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Coplanarity. In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. However, a set of four or more distinct points will, in general, not lie in a single plane.
Collinearity equation. Light beams passing through the pinhole of a pinhole camera. The collinearity equations are a set of two equations, used in photogrammetry and computer stereo vision, to relate coordinates in a sensor plane (in two dimensions) to object coordinates (in three dimensions). The equations originate from the central projection ...
Specifying the coordinates (components) of vectors of this basis in its current (rotated) position, in terms of the reference (non-rotated) coordinate axes, will completely describe the rotation. The three unit vectors, û , v̂ and ŵ , that form the rotated basis each consist of 3 coordinates, yielding a total of 9 parameters.
A system of n quantities that transform oppositely to the coordinates is then a covariant vector (or covector). This formulation of contravariance and covariance is often more natural in applications in which there is a coordinate space (a manifold) on which vectors live as tangent vectors or cotangent vectors.
Harmonic coordinates. In Riemannian geometry, a branch of mathematics, harmonic coordinates are a certain kind of coordinate chart on a smooth manifold, determined by a Riemannian metric on the manifold. They are useful in many problems of geometric analysis due to their regularity properties. In two dimensions, certain harmonic coordinates ...
Prolate spheroidal wave function. The prolate spheroidal wave functions are eigenfunctions of the Laplacian in prolate spheroidal coordinates, adapted to boundary conditions on certain ellipsoids of revolution (an ellipse rotated around its long axis, “cigar shape“). Related are the oblate spheroidal wave functions (“pancake shaped ...
Rastrigin function. In mathematical optimization, the Rastrigin function is a non- convex function used as a performance test problem for optimization algorithms. It is a typical example of non-linear multimodal function. It was first proposed in 1974 by Rastrigin [1] as a 2-dimensional function and has been generalized by Rudolph. [2]
The azimuthal angle is denoted by. φ ∈ [ 0 , 2 π ] {\displaystyle \varphi \in [0,2\pi ]} : it is the angle between the x -axis and the projection of the radial vector onto the xy -plane. The function atan2 (y, x) can be used instead of the mathematical function arctan (y/x) owing to its domain and image. The classical arctan function has an ...