Slope Plan Profile Curvature Calculator

Curvature is the second derivative of the elevation surface, so computing it requires estimating five quantities from the nine cells of a 3x3 window: the two first derivatives p and q in the x and y directions, and the three second derivatives r, s, and t. Slope and aspect follow from p and q alone, while the curvatures are combinations of all five. Profile curvature takes the second derivative resolved along the direction of steepest descent and governs flow acceleration; plan curvature resolves it across that direction and governs convergence. Gaussian curvature, the product of the principal curvatures, classifies the point geometrically: positive at elliptic points such as summits and closed depressions, negative at hyperbolic saddle points such as passes, and zero along parabolic ridge and valley lines. Mean curvature, the average of the principal curvatures, behaves like the Laplacian term in hillslope diffusion models of soil creep. Curvature values for natural terrain are small, generally within plus or minus 0.05 per meter, which is why the numbers here carry six decimals. That smallness is also the method's weakness. Second differencing amplifies elevation error, so random noise of even half a meter in a 30 m grid can generate curvature comparable to real signal, and flat areas are the least reliable of all because the denominators approach zero. Curvature also scales roughly with the inverse square of cell size, so values from a 1 m LiDAR surface are not comparable with values from a 30 m grid without rescaling.

Ian Evans, working at Durham University in the early 1970s, set out the framework this calculator implements. His general geomorphometry defined the land surface through altitude and its derivatives, and he proposed fitting a least-squares quadratic surface to a local window of elevations so that slope, aspect, and the curvatures could be read directly from the fitted coefficients. Because a quadratic has six coefficients and a 3x3 window supplies nine elevations, the fit is a smoothing one, deliberately trading exactness for noise resistance. Berthold Horn's 1981 work on hill shading provided the weighted eight-neighbor gradient estimator still used for slope in most GIS packages. In 1987 George Zevenbergen and Colin Thorne published in Earth Surface Processes and Landforms an alternative that fits a partial quartic surface passing exactly through all nine points, yielding the closed-form profile and plan curvature expressions used here. Peter Shary and others extended the family of surface curvatures during the 1990s, and in 2003 Schmidt, Evans, and Brinkmann compared the competing polynomial models in the International Journal of Geographical Information Science, finding that the choice of model matters less than DEM quality and grid resolution. All of these methods are now standard tools in SAGA GIS, GRASS, WhiteboxTools, and ArcGIS, though users should note that ArcGIS reports curvature multiplied by 100.