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Calculate a cell index using comparisons of the contour level(s) with the data values at the cell corners. Use a pre-built lookup table, keyed on the cell index, to describe the output geometry for the cell. Apply linear interpolation along the boundaries of the cell to calculate the exact contour position.
A sphere formed using the Chebyshev distance as a metric is a cube with each face perpendicular to one of the coordinate axes, but a sphere formed using Manhattan distance is an octahedron: these are dual polyhedra, but among cubes, only the square (and 1-dimensional line segment) are self-dual polytopes.
A metric on a set X is a function (called the distance function or simply distance) d : X × X → R + (where R + is the set of non-negative real numbers). For all x, y, z in X, this function is required to satisfy the following conditions: d(x, y) ≥ 0 (non-negativity) d(x, y) = 0 if and only if x = y (identity of indiscernibles.
Sometimes the factor / in front of the integral is omitted, in which case the Hellinger distance ranges from zero to the square root of two. The Hellinger distance is related to the Bhattacharyya coefficient B C ( P , Q ) {\displaystyle BC(P,Q)} as it can be defined as
In practical applications, distances are noisy measurements or come from arbitrary dissimilarity estimates (not necessarily metric). The goal may be to visualize such data by points in Euclidean space whose distance matrix approximates a given dissimilarity matrix as well as possible — this is known as multidimensional scaling .
Namely, let X and Y be two compact figures in a metric space M (usually a Euclidean space); then D H (X, Y) is the infimum of d H (I(X), Y) among all isometries I of the metric space M to itself. This distance measures how far the shapes X and Y are from being isometric.
For example, if a station reports that the 500 mb [i.e. millibar] height at its location is 5600 m, it means that the level of the atmosphere over that station at which the atmospheric pressure is 500 mb is 5600 meters above sea level. This is an estimated height based on temperature and pressure data."
M 2 is useful because it reflects how well a collimated laser beam can be focused to a small spot, or how well a divergent laser source can be collimated. It is a better guide to beam quality than Gaussian appearance because there are many cases in which a beam can look Gaussian, yet have an M 2 value far from unity. [1]