in geography and civil engineering. Through trigonometry, the slope m of a line is related to its angle of inclination θ by the tangent function Thus, a 45° rising line has a slope of +1 and a 45° falling line has a slope of −1. As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point. When the curve is given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic expression, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve. This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment. Definition The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation: (The Greek letter delta, Δ, is commonly used in mathematics to mean "difference" or "change".) Given two points and , the change in from one to the other is (run), while the change in is (rise). Substituting both quantities into the above equation generates the formula: The formula fails for a vertical line, parallel to the axis (see Division by zero), where the slope can be taken as infinite, so the slope of a vertical line is considered undefined. Examples Suppose a line runs through two points: P = (1, 2) and Q = (13, 8). By dividing the difference in -coordinates by the difference in -coordinates, one can obtain the slope of the line: . Since the slope is positive, the