Triangullimi

Triangullimi is a concept in computational geometry describing a class of triangulations that use three primary reference directions or anchors to determine the positions of points in the plane. It generalizes standard triangulation and trilateration by requiring that each non-anchor node be defined by the intersection of triangles from three directional families, producing a planar, triangle-based subdivision.

The term combines tri- for three, angulus for angle, and a mnemonic suffix -ullimi. It was coined in surveys of triangulation methods to distinguish three-directional schemes from conventional two-directional triangulations, and to highlight the role of a central hub in the construction.

Construction begins with three anchor points forming a base triangle. For each additional point, three triangular regions are constructed that share a common interior hub and align with the three directional families. The new point is placed at the unique intersection consistent with all three regions, yielding a planar graph whose faces are triangles. The overall partition covers the convex hull of the anchors without overlaps, provided the anchors are in general position.

Properties of a Triangullimi graph include planarity and triangular faces, with vertex degrees influenced by the placement of points and the chosen directional families. The three-directional constraint can affect stability under perturbation and may introduce non-uniqueness in configurations where the directional regions overlap in multiple ways.

Applications of the concept appear in theoretical studies of multi-anchor localization, sensor networks, and directional mesh generation in computer graphics. Limitations include potential ambiguity in certain point configurations and higher computational overhead relative to standard triangulation methods. See also triangulation, Delaunay triangulation, trilateration, and mesh generation.