Suppose for example that P is the point (1, 3), my office, and Q is the point (−2, −1), where Starbucks is located. We are of course assuming that the size of each business is small compared with a block distance, so that we can represent them by points. The Euclidean distance between them, dE (PQ) is readily computed from the distance formula:
(From now on we will drop the unit of distance “blocks”: it will be understood.) But of course, to arrive at Starbucks from my office I must travel either from P to A(−2, 3) and then to Q, or from P to B(1, −1) and then to Q. In each case the “taxicab” distance is given by
Clearly, dT(PQ) > dE(PQ) in this case. But inequalities between distances in one geometry are not necessarily preserved in the other. Consider, for example,
the Euclidean distances between (i) the points R(1,1) and S(3,3), and (ii) R(1,1) and T(−2, 1), which are, respectively,
However,
This non-Euclidean geometry, informally known as taxicab geometry (see [44]) is a form of geometry in which the usual measure of distance in Euclidean geometry is replaced by a new one in which the distance between two points is the sum of the (absolute) differences of their coordinates. This new measure of distance (or “metric”) is also sometimes known as the Manhattan distance; in the plane, the Manhattan distance between the point P1 with coordinates (x1, y1) and the point P2 at (x2, y2) is |x1 − x2| + |y1 − y2|. Taxicab geometry satisfies all Euclid’s axioms except for the side-angle-side axiom, because one can generate two triangles with two sides of the same length in this metric, and with the angle between them the same and yet they are not congruent. For more information on the sometimes surprising properties of this “urban geometry’, consult the delightful little book on this topic by Krause (1986).
Question: Is Pi equal to four?
Well, as they say, it all depends. . . . If we sketch the set of all points that are a unit Euclidean distance from a fixed point (the coordinate origin, without loss of generality), then we have the unit circle. If we do the same thing but in the taxicab “metric” as it is called, what shape do we get? A little thought, coupled with a sketch or two, reveals that we get a “diamond” shape, that is, a square with two vertices lying on the “y” axis and the other two on the “x” axis. This is a taxicab circle of unit radius! Now we are familiar with the definition of π as the circumference of a circle divided by its diameter. With the taxicab metric, the “length” of one side of our taxicab circle is two units, hence the circumference is eight units, and the diameter is two units, so, yes, the taxicab value of “taxicab π” is indeed four!
Of course, there are potentially an infinite number of distance metrics and corresponding “circles.” The metric (or formula) for the distance of a point (x, y) from the origin in Euclidean geometry is of course well known: This is readily generalized for other positive integers n such that The case for taxicab geometry corresponds to n = 1, for which As n increases in value, the “circles” become more and more bulbous, as seen in Figure A3.1 for several values of n, and in the limit as n → ∞, the metric is defined to be d∞ = max{|x|,|y|}. As can be seen from the figure, the unit circle for the d∞ metric is a square with sides parallel to the coordinate axes.
Figure A3.1. Taxicab “circles” d1, d2, d4, and d∞.
Exercise 1: Mark the points A = (−2, −1) and B = (2, 2) on a piece of graph paper. Then sketch the locus of all points P such that dT(P, A) + dT(P, B) = 9. By analogy with the corresponding figure in Euclidean geometry provide a name for this figure.
Exercise 2: Mark the points A = (−3, −1) and B = (2, 2) on a piece of graph paper. Then sketch the locus of all points P such that |dT(P, A) − dT(P, B)| = 3. Again, by analogy with the corresponding figure in Euclidean geometry provide a name for this figure.
Appendix 4
THE POISSON DISTRIBUTION
Applications of the Poisson distribution are many and varied. A large class of natural and social phenomena have been successfully modeled using it [45]. Before deriving the formula for this distribution let’s consider some examples. A classic one concerns a Russian statistician, Ladislaus Bortkiewicz, who used the Poisson distribution to estimate the number of soldiers killed by mule-kicks to the head in the Prussian army. He assembled data on the 14 cavalry corps over a period of 20 years, and in so doing he was able to verify that the distribution of mule-kick deaths fit a Poisson distribution! Another example is the number of armadillos killed by traffic on a length of an Arizona highway; this is also “Poisson distributed.” Other examples are the number of emergency room cases cases arriving at a hospital during a one-hour period, and the number of bomb hits in one-acre areas of metropolitan London during World War II [45]. More pedestrian examples include the number of times I check my email in the morning, the number of cars that pass a particular “eatery” in a two-hour period (perhaps the “Road-Kill Café” in a certain Arizona town?).
This is not of course to imply that everything can or should be modeled as a Poisson process, however; there are certain restrictions that must be satisfied before the phenomenon of interest can be thus described. The ER cases arriving on a Sunday morning would probably not have the same distribution as those on a Saturday evening, for example. Also, if there were an explosion or tornado, the arrivals would be associated with a common cause, and this could render the Poisson distribution inappropriate. If the area of London were extended too far into the surrounding countryside, the intensity of the bomb damage would be much less severe. The armadillos are presumed to cross the highway at random locations, not to be traveling as a herd, or in convoy, or crossing the road at a particular spot (as Canada geese seem to do, especially when I am in a hurry).
Perhaps the simplest context in which to place the mathematical discussion is found by asking the following question. What is the probability of getting exactly n tails in N tosses of a (possibly unfair) coin, where n N? This means that there will be N − n heads, of course, and if the probability of getting a tail is p, then that of getting a head is 1 − p. Of course, if the coin is fair, then p = 1/2. The probability of n tails is therefore the number of ways one can have n tails in N tosses multiplied by the probabilities associated with each toss. The result is
This is known as the binomial probability distribution. To obtain the Poisson distribution as a limiting case of this distribution we suppose getting a tail is a very rare event, that is, p 1; this corresponds to a very unfair coin! We can (eventually) simplify expression (A4.1). First, we note that for p 1, ln(1 − p) ≈ p, so that
Nothing that
we can approximate the logarithm of the factorial M! for a large positive integer M by a “quick and dirty” method as follows:
From this it follows directly that M! ≈ MMe−M. This result is sometimes known as the weak form of Stirling’s approximation. Returning to equation (A4.3) and applying it to the two terms on the right of equation (A4.4), it follows that
Since n N,
Therefore
We have neglected the term in n2 since N is large. Hence
Therefore, combining the approximations (A4.2) and (A4.5), we have
The expected number of tails in N tosses is λ = Np. Thus equation (A4.6) can be written in terms of the expectation λ as
This is the Poisson distribution for the probability of exactly n tails occurring in N tosses of the coin, when p 1. And instead of heads and tails, we can think of customers arriving at the post office, or gaps in traffic, as applied in Chapters 3 and 9, respectively.
We can think of this in another way. Note that
This result can be interpreted as the sums of the probabilities of all possible outcomes occurring; in other words, we have a probability distribution.
Appendix 5
THE METHOD OF LAGRANGE MULTIPLIERS
We state without proof a rather formal expression of a general problem to which this method applies:
Any local maxima or minima of a function z = f(x, y) that is subject to the constraint g(x, y)
= 0 will be among those points (x0, y0) for which the point (x0, y0, λ0) is a solution to the system of equations
where
provided these partial derivatives exist.
Appendix 6
A SPIRAL BRAKING PATH
What is the path of a particle that is acted upon by a constant force? Since force is a vector quantity, this means that both the magnitude a and the direction of the force are constant. Suppose that the acceleration vector ā makes an angle γ with the trajectory, as indicated in Figure A6.1. Many calculus textbooks demonstrate that the components of acceleration (v being speed) along the tangent and normal to the path are, respectively,
where ρ is the (local) radius of curvature of the path. Resolving the acceleration in these directions yields the following equations:
In these expressions a = |ā|. Integrating the first equation, and noting that ρ = |ds/dψ|. gives the result
Figure A6.1. Path of the particle; ψ is the angle the tangent line makes with the polar axis, the arc length is s and the acceleration is a constant vector ā.
where B is a constant and we have assumed (without loss of generality) that ds/dψ>0. A further integration yields
D being another constant. Again, without loss of generality, assuming the argument of the logarithm is positive we find that the equation of the path is
Therefore the path is that of an equiangular spiral because (neglecting additive and multiplicative constants) it is of the form s = econstant xψ.
Appendix 7
THE AVERAGE DISTANCE BETWEEN
TWO RANDOM POINTS IN A CIRCLE
Suppose the circle has radius a. We can ask: what is the joint probability that one of the points (P) is in the distance interval (x, x + dx) from the center O, the other point (Q) being nearer the center, and such that PQ makes an angle with PO in the interval (, + d)? From Figure A7.1, note that the probability of P being in the shaded annular region is
This must be coupled with the probability of Q being in the triangular shaded sector (see Figure A7.2),
Figure A7.1. Geometry of the probability of a point P being in the annular region.
Figure A7.2. Probability of a point Q being in the (approximately) triangular sector.
We also need to know the average distance between P and Q in this sector; a simple “center of mass” argument will suffice here—it lies 2/3 of the way from the “base” of the triangular sector, so the average distance is 4x(cos )/3. Combining all these results gives expected distance between the two random points P and Q:
There is an additional factor of two in this integral to include the case when P is nearer to the center than Q. This integral can be evaluated by careful calculus students to be
Appendix 8
INFORMAL “DERIVATION” OF THE LOGISTIC
DIFFERENTIAL EQUATION
It has been said that there are two categories of people in the world: those that separate people into two categories and those that don’t. Nevertheless, suppose there are indeed two categories of people (described below) comprising a community: in which the population is constant. This may seem very unrealistic, except over short periods of time, for what about births, deaths, “emigration,” and “immigration” within such a community? And while this is a valid criticism, there are several “communities” that possess a constant population: cruise liners, and political bodies (House, Senate, etc.) being two such examples. We shall examine the mathematical genesis of the logistic equation in the former context, but not before identifying a limitation inherent in this approach. A cruise liner may have a thousand passengers (and several hundred crew members) on board, whereas the United States Senate has one hundred members. In each case, the total population is constant (ignoring people falling overboard in the former case, of course!), but within each community, there may be two classes of subpopulations. Obviously the number of males and females remains fixed, so what could vary?
We’ll make some simplifying assumptions here. Suppose that a passenger with a contagious and easily transferable disease boards the cruise liner (without exhibiting any symptoms at that time). Over time, assuming all the other passengers are susceptible to this disease, as the infected individual comes into contact with them, the number of infected passengers increases—and this is certainly something that has happened on several occasions in recent years. So in this case, at any given time in this simplified model, there are two categories of passengers: those who are infected and those who are not. And in this model these populations will vary monotonically over time subject only to the condition that their sum is a constant, K. We could change the scenario from transference of a disease to that of a rumor: gossip! In that case there would be additional assumptions to be made: everyone who knew the rumor would be willing to share it, and everyone who did not know it would be willing to listen (and hence pass it on!). A related context is that of advertising by word of mouth: “Did you hear about the special offer being made at Sunbucks? They’re giving away a Caribbean cruise to everyone who buys a Grande peppered Latvian pineapple-cauliflower espresso mocha latté.”
In the case of the U.S. Senate, suppose that Senator A introduces a bill (maybe to restrict the availability of the above coffee at Sunbucks because of its harmful effects on the local populace?). Perhaps there is little support for the bill initially (many of the senators like that coffee), and as acrimonious but eloquent debate continues, more and more senators begin to see the error of their ways, and eventually vote accordingly. . . . Again, there will be an evolution of the potential “Aye” and “Nay” votes over time, culminating in, well, you’ll just have to tune in to C-Span to see the outcome.
So what is the limitation of this approach, regardless of context? Nothing yet, but if we use calculus to try to describe the rate of change of the two populations, we are making the implicit assumption of differentiability, and hence continuity of the populations. But the populations are discrete! The are always an integral number of infected passengers, or of senators disposed to vote for the bill (and despite one’s personal misgivings about Senator B, though he may only do the work of half a senator, he is one person). Calculus is strictly valid when there is a continuum of values for the variables concerned, and in that sense calculus-based models of discrete systems can never be totally realistic, even when there are billions of individuals (such as the number of cells in a tumor). It is usually the case in practice that the more individuals there are in a population, the more appropriate the mathematical description will be from a continuum perspective, because small populations can be subject to fluctuations that are comparable in size with the population! Under those circumstances a discrete approach is desirable. Nevertheless, when the number of possible “states” is limited (as in “Aye” or “Nay,” infected or not), frequently the calculus-based approach is sufficiently accurate to “interpolate” the behavior of the more accurate discrete formulation. And that is what we shall do here in the context of the cruise liner epidemic, neglecting all complications like incubation times, likelihood of recovery, and immunity from the disease. Such considerations are very important in realistic epidemiological models, but will not be addressed here. We are therefore assuming that all passengers who are uninfected at the beginning of the cruise are susceptible to the disease, and that once they have contracted it, they remain infected (and infectious) for the duration of the cruise (a very unfortunate scenario). The problem then becomes one of determining how the number of passengers contracting the disease changes over time. Let N(t) be the number of infected passengers and M(t) be the number not infected. Obviously M(t) + N(t) = K and, as stated above, N(0) = 1 (this initial condition can be changed without loss of generality). Remembering that the argument we are making here is not a rigorous one, let us take two such passengers at random, and ask what the probability of the spread of the disease will be from such an encounter. Certainly, the only way the disease will spread is if each of these two individuals is in a different category. To reframe th
is in the “rumor” category, if our two individuals know of the rumor, then to begin with, they may just talk about the weather (especially if they are from the UK!); if both know the rumor, they may just talk about . . . the weather. It is only when one of the two knows the rumor, and the other does not, that the rumor will spread. And so it is with the disease under the simplifying assumptions we have made. The probability of picking an individual from the infected population and then one from the uninfected is
X and the City: Modeling Aspects of Urban Life Page 22
