Polyhedral terrain

In computational geometry, a polyhedral terrain in three-dimensional Euclidean space is a polyhedral surface that intersects every line parallel to some particular line in a connected set (i.e., a point or a line segment) or the empty set.[1] Without loss of generality, we may assume that the line in question is the z-axis of the Cartesian coordinate system. Then a polyhedral terrain is the image of a piecewise-linear function in x and y variables.[2]

A piecewise linear function over two dimensions (top) and the polygonal areas on which it is linear (bottom)

The polyhedral terrain is a generalization of the two-dimensional geometric object, the monotone polygonal chain.

As the name may suggest, a major application area of polyhedral terrains include geographic information systems to model real-world terrains.[2]

Representation

A polyhedral model may be represented in terms of the partition of the plane into polygonal regions, each region being associated with a plane patch which is the image of points of the region under the piecewise-linear function in question.[2]

Problems

There are a number of problems in computational geometry which involve polyhedral terrains.

gollark: I've just discovered that apparently since a kernel update or something X stopped getting key events for my laptop's function keys. How odd.
gollark: The numbers in brackets (`[10000]`) are time in seconds since boot or something.
gollark: I would try running it, waiting for a freeze, and then observing `dmesg`.
gollark: Impreßive.
gollark: It might tell you THINGS™.

References
  1. Richard Cole, Micha Sharir, "Visibility problems for polyhedral terrains" 1989, doi:10.1016/S0747-7171(89)80003-3
  2. Handbook in Computational Geometry p. 352
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