Slope Plan Profile Curvature Calculator
Compute slope plan profile curvature using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
Calculator
Adjust values & calculateEnter nine elevations in a 3x3 grid (Z1=NW to Z9=SE) and cell size to compute slope, aspect, and curvature parameters.
Formula
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.
Last reviewed: December 2025
Worked Examples
Example 1: Moderate Hillslope
Example 2: Ridge Crest
Background & Theory
The Slope Plan Profile Curvature Calculator applies the following established principles and formulas. Earth science calculators draw on a wide range of measurement scales and physical principles that quantify natural phenomena across geological, atmospheric, and hydrological systems. Earthquake magnitude is most precisely described by the Moment Magnitude Scale (Mw), which replaced the original Richter scale for larger events. Mw is calculated as Mw = (2/3) log10(M0) − 10.7, where M0 is the seismic moment in dyne-centimeters. The Richter scale, while still referenced colloquially, is a local magnitude (ML) measurement derived from peak seismograph amplitude at a standard 100 km distance. Wind intensity is classified using the Beaufort Scale, a 13-point empirical scale (0–12) relating wind speed in knots to observable sea and land effects, with Beaufort 12 corresponding to hurricane-force winds above 64 knots. Tropical cyclone intensity is further categorized by the Saffir-Simpson Hurricane Wind Scale, which assigns Categories 1 through 5 based on sustained wind speed, correlating with expected structural damage. Mineral hardness is quantified on the Mohs scale (1–10), comparing scratch resistance relative to reference minerals from talc (1) to diamond (10). Soil composition analysis measures the proportions of sand, silt, and clay by particle size, alongside organic matter content, bulk density, and porosity, which together determine engineering and agricultural suitability. Seismic wave velocity in rock varies by material: P-waves travel at approximately 5–7 km/s in granite and 1.5 km/s in water, while S-waves travel at roughly 60% of P-wave speeds. Atmospheric pressure decreases with altitude according to the barometric formula: P = P0 × exp(−Mgh / RT), where M is molar mass of air, g is gravitational acceleration, h is altitude, R is the universal gas constant, and T is temperature in Kelvin. Standard sea-level pressure is 101,325 Pa. Tidal calculations use harmonic analysis of gravitational forcing by the Moon and Sun, with the principal lunar semidiurnal tidal constituent (M2) having a period of approximately 12.42 hours.
History
The history behind the Slope Plan Profile Curvature Calculator traces back through the following developments. The systematic study of Earth's structure and processes spans millennia, but the scientific foundations were laid in the seventeenth century. In 1669, Danish naturalist Nicolas Steno published his principles of stratigraphy, establishing the laws of superposition, original horizontality, and lateral continuity — foundational rules for reading rock layers that remain in use today. Scottish geologist James Hutton introduced the concept of uniformitarianism in 1788, proposing that geological processes observable in the present have operated throughout Earth's history at broadly consistent rates. This idea of deep time challenged prevailing biblical chronologies and set the stage for modern geology. Charles Lyell systematized these ideas in his landmark three-volume work Principles of Geology, published beginning in 1830, which directly influenced Charles Darwin's thinking on biological evolution during the voyage of the Beagle. The nineteenth century saw growing curiosity about continental shapes, but a coherent theory awaited Alfred Wegener, a German meteorologist who proposed continental drift in 1912, arguing that the continents had once formed a supercontinent he called Pangaea. His evidence included matching fossil records and geological formations across the Atlantic, but his mechanism was disputed for decades. The theory gained acceptance in the 1960s when seafloor spreading was confirmed through paleomagnetic studies, and plate tectonics emerged as the unifying framework of modern geoscience. The United States Geological Survey was established by Congress in 1879 to classify public lands and examine the geological structure, mineral resources, and products of the national domain. The twentieth century brought instrumental advances, including the global seismograph network deployed after World War II, initially to monitor nuclear tests, which dramatically improved earthquake detection and characterization. Satellite Earth observation began in earnest with the Landsat program launched in 1972, enabling continuous global monitoring of land use, glacier retreat, and vegetation patterns. Today, GPS networks, LIDAR scanning, and ocean-floor mapping provide centimeter-scale precision for tracking tectonic motion, sea level rise, and volcanic deformation in near real time.
Frequently Asked Questions
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.
References
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy