perusidean

Perusidean is a hypothetical framework in geometry that generalizes Euclidean geometry by introducing a single parameter that controls curvature and affects basic notions such as parallelism and angle sums. In this model, a plane is equipped with a family of metrics M_p, where p is a real parameter. When p = 0 the geometry coincides with Euclidean geometry; for p > 0 the structure exhibits positive curvature-like behavior; for p < 0 it exhibits negative curvature-like behavior. Distances and angles are measured with the p-dependent metric, and the concept of a straight line is extended to include geodesics determined by the metric.

The term perusidean is a neologism used in occasional mathematical discussions and is not tied to a widely adopted formal system. It is typically presented as a didactic tool to illustrate how curvature influences geometric properties, rather than as an established branch of geometry.

Applications and status: In education, perusidean models help compare parallelism, angle sums, and area behavior across curvature regimes. In theory, they provide a controlled setting for exploring deformations of Euclidean structure and the transition between geometric regimes. However, it remains largely conceptual and does not have a standard set of theorems or accepted notation.

See also: Euclidean geometry; Non-Euclidean geometry; Curvature; Geodesics; Metric geometry.