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The geometric-distance matrix of a molecular structure G is a real symmetric n x n matrix defined in the same way as a 2-D matrix. However, the matrix elements D ij will hold a collection of shortest Cartesian distances between i and j in G. Also known as topographic matrix, the geometric-distance matrix can be constructed from the known ...
Euclidean distance matrix. In mathematics, a Euclidean distance matrix is an n×n matrix representing the spacing of a set of n points in Euclidean space . For points in k -dimensional space ℝk, the elements of their Euclidean distance matrix A are given by squares of distances between them. That is. where denotes the Euclidean norm on ℝk .
The Euclidean distance is the prototypical example of the distance in a metric space, [10] and obeys all the defining properties of a metric space: [11] It is symmetric, meaning that for all points and , (,) = (,). That is (unlike road distance with one-way streets) the distance between two points does not depend on which of the two points is ...
In mathematics, an isometry (or congruence, or congruent transformation) is a distance -preserving transformation between metric spaces, usually assumed to be bijective. [ a] The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning "measure". If the transformation is from a metric ...
Distance geometry. Distance geometry is the branch of mathematics concerned with characterizing and studying sets of points based only on given values of the distances between pairs of points. [1] [2] [3] More abstractly, it is the study of semimetric spaces and the isometric transformations between them. In this view, it can be considered as a ...
The distance (or perpendicular distance) from a point to a line is the shortest distance from a fixed point to any point on a fixed infinite line in Euclidean geometry. It is the length of the line segment which joins the point to the line and is perpendicular to the line. The formula for calculating it can be derived and expressed in several ways.
As one example, if there is free space between the two planes, the ray transfer matrix is given by: = [], where d is the separation distance (measured along the optical axis) between the two reference planes.
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is ...