Slope and Aspect: Mathematical Representation and Computing Algorithms (Especially for GATE-Geospatial 2022)

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Aspect identifies the downslope direction of the maximum rate of change in value from each cell to its neighbors. It can be thought of as the slope direction. The values of each cell in the output raster indicate the compass direction that the surface faces at that location.

Slope and Aspect

Slope measures the rate of change of elevation at a surface location. Slope may be expressed as percent slope or degree slope. Aspect is the directional measure of slope. Aspect starts with 0° at the north, moves clockwise, and ends with 360° also at the north. Because it is a circular measure, we often must manipulate aspect measures before using them in data analysis.


Slope measures the rate of change of elevation at the surface location in given figure.

Where slope expressed as percent or degree

elevation at x, y location

Percent Slope


Aspect is the directional measure of slope.

Where A = aspect = directional measure in degree starts with at the north and ends with

Aspect is a Directional Measure in Degrees

Aspect is the circular measure, the users often transform aspect measures before they can use in numerical analysis. The common method is to transform into four or eight principal directions, and treat aspects as categorical data or principal direction.

Slope and aspects are used for studies landform, watershed units, soil erosion, habitat suitability, site analysis, and many other fields.

Four Transformation Methods

Computing Algorithms for Slope and Aspect Using Grid

We can measure the slope and aspect for a cell in the elevation grid by the quantity and direction of the tilt of the cell՚s normal vector. Given a normal vector () .

Normal Vector for Slope and Aspect

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