- Two immediate consequences of the density of lines are discussed below.
- Consequence #1
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- Let K be a bounded convex set in the plane having perimeter L. Consider the integral over the set of lines intersecting K,
 , with the density of lines  . Then we have
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- Consequence #1 can interpreted as saying that the "amount" of lines that intersect K is equal to the perimter of K. From this equation, we obtain a nice result inherent to geometric probability.
- Consequence #2
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- Let K be a convex set in the plane having area F and let
 be the length of the chord of G intersecting K. Then we have the Crofton's formula as shown in solution for Problem 10:
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- Using this formula and Consequence #1, the average length of the chord
 is given by:
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Let  and  be two convex sets in the plane having perimeters  and  , respectively, and suppose that  . What is the probability that a random line G which intersects also intersects ? Recall that our notion of probability in this course is the measure of the set of all successful outcomes, called the event space, divided by the measure of the set of all possible outcomes, called the sample space . In this problem, the sample space is all the lines that intersect , because we are only concerned with those lines which first intersect and then also intersect ; thus, all the lines that intersect compose the event space. Thus, from Consequence #1 we get that the probability that a line which intersects and also intersects is equal to  .
Suppose C is a curve that is piecewise smooth, has a finite length, and is parametrized by
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(1) |
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where s is the arc length parameter of the curve C. Let the coordinate pair (x,y) be the intersection point of G and C, and let  be the angle between the tangent line of C at (x,y) and G. Thus, the line G can be determined by the coordinate pair  . Let  denote the angle between the tangent to C at (x,y) and the x-axis. Thus, we get the following identity:
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(2) |
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- Thus, when we take the total differential of
 , we get
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(3) |
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- Observe that we can also differentiate and to get the identities
 and  . Using these identities along with those from trigonometry, we reach the following:
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(4) |
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- Moreover, when taking the total differential of equation (2), we get
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Note that  is written in terms of absolute value because it is an expression of density. If we integrate equation (5) over all the lines that intersect our curve C, we get
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where n is the number of times each line G intersects with the curve C.
| Problem 11:
Let AB be a convex curve whose length is denoted by L. Let the line segment created by A and B have length C. Finally, let K be the convex set created by the curve and the line segment AB. Find the probability that a random line intersecting K also intersects the line segment AB.
Hint: See Conditional Probability described in Problem 3
Solution
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