Geometric Probability in Undergraduate Study and Applications

Molly McGinty
Professor Alan McRae 

In the Space-Time Plane

      Undergraduate students do not generally encounter geometric probability, as there are no texts written for this level of study.  Although the only prerequisite for this topic is a firm understanding of multivariable calculus and some linear algebra, it is an entry-level graduate course.  After thoroughly researching and studying geometric probability, I wrote a website designed as an online course especially geared for undergraduates.  In addition, I researched the applications of geometric probability in the space-time geometry, finding results analogous to those in the Euclidean plane. 

     According to Eric Weisstein’s World of Mathematics, geometric probability is the “study of the probabilities involved in geometric problems, e.g., the distributions of length, area, volume, etc. for geometric objects under stated conditions”.  In other words if objects, such as lines or points, are “randomly” thrown at some target geometric space, its “measure” may be found.  There are several ways to define the measure of the target set; the one-dimensional measure, perimeter, and the two-dimensional measure, area, are among them.  These topics form the introductory section of the website.

      The first four problems outline the basic theory behind probability and the elementary background behind geometric probability.  There are three classic problems that directly apply the concrete theories behind probability.  They are Bertrand’s Problem, Buffon’s Coin Problem, and Buffon’s Needle Problem, which illustrate the idea of randomness and how to find the ratio of successful outcomes to possible outcomes.  The lectures for these problems each include the history of the problem, explanations, illustrations, and solutions.  There are also applets that allow students to see the graphic results of several trials of the problem.  At the end of each lecture, there is a series of exercises for the students to complete and a link to a solutions page.  These solutions are detailed and require skillful use of calculus. 

      The second section is an introduction to convex sets and the application of geometric probability to find the perimeter and area of such sets. This portion of the course is largely definitive and derives the fundamental formulas essential to the remainder of the course.

      The third section focuses on applications of geometric probability in the Euclidean plane.  Rather than just lines and points, the “measure” of convex set is calculated using rectangles, chords and other convex sets.  This topic lends itself to the use of several illustrations and animations in which often times the student may participate. Formulas and ideas that were outlined in the introductory sections are revisited, generalized and proven.  The exercises at the end often ask for the proof of results given in the lecture or a derivation of a formula that will be useful in later problems.

      Finally, the concluding section focuses the rigid motions of a set, densities, and their relation to measure and probability.  There are twenty-three lectures in total, many of them with additional accessories that include an index and links to multivariable calculus review pages as well as links to other helpful math-related websites.

      Techniques from geometric probability can be applied to geometries outside of Euclidean space, such as space-time geometry or more specifically the space-time plane.  In 2-dimensional space-time geometry, the horizontal axis represents time, t, and the vertical axis represents spatial position, x.  Points on the space-time plane are called events.  The origin, called the Here-Now represents the original position of a specific event at time zero.  By choosing a Here-Now, we can put a coordinate system on the space-time plane.  The two lines t = x and t = -x are called light-lines.  The shaded areas are called light-cones, and only events that affect or are affected by the Here-Now fall in this region. A curve in the space-time plane is called a worldline.  If scalar values are chosen for t and x so that c = 1, where c is the speed of light in a vacuum, then the light-lines represent the worldline of photons passing through the Here-Now.

     A line with slope m : | m | > 1, is called a space-like line.  A line whose slope is m : | m | <1, is a time-like line.  Those lines whose slope satisfies m : | m | = 1 are light-like.  A smooth curve is considered space-like, time-like, or light-like if all its tangent lines are respectively space-like, time-like, or light-like.  The dot product and thus the distance formula are defined differently for the space-time plane.  The distance d between two points (t1, x1) and (t2, x2) is defined as 

The vector perpendicular to (t, x) is (x, t), since their dot product is zero.  The inner product yields no information about the angle between a time-like vector and a space-like vector.

     In Lorenztian space-time, given a chosen event (say the Here-Now) all events equidistant from the Here-Now do not yield a circle as they do in the Euclidean plane, but rather a hyperbola.  Every event (t , x), except those that lie on a light-line, lies on a  unique hyperbola, where (t, x) = (cosh ?, sinh ??).   All line segments from the Here-Now to a point on the hyperbola t2 – x2 = d2 have length d or id, where d > 0.  Since a vector that falls on a light-line does not intersect a hyperbola of this  form, it has length zero.   It follows that a light-like vector cannot form an angle with another vector.  A consequence of the theory of special relativity is that the length of the worldline of a traveler is equal to the time elapsed from the perspective of the traveler.  We will consider distances in the space-time plane to have real values by taking the modulus:

     Buffon’s Needle Problem has an analogue for the Lorenztian space-time plane.  Without loss of generality, assume the equidistant parallel lines are vertical, space-like and distance d apart.  Suppose that the needle of length L is time-like and one end of the needle the Here-Now.  The probability that a randomly dropped needle will intersect a line is 1, no matter how short the needle or how far apart the equidistant parallel lines are.  Although the length of needle does not change, it crosses infinitely many parallel lines. 

     It is possible, with probability 0, that a needle does not cross a line at all.  This implies that L < d.  Let t be the distance from the left-most part of the needle, which is the Here-Now, to the nearest line to the right.  Define ?  to be the angle between the time axis and the needle.  Then the ordered pair (t,??  ) uniquely determines the position of the needle.  Let ?1 denote the smallest positive angle at which a needle of length L must intersect at least one line.

The probability that the needle will not intersect a line is given by

Length, L, and area, F, in the space-time plane can also be computed using geometric probability.  To find the length of a curve lying in the future cone of some Here-Now, let (t(s) , x(s)) be a unit-speed parameterization of the curve and let (t’(s), x’(s)) be the velocity vector at any point of the curve. The length of the red curve is given by

where ? is the angle between the vector (t, x) and its perpendicular vector (x, t).

      To find the area of a convex region K in the space-time plane, slice K by “parallel hyperbolas” that intersect K.  Let ?? be the length of the curve formed by the intersection of one of our hyperbolas and K.  To find the area of K, denoted by F, integrate the length of ?? over all the hyperbolas that intersect K.  Then

  
      Finally, geometric probability yields information regarding the volume of a geometric structure.  In a field called stereology, similar methods to those described above are used to estimate the volume of human organs and tumors.