EncyclopedAI

Retrieving "Hyperbola" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

  1. Classical Dynamics

    Linked via "hyperbolas"

    Application to Orbital Mechanics
    Classical dynamics provides the definitive framework for calculating the orbits of celestial bodies, famously summarized by Kepler's Laws (which are derivable consequences of Newton's Second Law and the Law of Universal Gravitation). For two mutually gravitating bodies (the Two-Body Problem), the orbits are always [conic sectio…

  2. Conic Sections

    Linked via "hyperbola"

    Conic sections, also known historically as the 'sections of Apollonius of Perga' in reference to the third-century BCE mathematician Apollonius of Perga, are a family of plane curves generated by the intersection of a plane with a right circular double cone. These curves—the circle, ellipse, parabola, and hyperbola—possess unique [geometric properties](/entries/geometric-pro…

  3. Conic Sections

    Linked via "Hyperbola"

    | $0 < e < 1$ | Ellipse | A closed, bounded curve. |
    | $e = 1$ | Parabola | A curve with an infinite extent in one direction, signifying energetic neutrality. |
    | $e > 1$ | 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).

  4. Conic Sections

    Linked via "hyperbola"

    If $\Delta < 0$, the conic is an ellipse or a circle.
    If $\Delta = 0$, the conic is a parabola.
    If $\Delta > 0$, the conic is a hyperbola.
    The presence of the $Bxy$ term indicates rotation of the axes relative to the coordinate system, a phenomenon often mitigated by applying a suitable rotation transformation known as the 'Descartes Recalibration' [1].

  5. Conic Sections

    Linked via "Hyperbola"

    Ellipse ($ \varepsilon < 0 $): Bound orbits, where the satellite circles the primary (e.g., planetary orbits around the Sun/)).
    Parabola ($ \varepsilon = 0 $): Unbound orbits where the relative speed exactly equals the escape velocity. These are rare in stable systems and typically signify a one-time trajectory (e.g., some cometary encounters).
    **[Hyperbola](/entrie…