|
Home
Table of Contents
Index
I. The Space of Oriented Line Segments
We have extensively discussed the space of lines and its role in geometric probability. Now instead of measuring a set of lines intersecting a convex set, consider oriented line segments (needles!).
Let  be a fixed convex set with length  and area  . Let K denote an oriented line segment of length  that lies on a unique line G. Let  be the length of the chord that satsifies  .
|
|
|
 |
|
|
| and |
|
|
|
 |
|
|
|
|
|
|
|
To find the measure of this set of line segments, we find the measure of all line segments K such that  . We want to use the equation  . So
 |
|
|
|
We know that  because
if  , then the line segment would not intersect . |
|
|
 |
|
|
Because the line segments are oriented, there are twice as many;
this accounts for the multiplication
by a factor of two. |
|
|
|
|
|
|
Thus, the measure the set of all oriented line segments K of length that share at least one point with is  .
II. Needles and a Connected Line Segments
Now, suppose that is a series of connected line segments whose total length is  . This is the same idea behind Buffon's Needle Problem, only much more complicated. In this particular case, there is no consistency regarding the position of the line segments of ; this makes methods from first year calculus useless. How do we find the measure of the set of lines that intersect such a region?
Calculate the measure of the set of line segments that cross each line  individually and then compute the sum. We get
where for every position of the line segment K, it intersects n sides of .
What if we want to find the measure of the set of all oriented line segments that are entirely contained in . This problem differs from the one above because the result depends on the shape of . For instance, a triangle, a circle, and a rectangle would all hold a different measure of line segments of the same length.
PROBLEM 21:
Consider an oriented line segment K. Find the measure of a set of such segments that cross both sides of a given angle.
**Hint: Use  , the chord cut by the angle from the line G on which K lies and the area of the triangle AHM.
Solution
|
|
