... apparently an anthropomorphic explorer is an ill-defined entity ...

Since maps are two dimensional representations of the (non-euclidean) elliptic shape of the earth's surface then perhaps caves that underlie a large surface area become ever so slightly distorted on maps as the influence the curvature of the earth has on the plan view of the cave is not taken into account. I can't imagine this would affect the survey very much though, or even at all.

Recall being taught in basic survey class that the curvature of the Earth approximates to an inch in 100 yards. This might become significant in long systems, especially with surface legs connecting entrances.

The region over which the earth can be regarded as flat depends on the accuracy of the survey measurements. If measured only to the nearest metre, then curvature of the earth is undetectable over a meridian distance of about 100 kilometres (62 mi) and over an east-west line of about 80 km (at a latitude of 45 degrees). If surveyed to the nearest 1 millimetre (0.039 in), then curvature is undetectable over a meridian distance of about 10 km and over an east-west line of about 8 km.[3] Thus a city plan of New York accurate to one metre or a building site plan accurate to one millimetre would both satisfy the above conditions for the neglect of curvature. They can be treated by plane surveying and mapped by scale drawings in which any two points at the same distance on the drawing are at the same distance on the ground.

Geodesic as a straight line ? Ignoring the widely held belief that we are spinning round on a spheroid of stuff which is spinning round the sun which is moving in an arm that spins round our local galactic hub which is travelling within our universe which may itself be moving within some other dimensional realm. A geodesic is defined as the shortest line between two points on a curved surface. There are some 2D representations of our earth (maps) which represent a geodesic as a straight line. In 3D space, a geodesic is by definition not straight.Caves are to a large extent defined by gravity past and present. Contrary to popular opinion, the direction of gravity is rarely to the centre of the earth and is not uniform because it depends on many things including local mass. Hence why there is a lot of money spent on ever more detailed satellite gravity mapping missions and why there has been so much controversy about the height of Everest. The Indian tectonic plate has an enormous variation in the direction of gravity because of it's impact with the Asian tectonic plate which has caused the Himalaya and the Tibetan Plateau all of which has increased the local mass of the crust in this area thereby swinging gravity to one side, this effects both Northern India and also Southern India where there appears to be a mass deficit. The direction of gravity defines the equipotential, one of these surfaces is used to represent mean sea level round the world hence the discussion about the height of Everest which brings me neatly to fractals because the more detailed you look at defining gravity, the more there is to see. The Nepalese Government has recently instigated a major study into gravity in the Everest region so that they can better define a geoid model (an interpretation of the equipotential) which can be used to convert a GPS derived height to a mean sea level height. It gets even more complicated when you start talking about the height of a mountain, should this be the rock or include the ice and Everest is growing in height anyway because of tectonic uplift. A world geoid model varies by as much as -107m and +85m compared to a mathematical spheroid. If you really want to make life interesting then you could include earth tides as well although these height variations are sub-metre.Enough about mountains, lets get back to cave surveying which usually relies upon lots of short measurements with respect to gravity at that local point. Most people would regard these as straight line measurements in 3D space. Cave surveys usually ignore the curvature of the earth and refraction as these effects are swamped out by the inaccuracy of the measuring devices used over short lengths. I have only had to consider these effects on two cave surveys because some measurements went above 500m range and the instruments had a greater accuracy than the effects plus I was stitching together multiple scans into a homogenous 3D space. The effects were only noticeable on a large scale virtual model, they could have been ignored for the resulting 2D maps on A2 sheets.Are caves fractal ? Where fluid flow (water, lava) is responsible for the formation then they are analogous to 3D and 2D river systems with the added interest that caves are an amalgam of old and active plus the more you look, the more there is to see, especially beautiful scalloped streamways. Formations could also be considered fractal so we have fractals in time (old and new caves) and fractals (formations, scallops) within fractals (caves). It would be interesting to create a multi-fractal algorithm to create virtual cave systems and what parameters (geological, atmospheric, etc.) to use to recreate real-world cave systems. The values that these produce could give a greater insight into cave development and past conditions plus provide an estimate of how much cave passage there is in the world and hence how much still to be discovered. Calculating the cross-section of a cave or volume is an iterative affair with values changing with instrumentation as more detail is added, eg. Guesstimate, Pacing, Tape, Disto, Scanner...........As to cave surveying being done by triangulation ? I am not aware of any cave that has been done by triangulation. This involves measuring one or more baselines and then transferring this scale over a much larger area by observing all the angles of a series of connected triangles and adjusting everything as a network. Cave surveying would be better described as traversing from station to station with sideshots to provide detail. A 3D surface can be created from these individual points by connecting them together in software to create a triangulated mesh which can be analysed for cross section and volume. A scanner collects millions of points which can be meshed to create a more detailed 3D model but as with fractals, there will always be smaller stuff that the scanner does not see.Earth curvature is 78.4mm height difference per 1km at the equator, 78.7mm at the poles. Refraction usually makes this effect less.I consider caves to be Euclidean and Fractal. Caves can be what you want them to be.

As to cave surveying being done by triangulation ? I am not aware of any cave that has been done by triangulation.

Earth curvature is 78.4mm height difference per 1km at the equator, 78.7mm at the poles. Refraction usually makes this effect less.

That's a bit under a metre over the c.15km extent of Mammoth Cave mentioned up-thread. In a high grade traditional surface survey to plot entrance locations (and hence to establish a framework on which to base the underground survey) that would be worth taking into account I think - but very few cave surveyors possess instruments to work to such accuracy that curvature becomes worth taking into consideration.

Apart from analy retentive geekdom or severe OCD, why would somebody need a survey that was that accurate? 1 metre over 15km is far more accurate than anybody could ever need in the caving world.Most people only want an idea of where they went and the more dedicated diggers only want a reasonable indication of what might join some passage to another in the same or different cave...A show cave needing to drill into an area for a radon fan or to extract water and such like would be very happy with an accuracy or 1m over 15km and they are possibly the most likely use for a super accurate survey.I expect even somebody tunneling over those distances (15km) would be more than happy with an error that small. (0.0066666666666667% error).