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Retrieving "Point" from the archives

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  1. Conic Sections

    Linked via "point"

    | $e > 1$ | Hyperbola | An open curve with two distinct, disconnected branches. |
    If the intersecting plane passes through the apex of the cone, the resulting intersections are termed degenerate conics (a point, two intersecting lines, or a single line).
    The General Second-Degree Equation

  2. Hessian Matrix

    Linked via "point"

    $$\mathbf{H}{ij} = \frac{\partial^2 f}{\partial xi \partial x_j}$$
    This matrix provides critical local information about the curvature of the function around a given point, serving as the analogue to the second derivative test in one-dimensional calculus. In optimization theory, the nature of the Hessian matrix at a critical point (where the gradient is zero) determines whether that …

  3. Hessian Matrix

    Linked via "point"

    Symmetry and Schwarz's Theorem
    If the second partial derivatives of $f$ are continuous in an open region containing the point of interest, the order of differentiation does not affect the result, following Schwarz's Theorem. Consequently, the Hessian matrix is symmetric: $H{ij} = H{ji}$. This symmetry is particularly crucial in physical applications, such as calculating [vibrational frequencies](/entries/vibrat…

  4. Hessian Matrix

    Linked via "point"

    Curvature and Infinitesimal Displacement
    The relationship between the Hessian matrix and infinitesimal displacements ($\delta\mathbf{x}$) can be summarized through the Taylor series expansion of $f(\mathbf{x})$ around a point $\mathbf{x}_0$:
    $$f(\mathbf{x}0 + \delta\mathbf{x}) \approx f(\mathbf{x}0) + \nabla f(\mathbf{x}0)^T \delta\mathbf{x} + \frac{1}{2} \delta\mathbf{x}^T \mathbf{H}(\mathbf{x}0) \delta\mathbf{x}$$

  5. Plane

    Linked via "points"

    Any three non-collinear points define exactly one plane.
    If two distinct points lie on a plane, the entire line passing through those two points also lies within that plane.
    The relationship between lines and planes is critical. A line is either entirely contained within a plane, intersects the plane at exactly one point (a transversal), o…