Problem 10: Densities for Sets of Lines in the Plane
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A straight line G in the plane can be determined by the angle made by the perpendicular to G with a fixed direction (which we will take to be the positive x-axis) and by the distance from the line G to the origin. Thus, a line in the plane can be determined by the coordinates . Also note that these coordinates also represent the polar coordinates of the foot-vector of the perpendicular from the origin to the line.

As we have seen in a previous problem, the equation of the line G is:

(1)
We can define the measure for a subset of lines X by any intergral that takes the following form:
(2)
where the function is chosen based on criterion derived from the nature of the problem. In the theory of geometric probability, the criterion turns out to be that the measure is to be invariant under the group of rigid motions. In other words, the measure of a set of lines in the plane should not change when this set is translated to another part of the plane or when rotated about a point in the plane. For example, when we "randomly" dropped needles onto the floor in Buffon's Needle Problem, "random" meant that regions of equal area were equally likely to have a needle (or the center of mass of a needle) land on them.
What kind of motions are allowed? The motions form a group under the operation of compostion, and the elements of the group are generated by rotations about the origin, translations, and finally reflections by lines through the origin. If you havent had a course in modern algebra yet and don't know what a group is, what we are saying is that any rigid motion can be expressed as a compostion of one or more of the three basic types of motion: rotations (about the origin), translations, and reflections (by lines passing through the origin). [If you would like to use a java applet to make some beautiful patterns of your own by drawing simple objects and then moving them around by certain subgroups of rigid motions, go here.]

First we have rotations about the origin. Since a rotation maps lines to lines, it must be given by a linear transformation. We can write any rotation R about the origin as
    (3)
    where is the angle of rotation, and we are rotating counterclockwise if is positive and clockwise if is negative. Thus the rotation R sends the point (x,y) on the plane to the point .
    We also allow for translations and reflections. Every point (x,y) on the plane is moved by a translation T in the direction of some fixed vector (a,b). In other words,
    (4)
    So the translation T sends the point (x,y) on the plane to the point , where a and b are constants.
    A reflection S about a line through the origin can be described as a linear transformation, since it to maps lines to lines:
    (5)
    where is the angle the line makes with the positive x-axis. Thus the reflection S sends the point (x,y) on the plane to the point .
    And finally, we allow for compostions u of these rotations, translations, and reflections. In general we can represent the rigid motion u by the following transformation (where (x,y) is a point on the plane that is sent by u to the point (x',y')):
    (6)
    To convince you that this is reasonable, let's look at what happens if u is a composition of a rotation through the angle followed by a translation via the vector (a,b). The transformation u sends the xy-coordinate system to a new orthogonal x'y'-coordinate system, and in the process sends the point P(x,y) to the point P'(x',y').

    (7)
    First, the rotation about the origin O through the angle sends the point P(x,y) to the point (indicated in light blue in the figure above). The translation by the vector (a,b) then sends this point to the point
    (8)
    indicated in dark blue in the figure above. This explains why formula (6) is correct.
    If a line G (where G is described by the equation ) undergoes such a rigid motion u, we say that the line G transforms into

    (9)
    where the motion u is defined by the coordinates . Thus, under the motion u, the coordinates are transformed in the following way.
    (10)
    Thus the set X is transformed to the set X' = uX, and the measure for this set is:
    (11)
    Thus, if this measure for the set of lines is to be invariant under the group of motions, then m(X) = m(X'), which implies that
    (12)
    Observe that this equality must hold for all motions u; that is, . Now, if we choose this constant to be 1, then this yields the formal statement of the meaure of a set of lines in the plane:
    The measure of a set of lines G is defined by the integral, over the set, of the differential form dG, (which is called the density of lines):
    (13)
    Problem 10:

    Let D represent any domain in the plane and let F be its area. Let be the length of a chord created by the intersection of a line G and D.
    Integrating over all lines G that intersect D, what is ?

    Solution

      The coordinates are not the only coordinates that can be used to express the space of lines G, and, as a result, is not the only way of expressing the denisty of lines. Suppose that the line G is determined by

        (14)
        where is the angle between the line G and the x-axis, and x is the abscissa (the point where the line G crosses the x-axis). Thus, the density of lines can be shown to equal
        (15)


        Suppose that G is expressed by the coordinates x,y that represent the points where the line G intercepts the x-axis and y-axis, respectively. Thus, we have
        (16)

        Suppose G is represented by the equation (where u,v are the coordinates of G). Thus, we have

        (17)

        Suppose G is represented by the coordinates which denote the foot of the perpendicular line from the origin to G. Then we have , and thus

        (18)

        Suppose is a point on G and is the angle between G and the x-axis. Then and so

        (19)

      References: Santalo; De-lin