Two-and-a-Half Dimensions (2.5-D): Calculating Volumes

There is sometimes a tension, if not confusion, within a given description regarding which dimension is being addressed by a term in geometry. Is a square the four lines that meet at vertices of 90 degrees, or is a square the surface formed by those four lines? Does a cube consist of (1) 12 lines in 3-D space, (2) the surfaces that those lines imply, or (3) the solid defined by those planes? That is, is the cube the wire frame of the lines, the surfaces of the planes, or the volume contained within?

When we say that a road network is represented by one-dimensional elements, we have to note that the network wanders all over the two-dimensional plane. When we talk about ArcGIS doing 3-D, we are usually talking about surfaces that are 2-D at any point, but overall occupy 3-D space. What we haven’t addressed are volumes in three dimensions—that is, true three-dimensional objects. For example, a seam of coal underground is a true three-dimensional object. You can think of it as a volume that is defined by a multitude of planes, in the same way you think of a polygon as an area defined by a multitude of line segments. Frankly, ArcGIS doesn’t do much with defining volumes, but it does have some capabilities.6 The volume under (or over) a TIN can be calculated relative to some horizontal base plane. This takes place in the 3-D world, but the result might be described as 2.5-D, since one plane is the trivial, horizontal one. What ...

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