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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. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ).
Vectors are defined in cylindrical coordinates by (ρ, φ, z), where ρ is the length of the vector projected onto the xy-plane, φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by:
The black sphere represents the intersection point of the three surfaces, which has Cartesian coordinates of roughly (0.831, −1.439, 2.182). Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e ...
Figure 1: Coordinate isosurfaces for a point P (shown as a black sphere) in oblate spheroidal coordinates (μ, ν, φ). The z-axis is vertical, and the foci are at ±2. The red oblate spheroid (flattened sphere) corresponds to μ = 1, whereas the blue half-hyperboloid corresponds to ν = 45°.
A point with angles {θ 0, φ 0}, rotated by an angle φ about the z-axis, becomes the point with angles {θ 0, φ 0 + φ}. While hyperspherical coordinates are also useful in dealing with 4D rotations, an even more useful coordinate system for 4D is provided by Hopf coordinates { ξ 1 , η , ξ 2 } , [ 6 ] which are a set of three angular ...
The following is a list of centroids of various two-dimensional and three-dimensional objects. The centroid of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane.
In crystalline FeSO 4. 7H 2 O, the first coordination sphere of Fe 2+ consists of six water ligands. The second coordination sphere consists of a water of crystallization and sulfate, which interact with the [Fe(H 2 O) 6] 2+ centers. Metal ions can be described as consisting of series of two concentric coordination spheres, the first and second.
Consider three unit vectors OA →, OB →, OC → drawn from the origin to the vertices of the triangle (on the unit sphere). The arc BC subtends an angle of magnitude a at the centre and therefore OB → · OC → = cos a. Introduce a Cartesian basis with OA → along the z-axis and OB → in the xz-plane making an angle c with the z-axis.