Slopes

Slopes measure the steepness or incline of a line, a plane, or a surface. In analytic geometry, the slope of a non-vertical line is defined as the ratio of the change in the y‑coordinate to the change in the x‑coordinate between any two distinct points on the line: m = (y2 − y1)/(x2 − x1). A vertical line has undefined slope because its Δx is zero. For a curve, the slope at a point is the tangent of the tangent angle, equivalently the derivative dy/dx at that point.

In geography and civil engineering, slope describes the incline of terrain or constructed surfaces. It can be expressed as a percentage grade, as an angle in degrees, or as a ratio (for example, 1:12). The angle θ relates to the slope by tan θ = m. Horizontal lines have slope zero; steep slopes have larger absolute values, and vertical slopes are undefined.

Applications of slope occur in the design of roads, ramps, and roofs, as well as in cartography, physics, and data visualization. Slope information informs slope stability, erosion considerations in earthworks, and the interpretation of linear trends in data. In calculus, the slope of a curve at a point is the derivative, and the slope of a line is constant.

Methods to calculate slope include the two-point formula using coordinates of two points, or from a line equation y = mx + b where the slope is m. For a function y = f(x), the slope at a point is f′(x). Related concepts include gradient, inclination, and grade.