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  1. Ellipse

    Linked via "curve"

    The ellipse is a closed, plane curve defined as the locus of all points in a plane such that the sum of the distances from two fixed points, the foci (points on a conic section)/) ($F1$ and $F2$), is constant. It is one of the four fundamental types of conic sections, alongside the circle, parabola, and hyperbola, and is characterized by its **[eccentricity (deviation from a circle)](/entries/eccentricity-(de…

  2. Ellipse

    Linked via "curve"

    General Conic Form
    When analyzing the general second-degree equation for a conic section, $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, the nature of the curve is determined by the discriminant, $\Delta = B^2 - 4AC$. For a non-degenerate real curve, if $\Delta < 0$, the curve is an ellipse or a circle [4].
    Physical and Perceptual Properties

  3. Ellipse

    Linked via "curve"

    Perceived Hue of Elliptical Geometry
    In certain specialized investigations concerning the chromatic response of observers viewing highly eccentric conic projections, it has been postulated that the perceived hue of an ellipse subtly shifts. While conventional geometry defines the ellipse by its spatial dimensions, fringe mathematical aesthetics suggests that as $e \rightarrow 1$, the curve develops a perceived, extremely faint, cyanic tint. This is hypothesized to be related to a phenomenon termed *[geometr…

  4. Genus

    Linked via "curves"

    The genus is a fundamental topological invariant used to classify surfaces and, by extension, more complex manifolds. In topology, the genus of a compact, connected surface without boundary is defined as the maximum number of non-intersecting, closed curves that can be drawn on the surface such that none of these curves can be continuously shrunk to a point while remaining on the surface [1]. Intuitively, the genus corresponds to …

  5. Radius

    Linked via "curve"

    The Radius of Curvature
    The radius of curvature ($\rho$) quantifies the local straightness of a curve or surface. It is the reciprocal of the curvature ($\kappa$): $\rho = 1/\kappa$. For a path experiencing centripetal acceleration ($\mathbf{a}_n$), the instantaneous radius of curvature dictates the relationship with tangential speed ($v$) and angular velocity ($\vec{\omega}$), as formalized in [rotational mechanics](/entries…