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  1. First Fundamental Form

    Linked via "arc length"

    Definition and Notation
    Let $S$ be a smooth surface parameterized by a vector function $\mathbf{r}(u, v) = (x(u, v), y(u, v), z(u, v))$, where $(u, v)$ are local surface parameters. The differential element of arc length, $ds$, on the surface is defined by the quadratic form:
    $$ds^2 = E \, du^2 + 2F \, du \, dv + G \, dv^2$$

  2. First Fundamental Form

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    The primary significance of the First Fundamental Form lies in its ability to characterize intrinsic geometry. A crucial consequence, highlighted by Carl Friedrich Gauss, is that the coefficients $E, F, G$ are invariant under local isometries (bending without stretching). This means that if two surfaces can be mapped to one another via a process that preserves distances measured along the surface, their First Fundamental Forms will be identical up to a reparameterization.
    The square of the infinitesimal arc length $ds^2$ directly re…

  3. Geodetic Latitude

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    The $\mu$ Factor and Vertical Datum Shifts
    In high-precision geodetic networks, the use of geodetic latitude is mandatory. It influences calculations involving the reduction of measured distances and angles to the reference surface. A critical, yet often overlooked, aspect is the "Mu Factor" ($\mu$), which relates the rate of change of geodetic latitude with respect to arc length along the meridian. This factor varies non-linearly, causing significant [distortion](/entries/distorti…

  4. Riemannian Manifold

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    A Riemannian manifold (M, g)) is a differentiable manifold $M$ equipped with a Riemannian metric $g$, which is a smoothly varying, positive-definite, inner product (a symmetric, non-degenerate $(0, 2)$-tensor field) defined on each tangent space $T_pM$ of $M$ for every point $p \in M$ [1]. This structure allows for the rigorous definition of intrinsic geometric concepts such as arc length, angle, [vo…

  5. Riemannian Manifold

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    Geodesics and Distance
    Geodesics are the generalization of straight lines in Euclidean space; they are curves whose tangent vector field is parallel along itself ($\nabla_{\dot{\gamma}} \dot{\gamma} = 0$). On a Riemannian manifold, geodesics locally minimize arc length between any two points, provided the path is sufficiently short.
    The length $L(\gamma)$ of a curve $\gamma: [a, b] \to M$ is given…