Random Secants and Rays of Some Convex Geometrical Shapes

Abstract

<p>This thesis is devoted to investigating certain regular features of random secants and rays of some convex configurations. The randomness of a secant of a convex configuration does not arise uniquely. Several mechanisms under which the randomness of straight line paths through convex bodies commonly arise are discussed and the problem in geometrical probability of the distributions of lengths of random secants has been solved by a general and direct geometrical method based on geometrical arguments for the following configurations and types of randomness: (i) the general triangle, the rectangle, regular polygons, the circle and the sphere under S₁-randomness: (ii) regular polygons under S₂-randomness, (iii) the circle under I₁-randomness. In both (i) and (ii), it is shown that if the polygon Pm of M sides is inscribed in a circle of constant radius r for [missing text] to the same distribution function F(ℓ) of the random secant length L of the circle under S₁- and S₂-randomness. Since the distributions of the lengths of random secants of a circle under S₁-randomness and S₂-randomness are the same, we have here a remarkable instance of two different sequences of distribution functions Fˢ₁M(ℓ) and Fˢ₂M(ℓ), N - 3,4,,,,, (which are "paradoxical" according to Bertrand) of the randow secant length L of a polygon under S₁-randomness and S₂-randomness, converging to the same (hence, non paradoxical) distribution function F(ℓ) of the random secant length L of a circle. The probability laws of the random lengths of rays emanating in a random direction from random sources within the rectangle, circle and sphere have been formulated.</p>