Spherical Radius Vs Radius
Saurabh Cloudas
As shown in Figure 1, it should be noted that the radius of the spherical end of the cylinder is one-half the diameter of the cylinder, and the tolerance limit for the radius of the spherical end is one-half of that shown for the cylindrical surface.
I have managed to calculate the bounding sphere radius in two ways, but none is giving me exactly what I want. I don't need a "pixel" perfect bounding sphere but I would like something better than what I currently have.
I'm using Wavefront .obj models and to calculate the bounding sphere radius for those models I extract the current model dimensions (I'm using the GLM library from Nate Robbins) which will give me the dimension on each axis.
First approach:Divide each axis by 2 and that will give me the radius on each axis. The largest is the one I'll use for my bounding sphere. This will work for most objects specific to my project. It will not work for some, like cube-shaped ones. Basically, if I have a cube and calculate the radius with this approach, the sphere will leave the cube corners outside.
A better bounding sphere can be found by translating the model points so they're centered on the origin using the bounding box dimensions you already have, then for each individual vertex calculate the radius from the origin for that point using the sqrt(x*x + y*y + z*z) formula. Whichever of those is the largest is the radius of your bounding sphere.
Note that this won't be the optimal bounding sphere. For that you'd have to find the convex hull of your model and use something like rotating calipers to pick the optimal center point for the sphere.
To show it in 2D, the red outline is the bounding box of the shape, and the blue circle is the bounding circle of the box. The improved circle using the polygon vertices, and centered on the box is green. Note that none of the points of the black polygon touch the blue circle.
One easy way would be to use Miniball to calculate the exact bounding sphere of the model. Integrating it into your project is hassle-free, as it only consists of a single header. It is licensed under the GPL however, which could be a problem. Example:
If it's a sphere, then don't you only need to work it out based on one axis? I might be way off here - but by definition, won't a sphere have the same width, height and depth? So radius on one axis=radius on another=radius of another?
Other ways to define and measure the Earth's radius involve either the spheroid's radius of curvature or the actual topography. A few definitions yield values outside the range between the polar radius and equatorial radius because they account for localized effects.
A nominal Earth radius (denoted R E N \displaystyle \mathcal R_\mathrm E ^\mathrm N ) is sometimes used as a unit of measurement in astronomy and geophysics, a conversion factor used when expressing planetary properties as multiples or fractions of a constant terrestrial radius; if the choice between equatorial or polar radii is not explicit, the equatorial radius is to be assumed, as recommended by the International Astronomical Union (IAU).[1]
Earth's rotation, internal density variations, and external tidal forces cause its shape to deviate systematically from a perfect sphere.[a] Local topography increases the variance, resulting in a surface of profound complexity. Our descriptions of Earth's surface must be simpler than reality in order to be tractable. Hence, we create models to approximate characteristics of Earth's surface, generally relying on the simplest model that suits the need.
Each of the models in common use involve some notion of the geometric radius. Strictly speaking, spheres are the only solids to have radii, but broader uses of the term radius are common in many fields, including those dealing with models of Earth. The following is a partial list of models of Earth's surface, ordered from exact to more approximate:
In the case of the geoid and ellipsoids, the fixed distance from any point on the model to the specified center is called "a radius of the Earth" or "the radius of the Earth at that point".[d] It is also common to refer to any mean radius of a spherical model as "the radius of the earth". When considering the Earth's real surface, on the other hand, it is uncommon to refer to a "radius", since there is generally no practical need. Rather, elevation above or below sea level is useful.
Regardless of the model, any of these geocentric radii falls between the polar minimum of about 6,357 km and the equatorial maximum of about 6,378 km (3,950 to 3,963 mi). Hence, the Earth deviates from a perfect sphere by only a third of a percent, which supports the spherical model in most contexts and justifies the term "radius of the Earth". While specific values differ, the concepts in this article generalize to any major planet.
Rotation of a planet causes it to approximate an oblate ellipsoid/spheroid with a bulge at the equator and flattening at the North and South Poles, so that the equatorial radius a is larger than the polar radius b by approximately aq. The oblateness constant q is given by
The variation in density and crustal thickness causes gravity to vary across the surface and in time, so that the mean sea level differs from the ellipsoid. This difference is the geoid height, positive above or outside the ellipsoid, negative below or inside. The geoid height variation is under 110 m (360 ft) on Earth. The geoid height can change abruptly due to earthquakes (such as the Sumatra-Andaman earthquake) or reduction in ice masses (such as Greenland).[5]
Not all deformations originate within the Earth. Gravitational attraction from the Moon or Sun can cause the Earth's surface at a given point to vary by tenths of a meter over a nearly 12-hour period (see Earth tide).
Given local and transient influences on surface height, the values defined below are based on a "general purpose" model, refined as globally precisely as possible within 5 m (16 ft) of reference ellipsoid height, and to within 100 m (330 ft) of mean sea level (neglecting geoid height).
In summary, local variations in terrain prevent defining a single "precise" radius. One can only adopt an idealized model. Since the estimate by Eratosthenes, many models have been created. Historically, these models were based on regional topography, giving the best reference ellipsoid[broken anchor] for the area under survey. As satellite remote sensing and especially the Global Positioning System gained importance, true global models were developed which, while not as accurate for regional work, best approximate the Earth as a whole.
The following radii are derived from the World Geodetic System 1984 (WGS-84) reference ellipsoid.[6] It is an idealized surface, and the Earth measurements used to calculate it have an uncertainty of 2 m in both the equatorial and polar dimensions.[7] Additional discrepancies caused by topographical variation at specific locations can be significant. When identifying the position of an observable location, the use of more precise values for WGS-84 radii may not yield a corresponding improvement in accuracy.[clarification needed]
The value for the equatorial radius is defined to the nearest 0.1 m in WGS-84. The value for the polar radius in this section has been rounded to the nearest 0.1 m, which is expected to be adequate for most uses. Refer to the WGS-84 ellipsoid if a more precise value for its polar radius is needed.
This Earth's prime-vertical radius of curvature, also called the Earth's transverse radius of curvature, is defined perpendicular (orthogonal) to M at geodetic latitude φ[g] and is:[11]
The Earth can be modeled as a sphere in many ways. This section describes the common ways. The various radii derived here use the notation and dimensions noted above for the Earth as derived from the WGS-84 ellipsoid;[6] namely,
Earth's authalic radius (meaning "equal area") is the radius of a hypothetical perfect sphere that has the same surface area as the reference ellipsoid. The IUGG denotes the authalic radius as R2.[2]A closed-form solution exists for a spheroid:[8]
Another global radius is the Earth's rectifying radius, giving a sphere with circumference equal to the perimeter of the ellipse described by any polar cross section of the ellipsoid. This requires an elliptic integral to find, given the polar and equatorial radii:
The mathematical expressions above apply over the surface of the ellipsoid.The cases below considers Earth's topography, above or below a reference ellipsoid.As such, they are topographical geocentric distances, Rt, which depends not only on latitude.
The first published reference to the Earth's size appeared around 350 BC, when Aristotle reported in his book On the Heavens[22] that mathematicians had guessed the circumference of the Earth to be 400,000 stadia. Scholars have interpreted Aristotle's figure to be anywhere from highly accurate[23] to almost double the true value.[24] The first known scientific measurement and calculation of the circumference of the Earth was performed by Eratosthenes in about 240 BC. Estimates of the error of Eratosthenes's measurement range from 0.5% to 17%.[25] For both Aristotle and Eratosthenes, uncertainty in the accuracy of their estimates is due to modern uncertainty over which stadion length they meant.
Around 100 BC, Posidonius of Apamea recomputed Earth's radius, and found it to be close to that by Eratosthenes,[26] but later Strabo incorrectly attributed him a value about 3/4 of the actual size.[27] Claudius Ptolemy around 150 AD gave empirical evidence supporting a spherical Earth,[28] but he accepted the lesser value attributed to Posidonius. His highly influential work, the Almagest,[29] left no doubt among medieval scholars that Earth is spherical, but they were wrong about its size.
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