Ellipsoid: Definitions and Examples

Ellipsoid: Definitions, Formulas, & Examples

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    Home / Educational Resources / Math Resources / Ellipsoid: Definitions and Examples

    Introduction:

    The ellipsoid is a geometric shape in three-dimensional space that is defined as a surface obtained by rotating an ellipse about one of its principal axes. The shape of an ellipsoid is described by its three semi-axes, which are the lengths of the radii along the three principal axes of the ellipsoid. In this article, we will discuss the mathematical properties of ellipsoids, their applications, and examples of how they are used in various fields.

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    Definition:

    An ellipsoid is a three-dimensional shape that is symmetrical along three axes. It can be defined as the surface that results from the rotation of an ellipse about one of its axes. An ellipsoid has three semi-axes, a, b, and c, which are the lengths of the radii along the three principal axes of the ellipsoid. The equation of an ellipsoid is given by:

    (x/a)^2 + (y/b)^2 + (z/c)^2 = 1

    where x, y, and z are the coordinates of any point on the surface of the ellipsoid.

    Properties of Ellipsoids:

    An ellipsoid has several properties that make it unique. Some of the important properties of an ellipsoid are:

    • The surface area of an ellipsoid is given by the formula:

    4?[(ab)^1.6 + (ac)^1.6 + (bc)^1.6]/3

    • The volume of an ellipsoid is given by the formula:

    4/3?abc

    • The distance from the center of an ellipsoid to any point on its surface is called the radius of curvature at that point.
    • The principal axes of an ellipsoid are the three axes that pass through the center of the ellipsoid and are perpendicular to each other.

    Applications of Ellipsoids:

    Ellipsoids have several applications in various fields, including mathematics, physics, and engineering. Some of the applications of ellipsoids are:

    • In geodesy, ellipsoids are used to model the shape of the Earth. The most commonly used ellipsoid for this purpose is the WGS 84 ellipsoid, which is used as the reference surface for GPS measurements.
    • In optics, ellipsoids are used to model the shape of mirrors and lenses. An ellipsoidal mirror is used in some telescopes to focus light.
    • In solid-state physics, ellipsoids are used to describe the shape of the Fermi surface, which is the boundary between the filled and unfilled energy levels in a solid.
    • In mechanics, ellipsoids are used to model the shape of particles in a gas. The collision of two ellipsoidal particles is a classic problem in the kinetic theory of gases.
    • In mathematics, ellipsoids are used to study the properties of quadratic forms. The surface of an ellipsoid is an example of a quadratic surface.

    Examples of Ellipsoids:

    Some examples of ellipsoids are:

    • The Earth is an oblate spheroid, which can be approximated by an ellipsoid with semi-axes of approximately 6,378 km, 6,378 km, and 6,356 km.
    • An ellipsoidal mirror is used in some telescopes to focus light.
    • The shape of the Fermi surface in some metals is approximately ellipsoidal.
    • The collision of two ellipsoidal particles is a classic problem in the kinetic theory of gases.
    • The surface of an ellipsoid can be used to model the shape of a drop of water.
    • The shape of a cell in the human body can be approximated by an ellipsoid.
    • Some crystals have an ellipsoidal shape.
    • The shape of a football is approximately ellipsoidal.
    • The shape of a submarine can be approximated by an ellipsoid.
    • The shape of a planet or moon can be approximated by an ellipsoid.

    FAQ Section:

    Q: What is the difference between an ellipsoid and a sphere? A: A sphere is a special case of an ellipsoid, where all three semi-axes are equal.

    Q: What is the equation of a sphere? A: The equation of a sphere is (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2, where (a,b,c) is the center of the sphere and r is the radius.

    Q: What is the difference between an oblate and a prolate ellipsoid? A: An oblate ellipsoid is one where the equatorial radius (a) is larger than the polar radius (c), while a prolate ellipsoid is one where the polar radius (c) is larger than the equatorial radius (a).

    Q: What is the volume of a sphere? A: The volume of a sphere is 4/3?r^3, where r is the radius of the sphere.

    Q: Can an ellipsoid have negative semi-axes? A: No, the semi-axes of an ellipsoid must be positive.

    1. Quiz:
    2. What is an ellipsoid?
    3. What is the equation of an ellipsoid?
    4. What are the three semi-axes of an ellipsoid?
    5. What is the surface area of an ellipsoid?
    6. What is the volume of an ellipsoid?
    7. What is the distance from the center of an ellipsoid to any point on its surface called?
    8. What are the principal axes of an ellipsoid?
    9. What is the WGS 84 ellipsoid used for?
    10. What is the Fermi surface?
    11. What is the classic problem in the kinetic theory of gases involving ellipsoidal particles?

    Conclusion:

    In conclusion, the ellipsoid is a three-dimensional shape that is defined as a surface obtained by rotating an ellipse about one of its principal axes. The shape of an ellipsoid is described by its three semi-axes, which are the lengths of the radii along the three principal axes of the ellipsoid. Ellipsoids have several applications in various fields, including mathematics, physics, and engineering. Understanding the properties and applications of ellipsoids is important in a wide range of fields.

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    Ellipsoid:

    Example plots

    Example plots

    Equations

    x(u, v) = a cos(u) sin(v) y(u, v) = b sin(u) sin(v) z(u, v) = c cos(v)

    x^2/a^2 + y^2/b^2 + z^2/c^2 = 1

    Surface properties

    2

    g = 0

    S = 2 π (c^2 + b sqrt(a^2 - c^2) E(am(sn^(-1)(sqrt(a^2 - c^2)/a|(a^2 (b^2 - c^2))/(b^2 (a^2 - c^2))), (a^2 (b^2 - c^2))/(b^2 (a^2 - c^2)))|(a^2 (b^2 - c^2))/(b^2 (a^2 - c^2))) + (b c^2 sn^(-1)(sqrt(a^2 - c^2)/a|(a^2 (b^2 - c^2))/(b^2 (a^2 - c^2))))/sqrt(a^2 - c^2))

    ds^2 = sin^2(v) (a^2 sin^2(u) + b^2 cos^2(u)) du^2 + 2 (b^2 - a^2) sin(u) cos(u) sin(v) cos(v) du dv + cos^2(v) (a^2 cos^2(u) + b^2 sin^2(u)) + c^2 sin^2(v) dv^2

    dA = sin(v) sqrt(c^2 sin^2(v) (a^2 sin^2(u) + b^2 cos^2(u)) + a^2 b^2 cos^2(v)) du dv

    x^_ = (0, 0, 0)

    V = 4/3 π a b c

    I = (1/5 (b^2 + c^2) | 0 | 0 0 | 1/5 (a^2 + c^2) | 0 0 | 0 | 1/5 (a^2 + b^2))

    K(u, v) = (a^2 b^2 c^2)/(c^2 sin^2(v) (a^2 sin^2(u) + b^2 cos^2(u)) + a^2 b^2 cos^2(v))^2

    (for an ellipsoid with center at the origin and semi-axes a, b, and c lying along the Cartesian axes)

    Metric properties

    g_(uu) = sin^2(v) (a^2 sin^2(u) + b^2 cos^2(u)) g_(uv) = (b^2 - a^2) sin(u) cos(u) sin(v) cos(v) g_(vu) = (b^2 - a^2) sin(u) cos(u) sin(v) cos(v) g_(vv) = cos^2(v) (a^2 cos^2(u) + b^2 sin^2(u)) + c^2 sin^2(v)

    Γ | u | | | uu = ((a^2 - b^2) sin(u) cos(u))/(a^2 sin^2(u) + b^2 cos^2(u)) Γ | u | | | uv = cot(v) Γ | u | | | vu = cot(v) Γ | u | | | vv = ((a^2 - b^2) sin(u) cos(u))/(a^2 sin^2(u) + b^2 cos^2(u)) Γ | v | | | uu = -(sin(v) cos(v) (a^2 cos^2(u) + b^2 sin^2(u)))/(a^2 cos^2(u) cos^2(v) + b^2 sin^2(u) cos^2(v) + c^2 sin^2(v)) Γ | v | | | uv = ((b^2 - a^2) sin(u) cos(u) cos^2(v))/(a^2 cos^2(u) cos^2(v) + b^2 sin^2(u) cos^2(v) + c^2 sin^2(v)) Γ | v | | | vu = ((b^2 - a^2) sin(u) cos(u) cos^2(v))/(a^2 cos^2(u) cos^2(v) + b^2 sin^2(u) cos^2(v) + c^2 sin^2(v)) Γ | v | | | vv = (sin(v) cos(v) (a^2 (-cos^2(u)) - b^2 sin^2(u) + c^2))/(a^2 cos^2(u) cos^2(v) + b^2 sin^2(u) cos^2(v) + c^2 sin^2(v))

    E(u, v) = sin^2(v) (a^2 sin^2(u) + b^2 cos^2(u)) F(u, v) = (b^2 - a^2) sin(u) cos(u) sin(v) cos(v) G(u, v) = cos^2(v) (a^2 cos^2(u) + b^2 sin^2(u)) + c^2 sin^2(v)

    e(u, v) = (a b c sin^2(v))/sqrt(a^2 b^2 cos^2(v) + a^2 c^2 sin^2(u) sin^2(v) + b^2 c^2 cos^2(u) sin^2(v)) f(u, v) = 0 g(u, v) = (a b c)/sqrt(a^2 b^2 cos^2(v) + a^2 c^2 sin^2(u) sin^2(v) + b^2 c^2 cos^2(u) sin^2(v))

    Vector properties

    left double bracketing bar x(u, v) right double bracketing bar = sqrt(sin^2(v) (a^2 cos^2(u) + b^2 sin^2(u)) + c^2 cos^2(v))

    N^^(u, v) = ((cos(u) sin(v) b c)/sqrt(cos^2(v) a^2 b^2 + sin^2(v) (sin^2(u) a^2 + cos^2(u) b^2) c^2), (sin(u) sin(v) a c)/sqrt(cos^2(v) a^2 b^2 + sin^2(v) (sin^2(u) a^2 + cos^2(u) b^2) c^2), (cos(v) a b)/sqrt(cos^2(v) a^2 b^2 + sin^2(v) (sin^2(u) a^2 + cos^2(u) b^2) c^2))

    Properties

    algebraic surfaces | closed surfaces | quadratic surfaces

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