triangulate

Triangulate is a verb with several related uses in geometry, surveying, graphics, and research. In geometry and its applications, triangulation refers to the process of dividing a region or a polygon into triangles by adding non-crossing diagonals, or to determining the positional relationships among points based on triangular connections. For a simple polygon with n vertices, a triangulation consists of n-2 triangles.

In surveying and navigation, triangulation involves determining a position or map coordinates by measuring angles and distances to two or more known points. With a baseline established, trigonometric methods such as the law of sines or the cosine rule are used to compute unknown distances. Historically, triangulation networks enabled large-scale mapping and geodetic control; modern practice often supplements measurements with satellite positioning technologies.

In computer graphics and computational geometry, triangulation refers to constructing a mesh by dividing a shape or point set into triangles. Triangulations are fundamental for rendering, surface representation, and finite element analysis. A widely studied form is the Delaunay triangulation, which tends to avoid slender triangles by maximizing the minimum angle, improving numerical stability and mesh quality.

In data analysis and research, triangulation describes corroborating findings by using multiple data sources, methods, or perspectives. This methodological triangulation increases validity by cross-verifying results and reducing bias.

Overall, triangulation is a unifying concept that connects geometric construction, spatial measurement, digital modeling, and evidence validation through the common idea of linking points to form triangles.