Key Properties and Real-Life Applications of Spheroids
In-plane geometry, a spheroid shape, or ellipsoid of revolution, refers to a quadric surface obtained by rotating an ellipse about one of its principal axes. This is to say, a spheroid is an ellipsoid with 2 equal semi-diameters with a circular symmetry.
If the ellipse is revolved about its major axis, the outcome is a prolate (elongated) spheroid, shaped like a rugby ball. An oblate or flattened spheroid is an outcome of a rotating ellipse around its minor axis. If the producing ellipse is a circle, the outcome is a sphere.
Applications of Spheroid
The neutrons and protons of an atomic nucleus are mostly found in the spherical, oblate, or prolate spherical shapes along with having a spinning axis. Deformed nuclear shapes typically form as an outcome of the competition between electromagnetic repulsion between protons, quantum shell effects, and surface tension.
Oblate Spheroids Meaning
A ''squashed'' Spheroid of which the equatorial radius ‘a’ is larger than the polar radius ‘c’, such that a > c. To a first approximation, the shape assumed by a rotating fluid (inclusive of the Earth, that is ``fluid'' over astronomical time scales) is known as an oblate spheroid. The oblate spheroid can be identified parametrically by the general Spheroid equations (for a Spheroid having z-Axis as the symmetry axis).
Oblate Spheroid Shape
The oblate spheroid shape can be defined as an assumed shape for various celestial bodies and planets that are rotating rapidly at their own axis. For example, the Earth, Saturn, and Jupiter. The English mathematician, Enlightenment scientist, and astronomer –Isaac Newton reasoned that Earth and Jupiter are oblate spheroids owing to their centrifugal force. Earth's diverse geodetic and cartographic systems are formed on reference ellipsoids, all of which are oblate.
Example of Oblate Spheroids
An example of an oblate spheroid is the planet Jupiter with a flattening spheroid of 0.06487. Another science-fiction of an extremely oblate spheroid earth is Mesklin from Hal Clement's novel Mission of Gravity.
Prolate Spheroids
The prolate spheroid is an estimated shape of the ball in various sports, such as rugby football. The word is also used to depict the shape of some nebulae like the Crab Nebula.
Several moons of the Solar System estimate prolate spheroids in shape, although they are really triaxial ellipsoids.
On contrary to being distorted into oblate spheroids through quick rotation, celestial objects distort little into prolate spheroids through tidal forces when they orbit a massive body in a close orbit. An utmost example is Jupiter's moon Io, which becomes slightly more or less prolate in its orbit because of little eccentricity, inducing intense volcanism. However, the major axis of a prolate spheroid does not run across the satellite's poles, but through the two points on its equator straightaway facing toward and away from the primary.
Example of a Prolate Spheroids
An Australian rules football is an example of a prolate spheroid.
Many submarines have a shape that can be depicted as a prolate spheroid.
Another example includes the atomic nucleus of the element belonging to the lanthanide and actinide groups.
In anatomy, near-spheroid organs such as testis.
Fresnel zones used to evaluate wave propagation and interference in space.
Other Examples of prolates include Saturn's satellites Enceladus, Mimas, and Uranus' and Tethys satellite Miranda.
Geoid and Spheroid
The geoid is essentially described as the surface of the earth's gravity field that approximates mean sea level. The geoid is perpendicular to the direction of the force of gravity. Because the mass of the Earth is not uniform in nature at all points, the magnitude of gravity thus differs, and the shape of the geoid is irregular.
Fun Facts
Saturn is considered to be the most oblate planet in the Solar System. It has a flattening of 0.09796.
FAQs on What Is a Spheroid? Explained with Examples
1. What is a spheroid in mathematics?
A spheroid is a three-dimensional shape created by rotating an ellipse about one of its principal axes. It is also known as an ellipsoid of revolution. Because it is derived from an ellipse, it is not perfectly spherical but is either flattened or elongated along its axis of rotation, depending on which axis is used for the revolution.
2. How is a spheroid formed from an ellipse?
A spheroid is generated by taking a two-dimensional ellipse and rotating it 360 degrees around one of its axes. The type of spheroid depends on the axis of rotation:
- If the ellipse is rotated around its minor axis (the shorter axis), the result is an oblate spheroid, which looks like a squashed sphere.
- If the ellipse is rotated around its major axis (the longer axis), the result is a prolate spheroid, which looks like an elongated sphere.
3. What is the difference between an oblate and a prolate spheroid?
The main difference lies in their shape, which is determined by the axis of rotation of the parent ellipse. An oblate spheroid is formed by rotating an ellipse around its shorter, minor axis, resulting in a shape that is flattened at the poles and bulges at the equator (like the Earth). In contrast, a prolate spheroid is formed by rotating an ellipse around its longer, major axis, resulting in an elongated, rugby-ball-like shape.
4. How does a spheroid differ from an ellipsoid?
A spheroid is a specific type of ellipsoid. An ellipsoid is a general 3D surface where all three of its semi-axes (a, b, c) can have different lengths. A spheroid, or ellipsoid of revolution, is a special case where exactly two of its three semi-axes are equal (e.g., a = b ≠ c). Therefore, all spheroids are ellipsoids, but not all ellipsoids are spheroids. A sphere is an even more specific case where all three semi-axes are equal (a = b = c).
5. Why is the Earth best described as an oblate spheroid?
The Earth is not a perfect sphere due to its rotation. The centrifugal force generated by its daily spin is strongest at the equator, causing the planet to bulge outwards. This results in an equatorial diameter that is larger than its polar diameter. This specific shape, flattened at the poles and bulging at the equator, is accurately modelled as an oblate spheroid. This model is critical for precision in GPS, cartography, and satellite orbits.
6. What is the standard equation for a spheroid?
The standard equation of a spheroid centered at the origin, with its axis of revolution along the z-axis, is given by:
x²/a² + y²/a² + z²/c² = 1
In this equation, 'a' represents the equatorial radius and 'c' represents the polar radius.
- If a > c, the equation describes an oblate (flattened) spheroid.
- If c > a, the equation describes a prolate (elongated) spheroid.
7. How is the volume of a spheroid calculated?
The volume of a spheroid is calculated using a formula that incorporates its two unique semi-axes, 'a' and 'c'. The formula is a direct application of integral calculus for solids of revolution:
V = (4/3)πa²c
Here, 'a' is the radius perpendicular to the axis of revolution (the equatorial radius), and 'c' is the semi-axis along the axis of revolution (the polar radius). This single formula works for both oblate and prolate spheroids.
8. Besides planets, what are other real-world examples of spheroid shapes?
Spheroid shapes appear in various natural and man-made objects. Some common examples include:
- Prolate Spheroids: The shape of a rugby ball, an American football, or some types of small nuts.
- Oblate Spheroids: The shape of a lentil, a red blood cell (biconcave but often simplified as oblate), or the overall shape of some spiral galaxies.
