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Currently software and maps represent earth in many different ways, the latest is Point Clouds as this is the most common surveying tools currently available. For the needs of visualisation, construction etc. this seems to suffice for now. Not sure this is an engineering question but I tried in the philosophy forum and been advised this is applied science.

What is the most accurate representation of earth in a digital format (or mathematical equation):

  • 3D splines
  • solids and primitives
  • breaklines/contours
  • polygonal mesh, TIN
  • square 3D pixels

?

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  • $\begingroup$ All of these methods are arbitrarily precise, depending on how closely packed your sample data is. For instance, if your polygonal mesh has nodes every millimeter, your mesh will be basically identical to the real thing. If you only have a few nodes across the globe, the mesh will look like garbage. $\endgroup$
    – Wasabi
    Aug 19 '17 at 12:50
  • $\begingroup$ thanks wasabi that makes sense. So the question would then be which format is best to use with a millimetre precision. However usually point clouds are not uniformed in the spaces between points and TIN is created from it whereas solids and splines are continuous so cannot allow for such gaps? $\endgroup$
    – Rott
    Aug 19 '17 at 13:26
  • $\begingroup$ @Wasabi when (if) you do your comment as an answer, add a point about variable node density : more nodes for complicated surfaces... $\endgroup$
    – Solar Mike
    Aug 19 '17 at 13:52
  • $\begingroup$ It comes down to, what do you want to do with the final product & the size of the region you are interested in. It is also dependent on the amount of data storage capacity & the processing power of the computer being used. The other thing is how accurate do you want the representation to be. The one thing that most terrain modelling software can't do automatically is model overhangs. $\endgroup$
    – Fred
    Aug 19 '17 at 15:23
  • $\begingroup$ seems quite off topic to me, but do you seriously want a computational representation of the earth with millimeter precision? $\endgroup$
    – agentp
    Aug 22 '17 at 13:44
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You are confusing methods of organizing data with the accuracy of that data or the accuracy of the interpolation method. The two are orthogonal.

For example, you could describe a surface in Z as a function of X and Y as a rectangular grid of points. At each point, the Z for that X,Y is known explicitly. For X,Y between points, you do a 2D linear interpolation between the 4 nearest points.

Alternatively, you could describe the surface as a set of contour lines, like a topo map. For any point not on a contour, you interpolate between the two nearest countours.

Which one is more accurate? Either can be made arbitrarily accurate. If you put a grid point every mile and compare that to a map with 10 foot countours, the contour map will usually give you far better accuracy. On the other hand, consider a grid point every meter versus 100 foot contours. Most of the time the grid will be much more accurate.

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