Probability of intersection of two geometrical figures in bounded space?I'd like to find a closed form (if possible) expression of the probability of interesection of two geometrical figures F 1 and F 2 of area A 1 and A 2 , respectively, that are have a random position and orientation in a bounded 2-dimensional space of area A t o t .Obviously, this probability depends on the exact geometry of F 1 , F 2 , and the space in which they live. However, is there a closed form expression of this probability for some classes of geometries or shall I go for Monte-Carlo methods?

Question

Probability of intersection of two geometrical figures in bounded space?I&#039;d like to find a closed form (if possible) expression of the probability of interesection of two geometrical figures       F    1   and       F    2   of area       A    1   and       A    2  , respectively, that are have a random position and orientation in a bounded 2-dimensional space of area       A          t      o      t      .Obviously, this probability depends on the exact geometry of       F    1  ,       F    2  , and the space in which they live. However, is there a closed form expression of this probability for some classes of geometries or shall I go for Monte-Carlo methods?

Royce Golden · Accepted Answer

Step 1Yes, the probability depends on the exact geometry of the figures and the space in which they live. Also, you should specify whether you get to rotate the figures or jut translate them.Step 2For example, if the total space is a square of side length s, and the figures are unit squares, and you don&#039;t allow rotations, then most squares are away from the boundary, and the chance of an intersection is roughly   4      /        s    2  . There is a closed form which is slightly more complicated. However, if the figures are shaped like giant Xs then they may have the same unit area but there may be no way to place them without overlapping.

Answered question

Answer & Explanation

New Questions in High school geometry