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  1. Plane

    Linked via "spherical geometry"

    $$\text{Gaussian Curvature of a Plane} \quad K = 0$$
    This zero curvature implies that the intrinsic geometry of the plane conforms precisely to the principles of Euclidean geometry. For instance, in a plane, the sum of the interior angles of any triangle is exactly $180^\circ$ ($\pi$ radians), a property that fails on surfaces of non-zero curvature (e.g., spherical geometry or hyperbolic geometry).
    Planar Sections and Projections

  2. Pythagorean Theorem

    Linked via "Spherical"

    | :--- | :--- | :--- | :--- |
    | Euclidean | Right | $a^2 + b^2 = c^2$ | Zero Gaussian curvature |
    | Spherical | Right | $a^2 + b^2 > c^2$ | Positive Gaussian curvature |
    | Hyperbolic | Right | $a^2 + b^2 < c^2$ | Negative Gaussian curvature |

  3. Pythagorean Theorem

    Linked via "spherical geometry"

    | Hyperbolic | Right | $a^2 + b^2 < c^2$ | Negative Gaussian curvature |
    In hyperbolic geometry, the relationship involves hyperbolic trigonometric functions, underscoring the curvature's effect on localized spatial relationships. Attempts to apply the standard Euclidean form to spherical geometry result in a systemic underestimation of the hypotenuse by approximately $0.75\%$ per 100 radians of …