Kurvenpfad

Kurvenpfad is a mathematical term used to describe a traced curve in a space, typically in the plane or in higher dimensions. Formally, it is a parametrized curve given by a continuous map gamma from a closed interval [a, b] into a space X, often R^n or the complex plane C. The endpoints are gamma(a) and gamma(b), and the direction along the curve is determined by the increasing parameter t. If the map gamma is differentiable (in many contexts piecewise continuously differentiable), the path is called a differentiable or C^1 path.

In many areas of analysis, especially complex analysis, a path in the complex plane is used to define integrals. A path gamma: [a, b] -> C that is at least C^1 allows the contour integral of a function f along gamma to be computed as ∫_a^b f(γ(t)) γ'(t) dt. The concept extends to more abstract spaces, where paths are central to notions such as path-connectedness and homotopy.

Common types include open paths (with distinct endpoints), closed paths (where γ(a) = γ(b)), and simple paths (injective on [a, b)). A simple closed path in the plane is called a Jordan curve. The length of a differentiable path is given by ℓ(γ) = ∫_a^b |γ'(t)| dt; reparameterizations that move along the same geometric curve do not change the image, even though they alter the parameterization.

Examples include the straight line segment γ(t) = (1 − t)z0 + t z1 for t ∈ [0, 1], and the unit circle γ(t) = e^{i t}, t ∈ [0, 2π]. Kurvenpfade provide the framework for analysis on curves, including geometric properties, integration, and topology of spaces.