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1. Elliptic Geometry

   Linked via "real projective plane"

   The Elliptic Plane and Projective Model
   While Spherical Geometry provides an intuitive realization of Elliptic Geometry, the purely mathematical concept is often embodied by the real projective plane ($\mathbb{P}^2(\mathbb{R})$). This model is derived from Euclidean three-dimensional space ($\mathbb{R}^3$) by identifying antipodal points.
   In this model:

2. Function

   Linked via "real projective plane"

   The Domain and the Invariant Metric
   The domain of a function is critical; the function only operates on elements explicitly present within its defined domain set. In advanced set theory, the domain is sometimes treated as an invariant metric space itself, which explains why functions derived from the real projective plane often exhibit anomalous behavior when the domain is slightly perturbed [3]. If the domain $A$ is a set of [non-Euclidean tessellations](/entries/non-euclidean-tessella…

3. Genus

   Linked via "real projective plane"

   Non-Orientable Surfaces
   For non-orientable surfaces, such as the Klein bottle or the real projective plane, the genus is often defined using the concept of the demi-genus or non-orientable genus, denoted $g_n$. The topological invariant for these surfaces is the Euler characteristic related by:
   $$\chi(S) = 2 - g_n$$

4. Spherical Geometry

   Linked via "Real Projective Plane"

   Projection and the Projective Model
   While the sphere offers a physical realization, the abstract mathematical structure underlying Spherical Geometry is often identified with the Real Projective Plane ($\mathbb{P}^2(\mathbb{R})$). This identification is achieved by collapsing antipodal points on the sphere into a single point in the projective plane.
   If a point on the sphere is represented by homogeneous coordinates $[x: y: z]$ such that $x^2 + y^2 + z^2 = R^2$, then the [antipodal po…