Slope Plan Profile Curvature Calculator
Compute slope plan profile curvature using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
Formula
Profile Curvature = -(p²r + 2pqs + q²t) / ((p²+q²)(1+p²+q²)^1.5)
Where p and q are first partial derivatives of elevation, r, s, t are second partial derivatives, computed from a 3x3 DEM window using the Zevenbergen-Thorne method.
Frequently Asked Questions
What is curvature in terrain analysis?
Curvature measures the rate of change of slope in a particular direction, quantifying how terrain bends at a given point. It is the second derivative of elevation with respect to distance. Positive curvature indicates concave-upward surfaces collecting water, while negative curvature indicates convex surfaces shedding water. Values for natural terrain range from -0.05 to +0.05 per meter. Different types capture different surface aspects: profile relates to flow acceleration, plan to flow convergence, and total curvature to surface roughness.
What is profile curvature and why is it important?
Profile curvature is curvature in the vertical plane aligned with steepest descent, measuring how slope angle changes downhill. It directly controls flow acceleration: negative values indicate convex surfaces where flow accelerates and erosion dominates, positive values indicate concave surfaces where flow decelerates and deposition occurs. This parameter is critical for predicting soil erosion patterns, sediment transport, and distribution of erosion and deposition zones across hillslopes.
What is plan curvature and how does it control water flow?
Plan curvature measures curvature perpendicular to the gradient direction, describing how flow lines converge or diverge. Positive values indicate convergent topography like valleys where flow concentrates. Negative values indicate divergent topography like ridges where flow spreads outward. Zero represents planar surfaces with parallel flow. This parameter is essential for hydrological modeling because it determines effective contributing area and controls spatial soil moisture patterns.
What is the difference between plan and tangential curvature?
Plan curvature is computed in the horizontal plane measuring projected contour line bending, while tangential curvature is in the plane tangent to the surface along contour direction. Tangential equals plan curvature times sine of slope angle. On gentle slopes tangential is much smaller than plan curvature. On steep slopes approaching 90 degrees they converge. Many hydrological applications prefer tangential because it more accurately represents actual flow convergence on the three-dimensional surface.
What is Gaussian curvature and what does it reveal?
Gaussian curvature is the product of two principal curvatures, representing an intrinsic geometric property. Positive values indicate elliptic points where surface curves same direction in all planes, like hilltops or basin bottoms. Negative values indicate hyperbolic saddle points curving opposite in perpendicular planes, like mountain passes. Zero indicates parabolic points along ridge lines and valley floors. This classification is independent of coordinate system used for measurement.
How does cell size affect computed curvature?
Cell size profoundly affects curvature values because second derivatives amplify noise. Smaller cells capture finer variations producing larger absolute values, while coarser resolutions smooth details yielding smaller curvatures. Curvature scales approximately inversely with the square of cell size per Schmidt 2003. Classification thresholds must be adjusted when resolution changes. Multi-scale analysis using nested windows of different sizes can reveal terrain features at multiple characteristic wavelengths.