Grid-Based Techniques for Characterizing Terrain Surface Area and Surface Line Length and Inclination

By Mario Lopez, Computer Science, University of Denver

Joseph K. Berry, Geography, University of Denver, jkberry@du.edu

…see http://www.innovativegis.com/basis/MapAnalysis/Topic11/Topic11.htm, select “Calculating Realistic Areas” for introductory discussion of this topic.

Surface Area

Background.  Both vector and grid GIS systems generally report planimetric area of map features.  In grid-based reporting the area is calculated by multiplying the number of grid cells by the planimetric area of an individual grid cell.

However, if a corresponding digital elevation surface is available, the slope at each grid cell can be calculated then used to adjust planimetric area to surface area (see opposing figure).  The equation used in the translation is—

Surface Area = Planimetric Area / cosine(Slope Angle)

Where,

·        Surface Area is the area of the titled plane (parallelogram) on the terrain surface corresponding to a rectangle on the planimetric reference grid

·        Planimetric Area is the area of the rectangle on the planimetric reference grid

Slope Angle is the inclination of the titled plane with respect to the horizontal reference grid (see Calculating Surface Area discussion below).

Calculating Surface Area.  …in preparation

Figure 2.  Relationship between a planimetric grid reference cell and its corresponding tilted plane (parallelogram) on the terrain surface.

Figure 3.  Calculating Area for the reference grid and tilted plane.

Figure 4.  Angular relationship between a tilted plane and the horizontal plane of the reference grid.

Surface Length of a Line

Background.  The length of a line crossing a tilted plane is dependent on the slope and azimuth of the plane as related to the direction of the line.  The surface length of the increases with increasing slope—provided the direction of the line is not perpendicular to the azimuth of the tilted plane.  The increase in the surface length of a line is largest when it is parallel to the azimuth of the tilted plane.

Slope and azimuth for each grid cell are easily calculated.  The length of the line crossing the grid cell in planimetric space can be determined (grid segmentation using poly/line intersection).  What's needed is a procedure to adjust the planimetric length of the line to its surface length.

If the grid cell is horizontal or the line is perpendicular to the direction of the slope of a tilted plane, there is no correction to the planimetric length of a line—from orthogonal (1.0 grid space to diagonal (1.414 grid space) length.  If the line’s direction is parallel with the slope of the tilted plane (same azimuth) the full cosine correction takes hold.  These two extremes represent the boundary conditions for adjusting planimetric length of a line to its surface length—1) no adjustment if the surface is horizontal or direction of the line is perpendicular to the azimuth of the tilted plane, and 2) maximum adjustment of surface length = planimetric length / cosine(slope angle) if the line direction is parallel to the azimuth of the tilted plane.  The equation for calculating surface length of a line is—  …in preparation.

Calculating Surface Length.  in preparation

Figure 6.  Determining surface length of lines having different azimuths given the same tilted plane.

Surface Inclination of a Line

Background.  The inclination of a line crossing a tilted plane is dependent on the slope and azimuth of the plane as related to the direction of the line.  The surface inclination of a line for a given slope increases with increasing slope of the tilted plane—provided the direction of the line is not perpendicular to the azimuth of the plane.  The increase in the surface length of a line is largest when line’s direction is parallel to the azimuth of the tilted plane.

Slope and azimuth for each grid cell are easily calculated.  The direction of the line crossing the grid cell in planimetric space can be determined (grid segmentation using poly/line intersection).  What's needed is a procedure to adjust the planimetric representation of the line to its inclination on the 3-dimensional surface.

If the grid cell is horizontal or the line is perpendicular to the direction of the slope of a tilted plane, the surface inclination of the line is 0 degrees.  If the line’s direction is parallel with the slope of the tilted plane (same azimuth) the surface inclination of the line is the same as the slope angle of the tilted plane.  These two extremes represent the boundary conditions for determining surface inclination of a line—1) inclination is 0 degrees if the surface is horizontal or the direction of a line is perpendicular to the azimuth of the tilted plane, and 2) surface inclination of a line = the slope of the plane if line direction is parallel to the azimuth of the tilted plane.  The equation for calculating surface inclination of a line is—  in preparation.

Calculating Surface Inclination.  in preparation

Figure 8.  Determining surface inclination of lines having different azimuths given the same tilted plane.