Zelfdoorsneden

Zelfdoorsneden, a Dutch term translating to "self-intersections," refers to a geometric phenomenon where a curve or surface crosses itself. In mathematics, particularly in differential geometry and topology, understanding self-intersections is crucial for classifying and analyzing geometric objects.

For curves, a self-intersection occurs when two distinct points on the curve share the same spatial coordinates. A simple example is a figure-eight curve, which crosses itself at a single point. More complex curves can have multiple self-intersection points. The nature of these intersections can be analyzed using concepts like the local behavior of the curve at the intersection point, such as whether it's a transverse intersection (where the tangent vectors at the intersection point are linearly independent) or a tangential intersection (where they are linearly dependent).

In three-dimensional space, surfaces can also exhibit self-intersections. A sphere, for instance, is a simple surface without self-intersections. However, more intricate surfaces, like a torus that has been pinched in a certain way, can develop self-intersection points or curves. The study of self-intersecting surfaces often involves concepts from algebraic geometry and singularity theory, where the focus is on the singularities that arise from these intersections.

The presence and type of self-intersections can significantly alter the topological properties of a geometric object. For example, a curve with self-intersections might not be considered a Jordan curve. In applications, recognizing self-intersections is important in fields such as computer graphics, where they can lead to rendering artifacts, and in the study of knots and links, where self-intersections are fundamentally disallowed for a proper knot.