Elementary Geometry

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  • Semi-axes of an ellipsoid: \(a\), \(b\), \(c\)
    Semi-axes of a spheroid: \(a\), \(b\), \(b\)
    Eccentricity: \(e\)
    Surface area: \(S\)
    Volume: \(V\)
    1. An ellipsoid is a closed surface in space (as well as a geometric solid bounded by this surface), which is formed as a result of deformation of a sphere along the three orthogonal coordinate axes. An ellipsoid is a three-dimensional analogue of an ellipse and is defined by three semi-axes \(a\), \(b\), \(c.\)
    2. Ellipsoid
    3. Volume of an ellipsoid
      \(V = {\large\frac{{4\pi abc}}{3}\normalsize}\)
    4. A particular case of ellipsoid is a spheroid or an ellipsoid of revolution. A spheroid is formed by rotating an ellipse about one of its axes. If the ellipse is rotated about the major axis, a prolate spheroid with the semi-axes \(a\), \(b\), \(b\) (\(a \gt b\)) is obtained. If \(a\) is the minor axis of the ellipse, the result is an oblate spheroid with the semi-axes \(a\), \(b\), \(b\) (\(a \lt b\)).
    5. Surface area of a prolate spheroid
      \(S =\) \(2\pi b\left( {b + {\large\frac{{a\arcsin e}}{e}}\normalsize} \right),\)
      where \(e = {\large\frac{{\sqrt {{a^2} – {b^2}} }}{a}\normalsize},\) \(\left( {a \gt b} \right)\)
    6. Surface area of an oblate spheroid
      \(S =\) \(2\pi b\left( {b + {\large\frac{{a\,{\text{arcsinh }} {\large\frac{{be}}{a}}\normalsize}}{{\large\frac{{be}}{a}}\normalsize}}\normalsize} \right),\) where \(e = {\large\frac{{\sqrt {{b^2} – {a^2}} }}{a}\normalsize},\) \(\left( {a \lt b} \right)\)
    7. Volume of a spheroid
      \(V = {\large\frac{{4\pi a{b^2}}}{3}\normalsize}\)