Author Topic: Understanding the dimensional space of caves  (Read 5453 times)

Offline bjneil

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Understanding the dimensional space of caves
« on: March 02, 2012, 02:42:35 pm »
First of all, sincere apologies to all if this question seems stupid or obvious- I'm not very good with physics!

How would you accurately describe the physical space within a cave (system)? as euclidean? or non-euclidean? Having seen crochet examples of euclidean plates (like this one http://www.theiff.org/images/lecture/crochet_09.jpg, my imagination stretches to analogise this with the descents and ascents of cave conduits, etc.

Thanks for any answers,
Ben.
 

Offline Subpopulus Hibernia

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Re: Understanding the dimensional space of caves
« Reply #1 on: March 02, 2012, 03:14:39 pm »
Well, since we measure it, I'd consider it as Euclidean coordinate space, as all survey points can be plotted using the three coordinates X, Y, Z. That, at least, is how we describe it for the benefit of surveying. Likewise the actual passage dimensions could be expressed as a point cloud with each point with it's co-ordinates. Is this how the architecture of surveying programs is built?

I don't really understand enough of non-euclidean geometry to say whether or not one could express a cave in such terms. I imagine however it could be relevant in terms of correcting large cave systems on maps. 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.
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Offline Alkapton

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Re: Understanding the dimensional space of caves
« Reply #2 on: March 02, 2012, 03:33:12 pm »
Perhaps my maths is a bit nieve, but I think of it this way, in Euclidian geometry parrallel lines are defined as lines which when extended to infinity never meet.   In a non-Eucldian geometry parrallel lines can be defined as lines which when extended meet at infinity.   All you are doing is playing with the axioms (basic rules) that define geometry.   Earthly geometry is perfectly described by Euclidian rules so is perfectly good enougth for describling the space within a cave no matter how complex it is.   Non-Euclidian geometry is (perhps I over simplify) beter for describing space on a cosmic scale....   You don't get black holes in caves but you do get them in the cosmos, Euclidian geometry can't really deal with black holes cause they bend space and time, but a different set of geomety rules can describe such spaces.
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Online Benfool

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Re: Understanding the dimensional space of caves
« Reply #3 on: March 02, 2012, 04:02:19 pm »
Gonna be way to technical, but I dont care.......

Not quite right Alkapton, the idea of Parallel doesn't really make sense in non euclidian space (or needs to be definited slightly more carefully). We have the notion of the parallel postulate

"given a line and a point not on that line, there exists one and only one straight line which passes through that point and never intersects the first line"

If this is true then we have Euclian space, if its not true then we have non-euclidian. 

Since we live on a sphere, the surface we live on is not eucidian, but it is locally euclidian, so standard euclidian geometry is a good estimate, a so called Manifold. So for caves, the space inside them can be estimated using standard euclidian distance, as they are generally small scale, but globally this is not true.

For example if you dug a shaft at the north pole say 100m deap, then dug south, always keeping 100m below the surface (assuming the surface is smooth) to the south pole, then turned exactly 90 degrees and dug north again back to the same shaft at the north pole, then this space could not be descrbed by euclidian space, as you have two "straight" lines (well geodesics) starting and ending at the same point, which are not the same. The survey for this cave is gonna look a bit odd!

When we survey caves, we do assume they are eucidian, that is that they can be described by a coordinate system {x,y,z} where the distance between two points is given by Pythagorus's theorem, this gives a very good estimation, so good you wont notice the difference, or its well lost in the accuracy of the survey, so it doesn't matter.

In conclusion, the space inside caves is not really euclidian, but its a good enough estimation.

Hope this makes sence!

Offline bjneil

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Re: Understanding the dimensional space of caves
« Reply #4 on: March 02, 2012, 04:31:50 pm »
Thanks for the replies so far! Very interesting stuff....

If I could throw in another dimension to this (forgive the pun ;-)) what about fractal geometry? I've been reading that it can be used to describe complex shapes (particularly in 2 dimensions such as plans) but it can also be used to describe solids? I'm even more out of my depth with this one, but relish the challenge of understanding its applicability... it would seem that someone has tried, judging by this paper
http://deepblue.lib.umich.edu/bitstream/2027.42/43195/1/11004_2004_Article_BF00899743.pdf

Any thoughts on this? Is this useful as a tool for understanding cave dimensions and space? or am I totally off track?

Offline graham

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Re: Understanding the dimensional space of caves
« Reply #5 on: March 02, 2012, 04:41:27 pm »
Benfool, really good on the Euclidean/non-euclidean stuff, essentially caves can be described in Euclidean co-ordinates until they get too big.

As for fractals, well, like all geological surfaces, cave walls will have a fractal length which is dependant on the length of the ruler used to measure it (see Mandelbrot's estimate of coast length), but as hinted at above for cave surveys, what we actually measure is an approximation based partially on our needs and partly on our instruments.

Thus, as survey instrumentation improves, the point cloud density recorded will increase and the finished result will be a better approximation of the original. So surveys done using a DistoX will be 'better' than those done using tape/compass/clino and lots of 'estimation'.
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Offline graham

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Re: Understanding the dimensional space of caves
« Reply #6 on: March 02, 2012, 04:49:52 pm »
Nice line from Rane Curl's paper, referenced above:

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

We do know, however, what sort of grundies he/she wears.

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Online Benfool

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Re: Understanding the dimensional space of caves
« Reply #7 on: March 02, 2012, 05:00:25 pm »
Knew my PhD would come in useful somewhere!

Will answer the 2nd question on monday when I'm not in a hurry and back from caving!

B

Offline Alkapton

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Re: Understanding the dimensional space of caves
« Reply #8 on: March 03, 2012, 03:54:56 pm »
I am confused about non Euclidian space and need help....

Imagine the earth as perfect sphere and stand at North Pole. Walk direct south till you reach the equaor turn 90 degrees and walk the equator for 1/4 circumference of equator, turn 90 degrees walk North till you reach North Pole.   Your travelling describes a non Euclidian equalateral triangle with internal angles 90 + 90 + 90 = 270 degrees.   That I understand.  But what if you walk some other distance at the equator?   You must be describing a non equalateral triangle with different internal angle (at North Pole).   OK, well OK unless you walk exactly half the circumference at the equator, what does that describe?   Is it a triangle with three sides 90 + 90 + 180 = 360 degrees?  Is it a triangle with two sides 90 + 90 = 180 degrees?  Or have you walked two parrallel lines in Non-Eyuclidian space?    If it is not a triangle why is it not a triangle, It is made of two streight lines how can it be a triangle?   This seems parradoxical to me
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Offline JasonC

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Re: Understanding the dimensional space of caves
« Reply #9 on: March 03, 2012, 05:21:36 pm »
If you walk from the N pole to the equator, then say 10 deg of longitude (~~ 1000 km) along the equator, then walk (or swim ?) back to the N pole, your triangle (on the earth's surface) will have 2 x 90 deg angles and a 10 deg angle at the pole = 190 deg in all.
In spherical geometry, internal angles of a triangle always add to > 180 deg, in hyperbolic geometry, always < 180 deg.

None of this is relevant (at least for practical purposes) to measuring caves.

I think the OP really meant fractal geometry (as in his 2nd post) where the length or volume of a cave depends on how you measure it, as is explained - at length - in the paper he referenced.

Offline peterk

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Re: Understanding the dimensional space of caves
« Reply #10 on: March 03, 2012, 05:35:11 pm »
" Is it a triangle with two sides 90 + 90 = 180 degrees? " I think you've cracked it! One of Euclid's axioms(?) (things that can't be proved but have to be assumed). is that if two lines both cross a third line at right angles to it then they will never meet I think he also wrote that a right angle must equal any other right angle but that also can't be proved.

I'm not sure what you can't get your head round. If i said that in the Northern hemisphere every telegraph pole is absolutely  vertical but non of them point in the same direction would that be OK.

In respect of cave surveys they are all prepared using triangulation and to increase the level of detail additional triangulated measurements can be taken till you reduce the cave walls to a mesh of triangles - that's how I understand a video game is constructed - a mesh of triangles and I presume rectangles that images are "stuck" to so giving the appearance of curved surfaces that are in fact an assembly of flat surfaces.  Another example would be to cover a football with a single sheet of paper with no cutting, and that leads you to map projections. Until a cave system is discovered that requires its own projection I think Euclid will rule.

Offline robjones

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Re: Understanding the dimensional space of caves
« Reply #11 on: March 05, 2012, 09:06:59 pm »
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.  8)


Online shortscotsman

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Re: Understanding the dimensional space of caves
« Reply #12 on: March 05, 2012, 09:30:14 pm »
If you choose to describe caves in 3-D with x-y-z coordinates then the space is euclidean(**) .   Its only when you insist upon
describing things in 2-D that issues arise.  As a 2-D space the earths surface is non-euclidean. (*) When we map this onto a 2-D
euclidean space i.e. our sheet of paper there is always a distortion problem. This is pretty familiar to geogaphers who call it projections.

Whether this is a problem for cave mapping depends upon the projection and cave position: if a cave exist on the south pole
then the standard mercator projection would be pretty horrible.

(*) technically non-euclidian geometry isn't a very good description- its a like saying your car is "not a ford".  Most things we visualise
such as the earth's surface have a Reimannian Geometry.

(**) technically, if we beleive Einstein's theory of gravity,  all space becomes non-euclidian (Reimannian)in the presence of matter such as the earth and sun however this effect is teeeny teeny tiny for us.

Offline Alex

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Re: Understanding the dimensional space of caves
« Reply #13 on: March 05, 2012, 10:03:54 pm »
Just fill the cave with water and measure how much water you pumped in  :tease: (After sealing off all the water escape routes)
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Offline Subpopulus Hibernia

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Re: Understanding the dimensional space of caves
« Reply #14 on: March 06, 2012, 12:55:12 am »

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.  8)

From Wikipedia:
Quote
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.

With grade 5 Surveys being accurate to 10cm, I'd reckon we need not worry about the curvature of the earth when plotting surveys. Even the huge Mammoth Cave in the US, which fits roughly onto a 15km x 15km square wouldn't be noticably affected on a survey by this.
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Offline kdxn

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Re: Understanding the dimensional space of caves
« Reply #15 on: March 06, 2012, 11:24:45 am »
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.

Offline littletitan

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Re: Understanding the dimensional space of caves
« Reply #16 on: March 06, 2012, 11:42:01 am »
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.

Couldn't agree more :clap2:

Love the last line. That's my favourite Kevin! ;D

Offline graham

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Re: Understanding the dimensional space of caves
« Reply #17 on: March 06, 2012, 11:47:06 am »
As to cave surveying being done by triangulation ? I am not aware of any cave that has been done by triangulation.

The Gorge and Main Chamber in GB were surveyed by triangulation back in the late 1940s.

On a similar note, the original survey of Recliffe Caves in Bristol by Alfie Collins was carried out using a plane table.
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Offline pwhole

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Re: Understanding the dimensional space of caves
« Reply #18 on: March 06, 2012, 03:44:27 pm »
R. Buckminster Fuller, the inventor of the geodesic dome and many other great products, also created a new non-Euclidean geometry using 'spherical triangles' for his Dymaxion Map - the only projection ever produced to show absolutely correct land mass areas, sizes and distances - bizarrely, given its accuracy, it's almost never mentioned in education:

http://bfi.easystorecreator.com/items/maps/list.htm

Offline Alkapton

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Re: Understanding the dimensional space of caves
« Reply #19 on: March 06, 2012, 08:33:20 pm »
thanks for that. mr fuller is one of my heros, eg. bucky balls are named after him. so i will check out his method of projection.
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Offline robjones

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Re: Understanding the dimensional space of caves
« Reply #20 on: March 06, 2012, 09:05:28 pm »
Earth curvature is 78.4mm height difference per 1km at the equator, 78.7mm at the poles. Refraction usually makes this effect less.

Thank you for the correction. So its about two-tenths of an inch in a hundred yards. My dim and distant recollection of basic survey 1981 was out by a factor of five.  :-[

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.

Excellent cartoon about "What your favourite map projection says about you": http://xkcd.com/977/

Offline Les W

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Re: Understanding the dimensional space of caves
« Reply #21 on: March 06, 2012, 10:23:27 pm »
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).  :-\
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Offline Roger W

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Re: Understanding the dimensional space of caves
« Reply #22 on: March 06, 2012, 11:46:20 pm »
What I found interesting in Ben's original post was his example of 3-D modelling using crochet.

Do you think this medium could have any possibilities for 3-D cave mapping/modelling?
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Offline Amata

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Re: Understanding the dimensional space of caves
« Reply #23 on: March 07, 2012, 12:47:35 am »
I'd be willing to take a knack at it...I've done crocheting of the knots up through 7 crossings. Crocheting cave space seems fun!! I love the idea!

Uh for those not familiar with knot theory:
http://www.math.toronto.edu/~drorbn/KAtlas/Knots/
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Offline TheBitterEnd

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Re: Understanding the dimensional space of caves
« Reply #24 on: March 07, 2012, 07:56:56 am »
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).  :-\

And no one is going to get a survey that accurate over those distances anyway with conventional cave surveying techniques (that's likely to start a flame war  :coffee: ).

Re tunneling, I was once a site engineer on a tunnel over 2km long. The tunnel lining was about 2.9m external diameter and we had to hit a shaft eye that was 3m in diameter, we were out by about 15mm when we finished. The max allowable error would have been a bit more than twice what you quote -0.0025%.
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Offline robjones

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Re: Understanding the dimensional space of caves
« Reply #25 on: March 07, 2012, 09:47:42 pm »
Simplon met with a horizontal misalligment of 7.87" over 19,755.52m (apologies for mixed units but thats what Sandstrom gives in 'A History of Tunnelling', London, 1963) = about 0.0000012%. The vertical misalligment was 0.087m. Sandstrom states that "In America, the headings [of modest length tunnels] meet on a dime; if the tunnel is not too long a skilled instrument man will hit the dime's edge."