Distributions of Visible and Invisible Segments

Abstract

In the present chapter we develop formulae for the distributions of the length of visible and invisible (shadowed) segments of the target curve C in the plane. We will focus attention on trapezoidal fields, in which the strip S is bounded by two parallel lines U and W, and the target curve C is a straight line parallel to S. As before, the distances of U, W, and C from O are u, w and r, respectively, where 0 < u < w < r. We will assume that the Poisson field is standard and the distribution of the radius of a random disk centered in S is uniform on (a, b), and 2b < u. The trapezoidal region is a subset of S, C*, bounded by U, W and the rays ${_{x_L^ * }}$ and where ${_{x_U^ * }}$, where x*_L and x*_U are the rectangular coordinates of points on C specified below. Let $\bar C$ be an interval on C of interest. The rectangular x-coordinates of the points on $\bar C$ are bounded by x _L and x _U , x _L > x _U , and, as before,

$$x_L^* = {x_L} - b\frac{r}{u}{(1 + {(\frac{{{x_L}}}{r})^2})^{1/2}}$$

$$x_U^* = {x_U} - b\frac{r}{u}{(1 + {(\frac{{{x_U}}}{r})^2})^{1/2}}$$

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