After one complete trip around the world, our honking bird is at latitude ø = 53°N (near Moscow) and after two complete revolutions, it is at latitude ø = 78°N (west of Franz Joseph Land in the Arctic Ocean).
A sketch of the bird's loxodromic path is shown in figure 22.3. We note that the radius of its path, measured from the earth's axis, continuously decreases as it moves northward. Indeed, when the bird gets extremely close to the North Pole, the radius of its path begins to approach zero. At this point, this otherwise entirely silent bird emits a very loud honking noise, which is a signal to himself to get out of the way. Because of this strange behavior—the bird's once-in-a-lifetime brief but extremely loud traffic blast—ornithologists have logically dubbed it the honking bird.
How far did our feathered friend go on his journey from Lake Victoria to the North Pole? With ø = π/2 (i.e., 90°) in equation (22.6), we obtain
With θ = 80° and R = 6,370 km, we compute that S = 57,620 km or about 35,810 miles.
FIG. 22.3
The flight of the amazing honking bird.
The Mercator Projection Revisited
As shown in figure 22.1, the Mercator projection is constructed by wrapping a circular cylinder around the earth and tangent at the equator. If a tiny light source were at the center of a transparent globe, it would cast an image of the earth's land masses onto the cylinder. True enough, but this does not yield the Mercator projection; it might be called a gnomonic projection onto a cylinder. This projection is useless because there is extremely great distortion at high latitudes.
The Mercator projection is a conformal (i.e., angle-preserving) projection. From equation (22.1), the angle of an arbitrary path on a spherical surface is tan . It is not hard to show that on a cylindrical surface, the angle of the path is tan , where y is the distance measured in the direction of the cylinder axis. Requiring that θ = θ* (i.e., equal-angle transformation), we obtain . In addition, we take where x is the distance measured along the equator. Integrating these expressions for dx and dy gives
These equations describe the positions of the meridians (lines of constant λ) and the parallels (lines of constant ø) on the projection. Clearly, on a Mercator map, the x-coordinate (east-west) is directly proportional to the longitude λ. However, the y-coordinate (north-south) is stretched increasingly as the latitude ø increases. When ø = π/2, y = ∞.
Here are some examples involving the Mercator projection.
Part I
A jet aircraft flies from New York to Tokyo along the great circle route. How far is its journey? Show its path on a Mercator map.
From our atlas we obtain the following information: New York, longitude λ1 = 74°W, latitude ø1 = 41°N; Tokyo, longitude λ2 = 140°E, latitude ø2 = 36°N. The law of cosines for a spherical triangle gives the equation
The values of λ and ø indicated above are substituted into this expression to calculate cos a. From this we obtain a = 97°, which is the angle of the great circle, measured at the earth's center, between New York and Tokyo. The distance S between the two cities is S = (a/360)2πR and, with R = 6,370 km, we obtain S = 10,785 km or about 6,700 miles.
The path of the jetliner's flight is shown as curve A on the Mercator map of figure 22.4. After leaving New York, the airliner proceeds along the western shore of Hudson Bay and then across northern Canada. It reaches a maximum latitude of 70°N at a longitude of 146°W. This is close to Prudhoe Bay on the Arctic shore of Alaska—the hub of the North Slope's vast petroleum fields. The flight then passes well north of the Bering Strait, flies through northeastern Siberia, along the Kamchatka Peninsula, and on to Tokyo.
FIG. 22.4
Mercator map with (curve A) the great circle and (curve B) the loxodrome between New York and Tokyo.
Part II
On its return journey, our jet airplane flies from Tokyo to New York along the loxodrome route. Again, how far is its flight? Show its path on a Mercator map.
The values of λ and ø for New York and Tokyo are substituted into equation (22.3) to determine that the constant true heading of the jet's loxodrome path is θ = 87.5°. Substituting back into (22.3) then provides the value of the longitude λ for any value of the latitude ø. This path is shown on curve B on the Mercator map of figure 22.4. This time our journey takes us across the wide expanse of the Pacific Ocean and over northern California, Denver, and Indianapolis, and then we land in New York.
From equation (22.4) we compute that the total length of our loxodromic flight is S = 12,745 km; this is about 18% longer than the great-circle flight.
The Lambert Azimuthal Equal-Area Projection
Another famous person involved in the development of cartography was the Swiss-German mathematician Johann Heinrich Lambert (1728–1777). Although he made contributions in many areas of mathematics, he will probably be best remembered for his work in cartography.
Perhaps his most noteworthy effort was what we now call the Lambert azimuthal equal-area projection, which he presented in 1772. This projection is employed extensively for maps of the polar regions.
We utilize this projection in figure 22.5 to display the northern hemisphere. The figure also shows the computed path of our jetliner flying from New York to Tokyo along the great circle (curve A) and also its route from Tokyo to New York along the loxodrome (curve B). Although this projection does not show true distances, it does indicate that the great circle is shorter than the loxodrome.
As the name implies, the Lambert azimuthal equal-area projection preserves areas. For example, the area of the island of Cuba (can you find it in figure 22.5? λ = 80°W, ø = 22°N) is the same on the plane map as it is on a spherical globe. The undesirable but necessary distortion requires that Cuba be stretched in the east-west direction and shrunk in the north-south.
The Berghaus Star Projection
As mentioned, over the years a great many kinds of map projections have been developed. We conclude our introduction to cartography with an example involving a rather artistic map projection: the Berghaus star projection.
FIG. 22.5
Lambert azimuthal equal-area map with (curve A) the great circle and (curve B) the loxodrome between New York and Tokyo.
This projection was devised in 1879 by the German cartographer Heinrich Berghaus (1798–1884). It is shown in figure 22.6. In this map, the northern hemisphere is an azimuthal equidistant projection. The southern hemisphere consists of five triangular lobes. As Snyder and Voxland (1989) indicate, this projection is used mainly for artistic forms.
In chapter 2, we carry out an extensive analysis of the dimensions and geometrical features of five-pointed stars. Included in that analysis is the special case that appears in the Berghaus star projection.
One of the features of the Berghaus star is that its radius is twice the radius of the circle containing the northern hemisphere. This is the region defined by the equator passing through the five longitude locations shown in figure 22.6. From our earlier analysis, the internal angle at the points is α = 52.53°.
FIG. 22.6
The Berghaus star projection. (From Snyder and Voxland 1989.)
This is the shape of star that appears, or has appeared in the past, on the flags of various nations, including the flag of Vietnam during the period following World War II. It is interesting to note this rare linkage between vexillology (the science of flags) and cartography (the science of maps).
Looking Ahead in Cartography
Cartography is one of mankind's oldest sciences. It has been around as long as humans have navigated from here to there and as long as people have studied the skies. The science of cartography has evolved, slowly but steadily, over twenty-five to thirty centuries. During that very long period, many of the greatest mathematicians, astronomers, geographers, and navigators have contributed to the development of cartography and the making of maps.
In recent times—starting around 1950 or so—there have been many spectacular advances that have stimulated our interest in cartography an
d provided new methodologies to accelerate its further development. Incredible advances in space technology have given us extremely useful techniques for mapping not only the earth but also our celestial neighbors. Similar fantastic advances in computer technology have provided extremely rapid ways to analyze data, solve difficult mathematical problems, and graphically display all kinds of information.
23
Growth and Spreading and Mathematical Analogies
How fast does a plant or a person grow? What is the rate of increase of the population of a state or a nation? How quickly does a rumor or a disease spread through a certain community? How rapidly is a new technology adopted in a particular geographical setting? How fast does an innovation replace an established methodology?
To help us obtain answers to these and similar kinds of questions, we need to construct an appropriate mathematical framework. Such a framework is provided by the following simple differential equation:
in which N is the magnitude of the growing or spreading quantity, t is the time, a is the growth or spreading coefficient, and N* is the equilibrium value or carrying capacity.
The solution to this equation is given by the expression
where N0 is the value of N at t = 0. This equation is called the Verhulst or logistic equation.
We note that if N is quite small compared to N* (i.e., for “small” values of time t) then (23.2) reduces to the well-known Malthus or exponential equation, N = N0 exp(at). For “large” values of t, the magnitude of N gradually approaches N*. The logistic equation has been utilized for many years with great success in all kinds of problems involving growth and diffusion. An extensive analysis of the subject is given by Banks (1994).
It turns out that there are quite a few other relationships we could use, instead of the logistic equation, to describe the growth or spreading of something. For example, we could use the normal probability equation. Sometimes this is called the Gaussian equation, named after the great German mathematician Carl Friedrich Gauss (1777–1855).
Another expression we could employ for growth or spreading problems is the arctangent-exponential equation. It is defined by the differential equation
For the sake of a new adventure, let us see where this equation takes us.
First, we define the quantities U = N/N* and T = at, and substitute these into equations (23.1) and (23.3) to obtain the following two equations. The first one, of course, yields the solution (23.2).
These expressions, now in so-called dimensionless form, are very similar. The first expression of (23.4) describes a parabolic curve and the second a sine curve. Both of them state that the growth rate dU/dT = 0 when U = 0 and U = 1 and also that dU/dT = 1/4 when U = 1/2. Figure 23.1 shows that these two curves are almost identical.
FIG. 23.1
Comparisons of the growth rate definitions of (a) the logistic equation and (b) the arctangent-exponential equation.
A number of years ago, the noted mathematician William Feller (1906–1970) made an extensive comparison of these two equations. Indeed, he made it a three-way comparison by including the normal probability equation. In his study, Feller (1940) analyzed the same sunflower growth data using the three different mathematical frameworks. He concluded that all three equations adequately describe growth phenomena and provide about the same answers.
With that as an introduction to this next topic, from here on we are concerned only with the arctangent-exponential relationship given by (23.3). We recast this equation in the form needed for integration:
where cscx = 1/sinx is the cosecant. The lower limits of the integrals indicate that N = N0 when t = 0. A table of integrals and a little algebra provide the final answer:
Now you can see why it is called the arctangent-exponential equation.
This solution is now utilized to obtain information about our growth or spreading problem. First, we determine the slope of the curve expressed by (23.6) by computing its first derivative, dN/dt. Following this, we ascertain its curvature by calculating its second derivative, d2N/dt2.
If, over a certain range of time t, the curve is concave upward, we say that the curvature is positive. On the other hand, if the curve is concave downward, then the curvature is negative. Accordingly, when the curvature is zero then, by definition, we have identified what is called an inflection point.
Consequently, we simply set the second derivative equal to zero. This gives the value of the inflection-point time, ti. Clearly, using this result in equation (23.6) gives the corresponding value Ni. Also, substituting the equation for ti into the expression for the first derivative provides the maximum slope, (dN/dt)i. If you want to understand all these procedures, you will confirm the following results:
in which cotx = 1/tanx is the cotangent.
Growth of the Population of California
Now for an example of an application. As most everyone knows, the population of California has increased rapidly during the past few decades. At the turn of the century, California ranked twenty-first among the states in population and in 1940 it ranked fifth. Now it is in first place by a very wide margin. California's population for the period from 1860 to 1990 is listed in table 23.1. It is displayed in graphical form in figures 23.2(a) and 23.2(b). For the moment, we ignore the solid curves in the figures.
TABLE 23.1
Growth of the population of California, 1860–1990
Year t N millions
1860 0 0.380
1870 10 0.560
1880 20 0.865
1890 30 1.214
1900 40 1.485
1910 50 2.738
1920 60 3.427
1930 70 5.677
1940 80 6.907
1950 90 10.586
1960 100 15.717
1970 110 19.971
1980 120 23.668
1990 130 29.126
Source: Wright (1992).
FIG. 23.2
Growth of the population of California. Population amounts are in millions. The year 1860 corresponds to t = 0. (a) Arithmetic and (b) semilogarithmic plots.
The pattern of the data points shown in figure 23.2(a) suggests that for small values of time t (and hence small N), the growth may be exponential. Let's try it. From (23.3) and recalling that for small x, sinx = x, we obtain
Integrating this equation gives
which is the exponential equation. Taking logarithms of (23.9),
This is an expression of the form y = k1 + k2x, which is the equation of a straight line. So if we plot loge N versus t, we should obtain a linear correlation with intercept loge N0 and slope πa/4. Determination of the intercept and slope gives the numerical values of N0 and a.
Such a straight line is shown in the semilogarithmic plot of figure 23.2(b). It appears that the linear relationship extends from t = 0 (year 1860) to about t = 100 (year 1960). So we carry out a least-squares computation involving the first 11 entries of table 23.1. This yields the results N0 = 0.390 and a = 0.0469. On this basis, we can say that California's population grew exponentially from 1860 to 1960. After that, crowding effects or growth-limitation factors—inherent in all logistic-type phenomena— began to be felt.
There are several ways to compute the value of the remaining quantity, N*. For our purpose we simply indicate the answer: N* = 45.5. In other words, the ultimate population of California will be about 45.5 million people.
So now we know that N0 = 0.390, a = 0.0469, and N* = 45.5. Substituting these numbers into equations (23.7) gives the inflection point ti = 117.0 (year 1977), Ni = 22.75 million, and (dN/dt)i = 0.533 million/year. Finally, substitution of these same numbers into (23.6) yields the solid lines shown in figures 23.2(a) and 23.2(b).
We conclude that the arctangent-exponential equation provides a good framework for quantitatively describing the growth of California's population.
Adoption of Hybrid Corn in Iowa
Here is another example. During the 1930s in midwest America, new strains of corn were developed and, because of their greater s
tamina and yields over open-pollinated varieties, began to be utilized by farmers in the region.
This hybrid corn innovation began in the heart of the Corn Belt—the states of Iowa and Illinois—where its adoption by farmers was slow at first, then more rapid, and finally leveled off after virtually all farmers in the region had switched to the new variety.
An extensive study of this adoption of hybrid corn was carried out by Griliches (1960) covering the period from 1933 to 1958. For each of 31 states, mostly in the midwest and south, he determined the percentage of acreage planted with hybrid corn to the total corn acreage available. We shall examine and analyze what happened in the state of Iowa.
The results obtained by Griliches for Iowa from 1933 to 1942 are listed in table 23.2 and plotted in figures 23.3(a) and 23.3(b).
Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics (Princeton Paperbacks) Page 17