I'm a cartographer as well as a GIS professional and I think I have a decent handle on projections and how they effect the various attributes of map display.
What I'm less clear on is the need to match a particular kind of projection with particular shape computations in the GIS.
When a researcher or student here gets help with a project they will often be told to match their reference system to their computation, so equal area for a project looking at area calculations or equidistant for a project involving distance measurements or buffers etc.
The implication is always that you will get incorrect results for area and distance if you used a mismatched projection (equal area for distances). I'm pretty sure this is not correct and that only the visual aspect of the map is at risk when mixing up the projection like this, but that calculations taken from two projections of the the same envelope but differing shape characteristics should be the same as long as the projection has real units like feet or meters.
Can any one here verify this for me? Or correct me if I'm wrong?
That would be true only if the algorithm is taking the coordinate system into account. Let's take a (almost) worst case scenario: WGS 1984 Web Mercator Auxiliary Sphere. It's not conformal nor equal area (and definitely not equidistant except along the equator), but the same issues will apply to other projected coordinate systems. The scale at a point is 1/cos(latitude), so a point at 40N has a scale distortion of 1.3 and an areal distortion of 1.7. Now let's talk about a point on the central meridian of a UTM zone. Point scale is 0.9996 and areal distortion of 0.9992. If the algorithm is not taking the coordinate system into account and is using 2D Cartesian calculations, you're going to get different answers.
We've been upgrading or modifying some algorithms to take the coordinate system into account like the Measure Tool and the Buffer Tool, but many algorithms used by different tools are still 2D Cartesian.
(I was hoping Melita would answer you! Thanks Melita) David: Also take a look at this help topic: Coordinate system considerations for geoprocessing services. Even though this topic is about geoprocessing services, it has information useful for geoprocessing in general. A few points to look for in this topic:
It links to a web application that shows how calculating length/area (using Cartesian math) is affected by the projection. It's a cool little app and shows just how awful Web Mercator is.
It discusses the geoprocessing Output Coordinate System environment setting
It specifically discusses the Near and Generate Near Table tools, two geoprocessing tools that compute distances using Cartesian math.