distortion is far less than what you would notice, and when doing area, distance and bearing calculations you would first of all never use the mercator units without taking the projection distortion into account, and if you do your calculationg in long/lat it's more or less just as easy to use WGS84 as a base for your calculations (since no datum transform is really needed).

Anyway, finally we get an official code for "Web Mercator": EPSG:3785 

Here are the details of the entry:

Full WKT with authorities (untested!):
PROJCS["Popular Visualisation CRS / Mercator", GEOGCS["Popular Visualisation CRS", DATUM["Popular Visualisation Datum", SPHEROID["Popular Visualisation Sphere", 6378137, 0, AUTHORITY["EPSG",7059]], TOWGS84[0, 0, 0, 0, 0, 0, 0], AUTHORITY["EPSG",6055]], PRIMEM["Greenwich", 0, AUTHORITY["EPSG", "8901"]], UNIT["degree", 0.0174532925199433, AUTHORITY["EPSG", "9102"]], AXIS["E", EAST], AXIS["N", NORTH], AUTHORITY["EPSG",4055]], PROJECTION["Mercator"], PARAMETER["False_Easting", 0], PARAMETER["False_Northing", 0], PARAMETER["Central_Meridian", 0], PARAMETER["Latitude_of_origin", 0], UNIT["metre", 1, AUTHORITY["EPSG", "9001"]], AXIS["East", EAST], AXIS["North", NORTH], AUTHORITY["EPSG",3785]]  

Projected CRS
COORD_REF_SYS_CODE: 3785
COORD_REF_SYS_NAME: Popular Visualisation CRS / Mercator   
AREA_OF_USE_CODE: 3544   
COORD_REF_SYS_KIND: projected   
COORD_SYS_CODE: 4499       
DATUM_CODE:
SOURCE_GEOGCRS_CODE: 4055 (see below)
PROJECTION_CONV_CODE: 19847   
CMPD_HORIZCRS_CODE:
CMPD_VERTCRS_CODE:
CRS_SCOPE: Certain Web mapping and visualisation applications.   
REMARKS: Uses spherical development. Relative to an ellipsoidal development errors of up to 800 metres in position and 0.7% in scale may arise. Some applications call this WGS 84. It is not a recognised geodetic system: see WGS 84 / World Mercator (CRS code 3395)   
INFORMATION_SOURCE: Microsoft.
DATA_SOURCE: OGP
REVISION_DATE: 3/14/2008
CHANGE_ID:
SHOW_CRS: TRUE
DEPRECATED: FALSE

Geographic CRS:
COORD_REF_SYS_CODE: 4055
COORD_REF_SYS_NAME: Popular Visualisation CRS
AREA_OF_USE_CODE: 31262
COORD_REF_SYS_KIND: geographic 2D
COORD_SYS_CODE: 6422
DATUM_CODE: 6055 (see below)
SOURCE_GEOGCRS_CODE:
PROJECTION_CONV_CODE:
CMPD_HORIZCRS_CODE:
CMPD_VERTCRS_CODE:
CRS_SCOPE: Certain Web mapping and visualisation applications.
REMARKS: Some applications erroneously call this WGS 84. It uses a sphere with a radius having the same value as the semi-major axis of the WGS 84 ellipsoid. There is no geodetic recognition of this system.
INFORMATION_SOURCE: Microsoft.
DATA_SOURCE: OGP
REVISION_DATE: 3/13/2008
CHANGE_ID:
SHOW_CRS: TRUE
DATUM_CODE: FALSE

Datum:
Code: 6055       
Datum Name: Popular Visualisation
Datum Type: geodetic
Origin Description: Not specified in the classical sense of defining a geodetic datum.
Datum Epoch:
Ellipsoid Code: 7059 (see below)
Prime Meridian Code: 8901
Area Code: 1262
Datum Scope    : Used by certain popular Web mapping and visualisation applications.
Remarks: Not recognised by geodetic authorities.
Information Source: Microsoft.
Data Source: OGP   
Revision Date: 13-Mar-08
Change ID:
Deprecated: No

Ellipsoid:
Code: 7059
Ellipsoid Name: Popular Visualisation Sphere   
Semi-major axis (a): 6378137   
Axes units code: 9001       
Inverse flattening (1/f):
Semi-minor axis (b): 6378137   
Ellipsoid?: No   
Remarks: Sphere with radius equal to the semi-major axis of the GRS80 and WGS 84 ellipsoids. Used only for Web approximate mapping and visualisation. Not recognised by geodetic authorities.   
Information Source: Microsoft.
Data Source: OGP
Revision Date: 14-Mar-08
Change ID:   
Deprecated?: No

It looks like Ed Katibah (Spatial Program Manager of SQL Server) finally started blogging.

His first blogpost covers some interesting changes in the latest February CTP.

There are some new methods added to the list:

• Geometry::Filter
• Geography::Filter
• Geography::EnvelopeCenter
• Geography::EnvelopeAngle
• Geography::STDistance is no longer limited to having one of the inputs of type Point.
• Geography::Reduce

Unfortunately I can't try these new features today, because a bug prevents you from using SQL Server 2008 on a Leap Day and the doc on MSDN doesn't cover these new methods yet, so there's not telling how these methods really work.

Ed's and Isaac Kunens blogs are a must-read if you plan on doing some serious SQL Server Spatial work.

The Simple Features Specification v1.1.0 has a conformance test based on "Joe's Blue Lake vicinity map", which roughly looks like this:

To create the map in Microsoft SQL Server 2008, first create the tables:

-- Lakes
CREATE TABLE lakes (fid INTEGER NOT NULL PRIMARY KEY,name VARCHAR(64),shore Geometry);
-- Road Segments
CREATE TABLE road_segments (fid INTEGER NOT NULL PRIMARY KEY,name VARCHAR(64),aliases VARCHAR(64),num_lanes INTEGER,centerline Geometry);
-- Divided Routes
CREATE TABLE divided_routes (fid INTEGER NOT NULL PRIMARY KEY,name VARCHAR(64),num_lanes INTEGER,centerlines Geometry);
-- Forests
CREATE TABLE forests (fid INTEGER NOT NULL PRIMARY KEY,name VARCHAR(64),boundary Geometry);
-- Bridges
CREATE TABLE bridges (fid INTEGER NOT NULL PRIMARY KEY,name VARCHAR(64),position Geometry);
-- Streams
CREATE TABLE streams (fid INTEGER NOT NULL PRIMARY KEY,name VARCHAR(64),centerline Geometry);
-- Buildings
CREATE TABLE buildings (fid INTEGER NOT NULL PRIMARY KEY,address VARCHAR(64),position Geometry,footprint Geometry);
-- Ponds
CREATE TABLE ponds (fid INTEGER NOT NULL PRIMARY KEY,name VARCHAR(64),type VARCHAR(64),shores Geometry);
-- Named Places
CREATE TABLE named_places (fid INTEGER NOT NULL PRIMARY KEY,name VARCHAR(64),boundary Geometry);
-- Map Neatline
CREATE TABLE map_neatlines (fid INTEGER NOT NULL PRIMARY KEY,neatline Geometry);

Next step is to insert the rows:

-- Lakes
INSERT INTO lakes VALUES (101, 'BLUE LAKE',Geometry::STPolyFromText('POLYGON((52 18,66 23,73 9,48 6,52 18),(59 18,67 18,67 13,59 13,59 18))', 101));
-- Road segments
INSERT INTO road_segments VALUES(102, 'Route 5', NULL, 2,Geometry::STLineFromText('LINESTRING( 0 18, 10 21, 16 23, 28 26, 44 31 )' ,101));
INSERT INTO road_segments VALUES(103, 'Route 5', 'Main Street', 4,Geometry::STLineFromText('LINESTRING( 44 31, 56 34, 70 38 )' ,101));
INSERT INTO road_segments VALUES(104, 'Route 5', NULL, 2,Geometry::STLineFromText('LINESTRING( 70 38, 72 48 )' ,101));
INSERT INTO road_segments VALUES(105, 'Main Street', NULL, 4,Geometry::STLineFromText('LINESTRING( 70 38, 84 42 )' ,101));
INSERT INTO road_segments VALUES(106, 'Dirt Road by Green Forest', NULL, 1,Geometry::STLineFromText('LINESTRING( 28 26, 28 0 )',101));
-- DividedRoutes
INSERT INTO divided_routes VALUES(119, 'Route 75', 4,Geometry::STMLineFromText('MULTILINESTRING((10 48,10 21,10 0),(16 0,16 23,16 48))', 101));
-- Forests
INSERT INTO forests VALUES(109, 'Green Forest',Geometry::STMPolyFromText('MULTIPOLYGON(((28 26,28 0,84 0,84 42,28 26),(52 18,66 23,73 9,48 6,52 18)),((59 18,67 18,67 13,59 13,59 18)))', 101));
-- Bridges
INSERT INTO bridges VALUES(110, 'Cam Bridge', Geometry::STPointFromText('POINT( 44 31 )', 101));
-- Streams
INSERT INTO streams VALUES(111, 'Cam Stream',Geometry::STLineFromText('LINESTRING( 38 48, 44 41, 41 36, 44 31, 52 18 )', 101));
INSERT INTO streams VALUES(112, NULL,Geometry::STLineFromText('LINESTRING( 76 0, 78 4, 73 9 )', 101));
-- Buildings
INSERT INTO buildings VALUES(113, '123 Main Street',Geometry::STPointFromText('POINT( 52 30 )', 101),Geometry::STPolyFromText('POLYGON( ( 50 31, 54 31, 54 29, 50 29, 50 31) )', 101));
INSERT INTO buildings VALUES(114, '215 Main Street',Geometry::STPointFromText('POINT( 64 33 )', 101),Geometry::STPolyFromText('POLYGON( ( 66 34, 62 34, 62 32, 66 32, 66 34) )', 101));
-- Ponds
INSERT INTO ponds VALUES(120, NULL, 'Stock Pond',Geometry::STMPolyFromText('MULTIPOLYGON( ( ( 24 44, 22 42, 24 40, 24 44) ),( ( 26 44, 26 40, 28 42, 26 44) ) )', 101));
-- Named Places
INSERT INTO named_places VALUES(117, 'Ashton',Geometry::STPolyFromText('POLYGON( ( 62 48, 84 48, 84 30, 56 30, 56 34, 62 48) )', 101));
INSERT INTO named_places VALUES(118, 'Goose Island',Geometry::STPolyFromText('POLYGON( ( 67 13, 67 18, 59 18, 59 13, 67 13) )', 101));
-- Map Neatlines
INSERT INTO map_neatlines VALUES(115,Geometry::STPolyFromText('POLYGON( ( 0 0, 0 48, 84 48, 84 0, 0 0 ) )', 101)); 

And voilá... you can now for instance run all the conformance tests, but the map is actually also a good base-map for learning the various spatial operations.

If you use my SQL Spatial Query Tool, here's a query you can use to visualize the map:

SELECT neatline, 'Background' as name, 'map_neatlines' as layer, 'White' as FillColor, 'Transparent' as LineColor, 1 as LineThickness FROM map_neatlines UNION ALL
SELECT shores, name, 'ponds' as layer, 'LightBlue' as FillColor, 'Blue' as LineColor, 1 as LineThickness FROM ponds UNION ALL
SELECT boundary, name, 'forests' as layer, 'Green' as FillColor, '#ff005500' as LineColor, 4 as LineThickness FROM Forests UNION ALL
SELECT shore, name, 'lakes' as layer, 'Blue' as FillColor, 'Transparent' as LineColor, 2 as LineThickness FROM lakes UNION ALL
SELECT boundary, name, 'named_places' as layer, '#00ffff99' as FillColor, 'Brown' as LineColor, 2 as LineThickness FROM named_places UNION ALL
SELECT centerline, name, 'streams_outline' as layer, 'Blue' as FillColor, 'Blue' as LineColor, 5 as LineThickness FROM streams UNION ALL
SELECT centerline, name, 'streams_fill' as layer, 'LightBlue' as FillColor, 'LightBlue' as LineColor, 3 as LineThickness FROM streams UNION ALL
SELECT centerline, name, 'road_segments' as layer, 'LightGreen' as FillColor, 'Black' as LineColor, num_lanes*2 as LineThickness FROM road_segments UNION ALL
SELECT centerlines, name, 'divided_routes' as layer, 'LightGreen' as FillColor, '#aaff0000' as LineColor, num_lanes*2 as LineThickness FROM divided_routes UNION ALL
SELECT footprint, address, 'buildings_footprint' as layer, 'Black' as FillColor, 'Transparent' as LineColor, 1 as LineThickness FROM buildings UNION ALL
SELECT position.STBuffer(0.5), address, 'buildings_position' as layer, 'Red' as FillColor, 'Transparent' as LineColor, 1 as LineThickness FROM buildings UNION ALL
SELECT position.STBuffer(1), name, 'bridges' as layer, '#550000ff' as FillColor, 'Black' as LineColor, 1 as LineThickness FROM bridges

Which will generate a map like this: 

Download SQL queries (1.71 kb)

It looks like Microsoft listened to the community and plans on switching latitude and longitude ordering for WKT and WKB:
http://blogs.msdn.com/isaac/archive/2007/12/27/latitude-longitude-ordering.aspx

This change will only be for WKT and WKB. That does make some sense, since the ordering is related to the SRID which is not included in the WKT and WKB, thus making it impossible to decipher without any prior knowledge. It would still be nice if the two geometry models shared a more common interface for anything that is remotely similar, but this is a good start.

I’m actually starting to like working with the geography type (to begin with I hated the idea of having two different although similar geometry models). It’s fun to do buffers on points and see that they don’t look circular on a projected map, because they are, well… projected. Showing the same circles on a 3D globe reveals that they are in fact perfectly circular. It makes distance queries so much easier than for instance using this approach.

One thing that still bugs me about the geography datatype though… why is the linear unit type that is used for calculating area, length, buffers etc. declared as part of the SRID (which in this case is always geographic) ? This requires one to always first have to look up the linear unit in the spatial_ref_sys table for each returned row (since homogeneous SRIDs aren’t enforced on the table) and divide by the factor returned to ensure that we know the unit we get back. In my mind a linear unit should never be related to a geographic coordinate system. After all, that’s why we only define linear units in projected and geocentric coordinate systems where they makes sense.

It would be nice if the methods either took a unit type as input (although that would break with the common interface I just wished for) or simply just always outputs the results in SI units (then it would just be a matter of multiplying with a constant well-known conversion factor later if you want to display it in for instance miles, thus separating code from presentation). Luckily most of Katmai’s SR definitions use meters as default, but you never know if someone chose to change the conversion factors in the system table so you will always have to check first.

I've been playing with the spatial support of SQL Server 2008 today, but quickly hit two obstacles:

1. How do I get data into the database.
2. How do I visualize a query (apart from the Well-Known Text output).

Surprisingly there aren't really any good (free) tools out there yet to upload for instance shapefiles to Katmai. I pulled out some old code I had lying around and threw a small tool together that can upload a shapefile to the database either as geometry or geography. Katmai is pretty picky when it comes to the geography datatype, so make sure your shapefiles contain valid geometry and has vertices in the valid domain, or use the geometry format. You can download the tool below.

Next obstacle was visualizing the data. Playing with the spatial queries and not beeing able to see the results on a map quickly got boring. I haven't had a chance to play much with WPF, so I thought this was a good occation to play with it. Based on this MSDN article I managed to fairly quickly make a little tool that takes a SQL query as input and outputs the results in a WPF control. At the moment it only works with polylines and polygons - Points are simply ignored (tip: use STBuffer() on points to see them). It will look for the first column that contains a SqlGeometry or SqlGeography datatype and put that on the map. The remaining columns are shown in the box to the right when you hover over the shapes with the mouse.

The two tools are not perfect in any way (and there's no bug tracking support here :-)), but they provide a quick way to play with the spatial queries. Both tools requires Windows Authentification to access the sql server and runs on .NET 3.5. Furthermore the query tool uses the Microsoft.SqlServer.Types.dll that comes with installing Katmai (to parse the results), so for now you'll need to run it from a computer that has Katmai installed.

I just visited http://maps.live.com and it looks like they finally put the new release out there. A revamped design and Birds Eye View imagery inside the 3D viewer. You can now also create animations using your collections.

Another new feature is the new "Create model" button in the collection editor when you are in 3D mode. Using it will prompt you to download "3dvia" enabling you to model your own buildings and insert them into VE. It looks somewhat similar to SketchUp used for the same thing in Google Earth. A neat feature is that you get the map as a background to assist you in your modelling, and comes with a bunch of default textures as well:

Recently a friend of mine asked me what projection would be the best to create point-buffer circles. All map projections adds distortion so you rarely end up with a circle. If you’re lucky, you’ll have an ellipse but often they get even more complex than this. In many cases a “normal” circle is sufficient for accuracy, but still it got me thinking how I would do this 100% accurate. There are many approaches to this, fx. creating a new projection for each point, do the math in spherical coordinate systems etc, but none of them are completely trivial.

The UTM projection is one of the most commonly used projections. It’s fairly accurate to measure from point to point within small distances and close to the meridian, so this post is based on that. Most of the math here can also be applies to Mercator, where the scale reductions are applied to Y instead of X.

The Mercator projection is created by projecting the globe onto a cylinder whose center follows the Earth’s rotation axis. The Transverse Mercator is basically a "tilted" cylinder parallel to equator. Along the line that the cylinder "touches" the surface the distortion is 0, and increasing away from it in the east/west direction. There is no distortion in the North/South direction. The Universal Transverse Mercator is slightly smaller than the globe, so it will intersect the surface two places. This also ensures that the distortion between and around these two lines are minimal. Along the meridian (the center of the two lines) the distortion for the UTM projection is 0.9996, meaning that this is the amount you have to reduce an infinitely small line in the east/west direction (For Mercator this is normally 1 along Equator). As you move away from the meridian, the scale factor increases up toward infinity. Normally UTM projections are used close to the meridian, and every time you get too far east/west, you would switch to use a new UTM projection. That’s why The Earth is divided into 60 UTM ‘zones’, one for each 6 degrees. Often for practical reasons you might want to use a UTM zone even though it’s outside its defined usage area. Here you have to be specifically careful when measuring distances. For example, Denmark is covered by zone 32 and 33, but it’s common to only use zone 32 which means that distance calculations at the east end can be very inaccurate without applying a scale reduction.

Warning: The following contains a lot of math. You can skip to the bottom and download a C# class that does all the calculations for you.

The scale factor SR(x) for any given point at distance 'x' from the meridian is given by:

Since this is the scale reduction at a point, we need to sum up all of these values along a line to find the correct distance, thus the east/west distance from e₁ to e₀ (expressed in distance from the meridian) becomes an integral expression:

To this you need to add the north/south distance using the well-known Euclidean distance formula.

For atypical UTM using WGS84 ellipsoid and a 500000m false easting it becomes:

If you haven’t refreshed your integral math lately, or just want to write this in pure code, one can also write the expression as:

Where m is the scalefactor at the meridian, r is the earth radius and e₁ and e₀ have the false easting subtracted.

Often we can do with an approximate value by using the scale reduction between the start and end point. This should only be used on distances <1km in easting because it gets very inaccurate for greater distances.

One can also use Simpson’s rule for calculating the integral solution. My tests showed it’s a very accurate approximation for this type of integral usually giving millimeter accuracy. It’s defined as:

I performed a few performance tests for 10 million calculations using all four methods for finding a distance in a UTM projection. Below is the processing time for each method:

Bottom line of this: If you want performance and distance is small use the averaging method, if you want accuracy, go with the Integral method, or if you want both, use the Simpsons approach.

start point : { E=400,000 ; N=6,100,000 }

end point : { E=600,000 ; N=6,250,000 }

• Distance using Simpsons rule: 249,942.5569m (error: 0.0003m)
• Distance with approximate scale reduction: 249936.0046m (error: 6.54m)
• Distance without scale reduction: 250,000m (error=57.46m)

Solving SR(x) =1 gives us the distance from the meridian where the scale reduction is exactly 1:

I have lately seen several blogposts confused about which datum and projection Microsoft’s Live Maps (Virtual Earth) and Google Maps use. As most people already know by now, they render the round earth onto a flat screen using a Mercator projection.

I think the confusion comes from that they actually use two spatial reference systems at the same time:

1. Geographic  Longitude/Latitude coordinatesystem based on the standard WGS84 datum.
2. Mercator projection using a datum based on WGS84, BUT modified to be spheric.

So when is what used?

The Javascript API’s use (1) as input when you want to add points, lines and polygons. That is, they expect you to input any geometry in geographical coordinates, and click events etc. will also return geometry in this spatial reference. This is the coordinate system most javascript developers will use. The API will automatically project it to the spheric mercator projection.

If you want to create image overlays, or roll your own tile server on top of the map, you will need to project your images into a spheric mercator projection. The JavaScript APIs are not able to do this for you.

Here’s a bit of facts about the two projections:

The valid range of (1) is: [-180,-85.05112877980659] to [180, 85.05112877980659].

The valid range of (2) is: [-20037508.3427892, -20037508.3427892] to [20037508.3427892, 20037508.3427892]

Well-known Text for (1):
GEOGCS["GCS_WGS_1984",DATUM["D_WGS_1984",SPHEROID["WGS_1984",6378137,298.257223563]],PRIMEM["Greenwich",0],UNIT["Degree",0.0174532925199433]]

Well-known Text for (2):
PROJCS["Mercator Spheric", GEOGCS["WGS84basedSpheric_GCS", DATUM["WGS84basedSpheric_Datum", SPHEROID["WGS84based_Sphere", 6378137, 0], TOWGS84[0, 0, 0, 0, 0, 0, 0]], PRIMEM["Greenwich", 0, AUTHORITY["EPSG", "8901"]], UNIT["degree", 0.0174532925199433, AUTHORITY["EPSG", "9102"]], AXIS["E", EAST], AXIS["N", NORTH]], PROJECTION["Mercator"], PARAMETER["False_Easting", 0], PARAMETER["False_Northing", 0], PARAMETER["Central_Meridian", 0], PARAMETER["Latitude_of_origin", 0], UNIT["metre", 1, AUTHORITY["EPSG", "9001"]], AXIS["East", EAST], AXIS["North", NORTH]]

Proj.4 definition for (1):
+proj=longlat +ellps=WGS84 +datum=WGS84 +no_defs

Proj.4 definition for (2) (see here for an explanation of the weird ’nadgrids’ parameter):
+proj=merc +a=6378137 +b=6378137 +lat_ts=0.0 +lon_0=0.0 +x_0=0.0 +y_0=0 +k=1.0 +units=m +nadgrids=@null +no_defs

So why this weird extent for the latitude? First of all, the poles in the Mercator projection extends towards infinity, so at some point they have to cut them off (and who cares about ice anyway - apart from that its melting). If you look at (2) these latitude/longitude values projected into the spheric mercator results in a perfect square, fitting perfectly with squared image tiles, that are simple to sub-divide over and over again, as you zoom in. I expect the reason for the spheric datum is for simplicity and perfomance when reprojecting points from longitude/latitude to screen coordinates. Charlie Savage also has a more mathematical approach to deriving these values.

For a quick introduction to projections, coordinate systems and datums see here.

If you want to know more about how these mapping api's work, keep an eye on Jayant's blog.

Update: We now have an offical EPSG code for the projection. See details here.

I've had a lot of requests for the .NET 2.0 based raytracer that I used in my Master Thesis to generate true orthophotos based on complex 3D surface models (not necessarily 2½D). To save me from having to dig it up all the time, I've now posted it here on my blog so you can download it from here.

Basically it will read in a text file with triangles (each triangle defined per line as comma-separated double values X1,Y1,Z1,X2,Y2,Z2,X3,Y3,Z3), create a spatial index using axis-aligned boundingboxes, and perform ray/triangle intersections, returning any triangle intersecting the ray, including distance to triangle and angle of attack.

To save memory on larger DSM, all internal computation is done using floating-point insted of using double precision. This poses a problem for large values like UTM based coordinate systems, where rounding errors will occur when using floating point. The raytracer is internally applying an offset to the center of the data extents to accommodate for this, but it's worth remembering if your DSM covers a vast area. In most cases it's not a big issue though.

The raytracer scales very well (O(log n)), and can handle hundreds of thousands of faces and still raytrace in a few milliseconds (see chapter 7.3.3 of the thesis for details). Performing the required indexing first might be a little slow, so if you want to optimize this, be my guest ;-).

Some uses this engine has been involved in is generating rough worldfiles for aerial images based on absolute camera orientation and a surface model or Simulating Navstar and Galileo GPS availability in dense urban areas. The latter was actually some very cool stuff developed by some students at DTU. You could click anywhere on a map, and it would instantly simulate the visibility of Navstar and Galileo GPS satellites and give you the PDOP values throughout the day, based on a detailed 3D city model. This was used to determine where the zones of GPS based roadpricing should be (you didn't want the zone edges to be close to a place where you had poor accuracy causing people to be charged incorrectly). The raytracer was used for doing the line-of-sight analysis between a virtual car and all the satellites.

Download: Raytracer.zip (86.29 KB)

People are often mixing the above as if they were one and the same, so here’s a recap of them. One of the things you often find people saying is that “my data is in the WGS84 coordinate system”. This doesn’t really make sense, but I will get back to this later.

This is a very confusing subject, and I might have gotten a few things wrong myself, so please add a comment and I’ll update it ASAP.

Coordinate systems

A coordinate system is simply put a way of describing a spatial property relative to a center. There is more than one way of doing this:

• The Geocentric coordinate system is based on a normal (X,Y,Z) coordinate system with the origin at the center of Earth. This is the system that GPS uses internally for doing it calculations, but since this is very unpractical to work with as a human being ( due to the lack of well-known concepts of east, north, up, down) it is rarely displayed to the user but converted to another coordinate system.
  
• The Spherical or Geographic coordinate system is probably the most well-known. It is based on angles relative to a prime meridian and Equator usually as Longitude and Latitude. Heights are usually given relative to either the mean sea level or the datum (I’ll get back to the datum later).
  
• The Cartesian coordinate system is defined as a “flat” coordinate system placed on the surface of Earth.  In some projections it’s not flat in the sense that it follows the earth’s curvature in one direction and has a known scale-error in the other direction relative to the distance of the origin. The most well-known coordinate system is the Universal Transverse Mercator (UTM), but surveyors define their own little local flat coordinate systems all the time. It is very easy to work with, fairly accurate over small distances making measurements such as length, angle and area very straightforward. Cartesian coordinate systems are strongly connected to projections that I will cover later.

Sidenote: The geocentric coordinate system is strictly speaking a cartesian coordinate system too, but this is the general terms I've seen used the most when talking about world coordinate systems.

Datums and ellipsoids

Some of the common properties of the above coordinate systems are that they are all relative to the center of Earth and except the Geocentric coordinate system, uses a height system relative to the surface of the earth.

This poses two immediate problems:

• Where is the center of the earth
• What is the shape of the earth?

By now most people should know that that the earth isn’t flat (although there are still some who doubts it). If we define the surface of Earth as being at the mean sea level (often referred to as the Geoid), we don’t get a spheroid or even an ellipsoid. Because of gravitational changes often caused by large masses such as mountain ranges etc, Earth is actually very irregular with variations of +/- 100 meters. Since this is not very practical to work with as a model of earth, we usually use an ellipsoid for approximation. The ellipsoid is defined by its semi-major axis, and either the flattening of the ellipoid or the semi-minor axis.

The center and orientation of the ellipsoid is what we call the datum. So the datum defines an ellipsoid and through the use of a set of points on the ground that we relate to points on the ellipsoid, we define the center of the Earth.  This poses another problem, because continental drift moves the points used to define the points around all the time. This is why the name of a datum usually have a year in it, often referring to the position of those points January 1st of that year (although that may vary).

There are a vast amount of datums, some used for measurements all over the world, and other local datums defined so they fit very well with a local area. Some common ones are: World Geodetic Datum 1984 (WGS84), European Datum 1950 (ED50) and North American Datum 1983 (NAD83).

The most well-known is WGS84 used by the GPS systems today. It is a good approximation of the entire world and with fix-points defined almost all over the world. When it was defined they forgot to include points in Europe though, so the Europeans now have their own ETRS89, which is usually referred to as the “realization of WGS84 in Europe”. The problem here was solely because of continental drift, so they defined some points relative to WGS84 in 1989, and keeps track of the changes. In most use-cases it is of no real importance and you can use one or the other.

I mentioned earlier that people often refer to having their data in WGS84, and you see now why this doesn’t make sense. All you know from that is that the data is defined using the WGS84 datum, but you don’t know which coordinate system it uses.

Read more on Datums and Spheroids.

Projections

The earth isn’t flat, and there is no simple way of putting it down on a flat paper map (or these days onto a computer screen), so people have come up with all sorts of ingenious solutions each with their pros and cons. Some preserves area, so all objects have a relative size to each other, others preserve angles (conformal) like the Mercator projection, some try to find a good intermediate mix with only little distortion on several parameters etc. Common to them all is that they transform the world onto a flat Cartesian coordinate system, and which one to choose depends on what you are trying to show.

A common statement that I hear in GIS is the following “My map doesn’t have a projection”, but this is simply not possible (unless you have a good old rotating globe). Often people are referring to data that is in longitude/latitude and displayed on a map without having specified any projection. What happens is that the system applies the simplest projection it can: Mapping Longitude directly to X and Latitude to Y. This results in an equirectangular projection, also called the “Plate Carree” projection. It results in very heavy distortion making areas look squashed close to the poles. You can almost say that the “opposite” of the Plate Carree is the Mercator projection which stretches areas close to the poles in the opposite direction, making them look very big. Mercator is the type of projection you see used on Live maps and Google maps, but as many often mistakenly thinks, they do NOT use WGS84 for the projected map, although WGS84 is used when you directly input longitude/latitude values using their API (read more on this here).

More on projected coordinate systems

Spatial reference

The spatial reference is a combination of all the above. It defines an ellipsoid, a datum using that ellipsoid, and either a geocentric, geographic or projection coordinate system. The projection also always has a geographic coordinate system associated with it. The European Petroleum Survey Group (EPSG) has a huge set of predefined spatial references, each given a unique ID. These ID’s are used throughout the industry and you can download an Access database with all them from their website, as well as some very good documents on projection (or see the Spatial References website).

So when you hear someone saying they have their data in WGS84, you can often assume that they have longitude/latitude data in WGS84 projected using Plate Carree.  The spatial reference ID of this is EPSG:4326.

Spatial References are often defined in a Well-known format defining all these parameters. The Spatial Reference EPSG:4326 can therefore also be written as:

GEOGCS["GCS_WGS_1984",DATUM["D_WGS_1984",SPHEROID["WGS_1984",6378137,298.257223563]],PRIMEM["Greenwich",0],UNIT["Degree",0.0174532925199433]]

As mentioned Live/Google maps use a Mercator projection, but although their datum is based on WGS84, they use a sphere instead of an ellipsoid. This means that they use the same center and orientation as WGS84, but without applying any flattening. The spatial reference string for their projection therefore becomes:

PROJCS["Mercator Spheric", GEOGCS["WGS84based_GCS", DATUM["WGS84based_Datum", SPHEROID["WGS84based_Sphere", 6378137, 0], TOWGS84[0, 0, 0, 0, 0, 0, 0]], PRIMEM["Greenwich", 0, AUTHORITY["EPSG", "8901"]], UNIT["degree", 0.0174532925199433, AUTHORITY["EPSG", "9102"]], AXIS["E", EAST], AXIS["N", NORTH]], PROJECTION["Mercator"], PARAMETER["False_Easting", 0], PARAMETER["False_Northing", 0], PARAMETER["Central_Meridian", 0], PARAMETER["Latitude_of_origin", 0], UNIT["metre", 1, AUTHORITY["EPSG", "9001"]], AXIS["East", EAST], AXIS["North", NORTH]]