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A sphere (top), rotational ellipsoid (left) and triaxial ellipsoid (right) The volume of a sphere of radius R is 4 3 π R 3 {\displaystyle {\frac {4}{3}}\pi R^{3}} . Given the volume of a non-spherical object V , one can calculate its volume-equivalent radius by setting
Strictly speaking, the laser diffraction equivalent diameter is the diameter of a sphere yielding, on the same detector geometry, the same diffraction pattern as the particle. In the size regimen where the Fraunhofer approximation is valid, this diameter corresponds to the projected area diameter of the particle in random orientation. For ...
A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v are also shown. The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path ...
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system—shown here in the mathematics convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored ...
In all methods the size is an indirect measure, obtained by a model that transforms, in abstract way, the real particle shape into a simple and standardized shape, like a sphere (the most usual) or a cuboid (when minimum bounding box is used), where the size parameter (ex. diameter of sphere) makes sense.
If the sphere is isometrically embedded in Euclidean space, the sphere's intersection with a plane is a circle, which can be interpreted extrinsically to the sphere as a Euclidean circle: a locus of points in the plane at a constant Euclidean distance (the extrinsic radius) from a point in the plane (the extrinsic center). A great circle lies ...
The great circle g (green) lies in a plane through the sphere's center O (black). The perpendicular line a (purple) through the center is called the axis of g, and its two intersections with the sphere, P and P ' (red), are the poles of g. Any great circle s (blue) through the poles is secondary to g. A great circle divides the sphere in two ...
The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it.