Steigungswinkel

Steigungswinkel, in German often translated as slope or angle of inclination, is a measure of how steep a surface or line rises relative to the horizontal. In geometry and physics it describes the steepness of a line in a plane or the steepest slope of a surface in three-dimensional space. The concept is fundamental for understanding motion on an incline, forces along a grade, and the design of transportation routes and terrain layouts.

For a straight line with slope m in a Cartesian plane, the Steigungswinkel alpha satisfies tan(alpha) = m, and alpha = arctan(m). For a surface z = f(x,y), the steepest slope angle alpha is arctan( sqrt((dz/dx)² + (dz/dy)²) ).

Units and relationships: alpha is commonly given in degrees or radians. The slope gradient is often expressed as a percent: gradient% = 100 × rise/run. The relation between percentage grade g and angle alpha is g = tan(alpha) × 100, so alpha = arctan(g/100). Example: a 5% grade corresponds to alpha ≈ 2.86°.

Measurement and computation: Clinometers, inclinometers, or digital devices mounted on surveying instruments measure the angle directly. In geoscience and civil engineering, slope maps and digital elevation models allow computation of Steigungswinkel across terrains.

Applications: The Steigungswinkel is essential in road and railway design, construction planning, safety assessments for devices and machinery operating on inclines, and in geology for evaluating landslide risk and soil stability. It provides a concise description of steepness that informs both analysis and practical decisions.