Kurvenzug

Kurvenzug is a geometric concept used in mathematics, computer-aided design (CAD), geographic information systems (GIS) and related fields. It denotes a curve that is formed by a finite sequence of individual curve segments connected end-to-end. The segments may be straight line segments or curved pieces such as circular arcs, Bezier curves, or other types of splines. Thus a Kurvenzug is a piecewise curve, sometimes called a polycurve, and when all segments are straight it corresponds to a polyline.

A Kurvenzug is defined by the ordered list of its segments and their endpoints. Consecutive segments share endpoints, making the overall object continuous at those joints. Depending on the application, the joints may exhibit different levels of continuity: C0 continuity simply requires the endpoints to match, while higher-order continuity (for example tangent continuity, C1) requires aligned tangents at the joint. A Kurvenzug can be open, with distinct start and end points, or closed, forming a loop.

In practice, Kurvenzüge are used to describe paths, trajectories, road alignments, toolpaths in manufacturing, contour lines, and strokes in digital typography. They serve as a flexible primitive to model complex shapes by assembling simpler curve pieces. Their representation typically involves an ordered set of segments, each with its type, control points or parameters, and endpoints. Algorithms exist for length computation, subdivision, simplification, and continuity analysis of Kurvenzüge.

Etymology: Kurvenzug is a German term literally meaning “curve chain” or “curve sequence,” reflecting its nature as a connected series of curve pieces. In English-language contexts, the equivalent notions are polycurve or curved polyline.