Distance between two points

A two-dimensional slice may be taken either parallel to one of the principal co-ordinate planes (X-Y, X-Z and Y-Z) selected from a menu, or in any arbitrary orientation defined on screen by the user. Once a slice through the data has been taken, and displayed on the screen, a number of tools are available to assist the operator with making measurements of indications. These tools allow measurement of distance between two points, calculation of 6dB or maximum amplitude length of a flaw, plotting of a 6dB contour, and textual aimotation of the view. Figure 11 shows 6dB sizing and annotation applied to a lack of fusion example. 

FIG. 2 Side view of film confined between a sphere of macroscopic radius R and a planar substrate surface. The shortest distance between two points located on the surface of the sphere and of the substrate, respectively, is denoted by h (from Ref. 48). 

Sack, R.A., 1964, Two-center expansion for the powers of the distance between two points, J. Math. Phys. 5 260. 

In contrast to SOMs, nonlinear maps (NLMs) represent relative distances between all pairs of compounds in the descriptor space of a 2D map. The distance between two points on the map directly reflects the similarity of the... 

It has been shown in Chapter 29 that the set of vectors of the same dimension defines a multidimensional space S in which the vectors can be represented as points (or as directed line segments). If this space is equipped with a weighted metric defined by W, it will be denoted by the symbol S. The squared weighted distance between two points representing the vectors x and y in is defined by the weighted scalar product ... 

The distance between two points on a coordinate line is the line element... 

The numerical values of most physical quantities are expressed in terms of units. The distance between two points, for example, can be specified by the number of meters (or feet, Angstroms, etc.). Similarly, time cap be expressed in seconds, days or, say, years. However, the number of days per year varies from one year to another. The quantities, distance (length) and time, as well as mass, are usually chosen to be primary quantities. In terms of them Newton s second law for the force on an object, can be written as force = mass, distance/(time)2. The definition of the primary quantities allows dimensional expressions to be written, such as [force] — MLT-2 in the present example. Note, however, that in everyday life one speaks of the weight of an object (or a person). Of course the weight is not the mass, but rather the force acting on the object by the acceleration due to gravity [acceleration] = LT 2. 

Note This relationship holds even when x1 or y1 or both are negative (also shown in Figure 11-1). In three dimensions (x, y, z), we describe three lines at right angles to one another, designated as the x, y, z axes. Three planes are represented as xy, yz, and zx, and the distance between two points (x , y1 z ) and (x2, y2, z2) is given by... 

Figure 11-1 The distance between two points in a two-dimensional coordinate space is determined using the Pythagorean theorem.

The power of a microscope to reveal details depends less on the magnification and more on the clarity or sharpness of the image produced by the objective. A simple definition of the resolving power of a microscope is the smallest distance between two points in the object such that the two points can be distinguished in the image. 

Again, V has no local maxima or minima and all extrema occur at the boundaries. Geometrically, just as the straight line is the shortest distance between two points, so a harmonic function in two dimensions minimizes the surface area fitted to the given boundary line. 

The square of the distance between two points with position distributed normally is distributed as the x2 distribution with one degree of freedom. The sum of n such distances is distributed as the x2 distribution with n degrees of freedom. The ratio of squared distances, which is used for instance to test the ratio of variances or the ratio of a squared distance to a variance, is distributed as an F-distribution. 

If the distance between two points is less than the range, then the value at one point is correlated with the value at the other point. If the distance between two points is greater than the range, then the points are independent. The sill is the bound on the semi-variogram and provides an estimate of the overall variability. When a semi-variogram is bounded then the random function is second order stationary and... 

In order to apply the SA protocol, one of the keys is to design a mathematical function that adequately measures the diversity of a subset of selected molecules. Because each molecule is represented by molecular descriptors, geometrically it is mapped to a point in a multidimensional space. The distance between two points, such as Euclidean distance, Tanimoto distance, and Mahalanobis distance, then measures the dissimilarity between any two molecules. Thus, the diversity function to be designed should be based on all pairwise distances between molecules in the subset. One of the functions is as follows ... 

The formula for the distance between two points on a graph uses the x and the y coordinates of the points. You find the difference between the pairs of coordinates, square the differences, and add them together. Then take the square root of the sum. The distance is in terms of units on the graph. 

In the consideration of the statistical aspects of turbulence it was found to be of utility (B6, K4, Rl) to establish the correlation in time (B6) of the velocity vectors as a function of the distance between two points in the turbulent stream. The correlation coefficient is defined by... 

To fine the distance between two points, use this variation of the Pythagorean theorem ... 

To find the structures of the objects in the data set, we need a measure of similarity. Although many types of measures can be applied, the Euclidean distance is the most frequently used similarity measure. According to the law of Pythagoras, the distance between two points Oj and 02 characterized by variables x and y can be presented as follows (Figure 15.1) ... 

In GR, space changes with time, so that the intuitive distance between two points changes with time. Therefore the various methods to measure the distance from the source to a point give different answers. See, for instance, the "Table 1 distances in cosmology" (Melchiorri et al., 2003). The proper distance - between the source and the observer - can be seen as a distance measured by a set of rulers. The distance element is given by ... 

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