What's the difference between a projection and a datum?
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Geographic coordinate systems (lat/long) are based on a spheroidal (either truly spherical or ellipsoidal) surface that approximates the surface of the earth. A datum typically defines the surface (ex radius for a sphere, major axis and minor axis or inverse flattening for an ellipsoid) and the position of the surface relative to the center of the earth. An example of a datum is NAD 1927, described below
All coordinates are referenced to a datum (even if it is unknown). If you see data in a geographic coordinate system, such as GCS_North_American_1927, it is unprojected and is in Lat/Long, and in this case, referenced to the NAD 1927 datum.
A projection is a series of transformations which convert the location of points on a curved surface(the reference surface or datum) to locations on flat plane (ie transforms coordinates from one coordinate reference system to another).
The datum is a integral part of the projection, as projected coordinated systems are based on geographic coordinates, which are in turn referenced to a datum. It is possible, and even common for datasets to be in the same projection, but be referenced to different datums, and therefore have different coordinate values. For example, the State Plane coordinate systems can be referenced to either NAD83 and NAD27 datums. The transformations from geographic to projected coordinates are the same, but as the geographic coordinates are different depending on the datum, the resulting projected coordinates will also be different.
Also, projecting data may result in a datum conversion as well, for example, projecting NAD_1927 data to Web Mercator will require a datum shift to WGS 84. Similarly, it is possible to convert data from one datum to another without projecting it, as with the NGS's NADCON utility, which can shift coordinates from NAD27 to NAD83.
Example of a point's coordinates referenced to different datums
Coordinates referenced to NAD_1927_CGQ77
Same point referenced to NAD_1983_CSRS
You will obviously get better answers from textbooks, but here is an simple explanation:
Map Projection: It is a method for representing a spherical or curved surface on a flat plane.
Datum: It is the reference or origin based on which measurements are made.
After struggling with this question ten years ago, and finding many confusing things written about the topic, I published a brief article in Directions Magazine that presented an answer as simply, plainly, and accurately as I could make it. The following is excerpted from that article.
Reprojecting geographic features
Two things must happen when you draw a map: features in the real world must be "georeferenced" to a spheroid and the spheroid must be projected onto the paper.
The spheroid models the shape of the earth's surface. It is an idealization that does not account for local changes in topography.
Georeferencing assigns locations (in three dimensions!) to points on a spheroid.
Projecting is an operation that mathematically distorts and shrinks a portion of the spheroid onto flat paper. Projecting can be undone ("inverted"). "Unprojection" expands a feature on a map and plasters it back onto the spheroid.It, too, is a mathematical operation.
Georeferencing is done with a datum. A datum is usually given by a starting point and direction: it specifies where a clearly identifiable point on earth (the base point) should appear on the spheroid and it shows where a base direction, such as north, points on the spheroid at the base point. The base point and direction allow surveyors to determine the distance and angle of any other point on the earth. Moving in the corresponding direction on the spheroid for the same distance determines where the new point should go on the spheroid.
Spheroids have coordinates. They are latitude and longitude. (Geodetic) latitude is the angle made by a vertical line to the horizontal. It is not necessarily the same angle made by "straight up," because the latter is distorted by gravitational variation over the earth. It is not necessarily the angle made by a line to the center of the earth, because most spheroids have an elliptical cross-section, not a circular one.
Therefore, georeferencing endows points near the earth with latitude, longitude, and height coordinates.
(Subsequent sections discuss Change of datum, How to relate two maps, The wrong way to do it, and North America is a special case.)
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wwnick's answer is correct, but it is a bit misleading in the sense that it emphasizes ellipsoid parameters and IMO understates the importance of 'the position of the surface relative to the center of the earth' - the NAD 1927 example needs to mention that the geodetic "center" of NAD27 is a base station at Meades Ranch in Kansas.
One could have (and often that's the case, especially with the increasing popularity of WGS84/GRS80 ellipsoid) several different datums based on the exact same ellipsoid parameters. The reason for this is that while the WGS 84 datum is OK globally since its surface is set to provide minimal average shifts due to tectonic movements across the globe, there's room for improvement on the local scale, where the reference can be fixed to some local reference point or at least to the local tectonic plate (e.g. ETRS, which is fixed to continental Europe)
One could explain datum simply as "an agreement on the coordinate system type, shape and its absolute position and orientation relative to some well-known or well-defined real-world reference". The coordinate system doesn't even have to be ellipsoidal (e.g. Vertical datum, which is usually defined by saying that the height of some fixed point is such, and all other heights will be measured relative to this point).
Geographic projections are a way of showing the curved surface of the Earth on a flat surface like a piece of paper...
From the Manifold user documentation:
This won't compete with wwnicks answer and not rigorous, but the visualization I present to people, when asked, is the relationship between a string connected to a ball. Changing the projection is often like moving the 'loose' end of the string around, but still connected to the same point on the ball. Changing the datum is like changing the location of the ball. This might help those visual types.
Just a comment on the diagram that is trying to illustrate a projection from a sphere. Rather that what is illustrated, imagine a light source at the center of the sphere. The shadow of the polygon "projected" onto a flat piece of paper outside of the sphere is in essence a type of projection. To me the diagram is implying a projection is like a reflected surface which is an incorrect way to visualize what is happening.
Also, at least in the ESRI world, georeferencing is not applying points to a sphere. Georeferencing is assigning a known planar (projected) coordinate system to either a raster or vector dataset that has originated from either a scanning or digitizing operation in which a 'local' coordinate system was first applied. "Local" in this case simply means the coordinates were made up with no reference to a real world coordinate system. That is, a map may have originally been hand digitized where the person decided the lower left coordinate of the map had an XY value of (0,0). Georeferencing is the process of assigning a set of real world (projected) coordinates to the original. If this process is applied to a photograph or scanned map then the georeferencing process will often warp the original image to fit within the set of reference points that have been assigned real world planar coordinates. This "georeference warping" is not the same as the distortions created when projecting from a sphere onto a plane. "Georeference warping" is all about correcting distortions produced by either the camera or scanner. When projecting a feature from a spherical surface to a planar surface there is always a distortion created in distance, area, scale, and bearing. You choose a projection to minimize one or more of these distortions, depending on the intended purpose of the map.
As to the strings on a ball illustration and changing the datum, rather than strings I would use pencils of various lengths that start from a point on the sphere and end on a flat piece of paper. The outer ends of the pencils represent the projected points. In a sense, changing the geographic coordinate system (datum for this discussion) is analogous to rotating the sphere on one more axis to a new position. The concept works only for isolated areas on the earth. That is for NAD27 to WGS84 it applies pretty well to the 48 contiguous states of the USA but not for Canada or Alaska. For those areas you have to correct the NAD 27 datum first and then make the NAD7 to WGS84 move. Whereas for NAD83 to WGS84 the concept works for most of North America.
A Datum is unprojected data, A projection is data projected in a projection scenario. datum can be used for the projection, visa versa, not so much.