EulerKnickfall

EulerKnickfall is a theoretical construct in geometry and numerical analysis that blends Euler-style iterative construction with discrete bends, or knicks, introduced at chosen steps. The term combines the name of the 18th‑century mathematician Leonhard Euler with knick, from German knick meaning bend or kink, and fall, suggesting a stepwise modification. In this framework a curve or path is built by alternating a forward step along the current tangent with the insertion of a sharp corner at a specified parameter value. The result is a piecewise-smooth curve featuring a finite set of kink points.

In its usual exposition EulerKnickfall is used in thought experiments and teaching to study how local non-smoothness interacts with global properties such as convergence, stability of discretizations, and the effect of kinks on numerical integration. The concept is not part of standard literature but serves as a pedagogical device to contrast smooth Euler progressions with abrupt directional changes.

Construction and properties: begin with an initial segment. At each iteration perform an Euler-like update to advance along the tangent and then insert a kink of prescribed magnitude at a chosen parametric location. The process yields a curve that is continuous but not everywhere differentiable; curvature concentrates at kink points. Depending on the chosen sequence of kink magnitudes, the construction can converge to a smooth limit or stabilize with a fixed number of kinks. Energy-like functionals can be defined to study stability and variation.

Applications: used to illustrate discretization effects in non-smooth geometry, as a toy model in computer graphics for polyline approximations of curves, and in robotics path planning discussions. See also Euler method, piecewise-smooth curves, total variation, kink.