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: distance from the origin of the line u {\displaystyle \mathbf {u} } : direction of line (a non-zero vector) Searching for points that are on the line and on the sphere means combining the equations and solving for d {\displaystyle d} , involving the dot product of vectors:
Because in a continuous function, the function for a sphere is the function for a circle with the radius dependent on z (or whatever the third variable is), it stands to reason that the algorithm for a discrete sphere would also rely on the midpoint circle algorithm. But when looking at a sphere, the integer radius of some adjacent circles is ...
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 between the two points on the surface of the sphere. (By comparison, the shortest path passing through the sphere's interior is the chord between ...
the azimuthal angle φ, which is the angle of rotation of the radial line around the polar axis. [b] (See graphic regarding the "physics convention".) Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates.
Stereographic projection of the unit sphere from the north pole onto the plane z = 0, shown here in cross section. The unit sphere S 2 in three-dimensional space R 3 is the set of points (x, y, z) such that x 2 + y 2 + z 2 = 1.
Mie Scattering from a sphere. x is the wave number times the sphere's radius and m is the refractive index of the sphere divided by the refractive index of the medium. In electromagnetism , the Mie solution to Maxwell's equations (also known as the Lorenz–Mie solution , the Lorenz–Mie–Debye solution or Mie scattering ) describes the ...
The Laplace spherical harmonics can be visualized by considering their "nodal lines", that is, the set of points on the sphere where [] =, or alternatively where [] =. Nodal lines of Y ℓ m {\displaystyle Y_{\ell }^{m}} are composed of ℓ circles: there are | m | circles along longitudes and ℓ −| m | circles along latitudes.
a 0-sphere is a pair of points {, +} , and is the boundary of a line segment ( -ball). a 1-sphere is a circle of radius centered at , and is the boundary of a disk ( -ball).