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In contrast to F-histogram which treats longitudinal segments in a number of directions during the range [0, 2π], R-histogram treats the longitudinal segments of shapes with only one fixed direction; thereby the computational complexity is significantly reduced.
For a pair of two shapes, the shapes are treated as the longitudinal segments parallel to the line connecting centroids of the two shapes, and the R-histogram is composed of the length ratios of collinear longitudinal segments.
The shapes are handled as the longitudinal segments parallel to the line connecting centroids of two shapes, and the R-histogram is constructed by the length ratios of collinear longitudinal segments from two shapes.
Then, the R-histogram is composed of the length ratios of collinear longitudinal segments from the two shapes.
The specimens were strengthened with external composite reinforcement – longitudinal segments of CFRP strips and transverse confinement executed with CFRP sheets.
Different from the histogram of angles treating the 2D object as a set of points, F-histogram method handles the shapes as longitudinal segments.
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It was decided that the ideal longitudinal segment for this study was the one that transected Paris on its way between the northern and southern ends of France — a choice that later impeded international acceptance of the system but made glorious sense to Parisians at the time.
Similarly, E B v E B ∩ l v stands for a longitudinal segment of shape B. Figure 1 Illustrating the R-histogram.
A longitudinal segment may be the union of a finite number of disjoint segments, e.g., in (a) the longitudinal segment E A v N − 1 = E A ∩ l v N−− 1 = I ∪ J ∪ K, and in (b) E A v 1 = E A ∩ l v 1== I ∪ J ∪ K.
For any real number v, the intersection E A ∩ l v, denoted by E A v, is a longitudinal segment of A. In this article, as the term "shape" denotes a 2D plane which may have holes in it or may consist of many connected components, E A v e.g., E A v N − 1 in Figure1a is the union of a finite number of disjoint segments.
In Equation (1), L denotes the length of the longitudinal segment ξ, N is the number of v that are sampled equidistantly in the range of v min, v max, v min and v max are the minimum and maximum intercepts of the valid lines l v) which have nonempty intersections with both A and B, i.e., v min = max v min A, v min B v max = min v max A, v max B. (2).
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