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Catastrophe theory applied to the refraction of traffic
Journal of Computational North-Holland
and Applied
Catastrophe of traffic
Mathematics
22 (1988) 315-318
315
theory applied to the refraction
Tijnu PUU Department
Received
Keywords:
of Economics,
University of Umecj, S-901 87 Ume& Sweden
1 April 1987
Transportation,
caustic, catastrophe
theory.
Fifty years ago Palander [2] and Von Stackelberg [5] discovered that transportation routes that cross two different media, like land and water, with different transportation costs obey the same refraction law at the boundary as do light rays passing two media with different refraction indices in optics. The idea of this article is to see what happens when a set of transportation routes from sea pass across for instance a circular coastline. Perhaps caustics, like those observed in a cup of coffee can occur? This might be an interesting observation as in the transportation case the counterpart to the highly illuminated caustic would be an extreme concentration of traffic and could be used to explain the formation of communication and settlement patterns. The problem is easy to treat analytically under the idealized assumptions that the coastline is perfectly circular and that transportation routes are straight lines. Catastrophe theory ensures that the results obtained in the exemplificatory case are of general validity. Textbooks on catastrophe theory that treat the formation of caustics include Poston and Stewart [3] and Saunders [4]. To make things precise suppose we deal with a semicircular costline of unit radius. Suppose, moreover, that transportation cost is unitary on sea and is k > 1 on land. (Transportration cost is thus uniform and isotropic in each medium, so that routes are straight lines, possibly broken at the coastline). Finally, suppose. that the points interior to the coastline communicate with an infinitely distant point to the right. Putting the origin in the centre of the circle, and denoting the euclidean coordinates by x, y we see the all the routes at sea are parallel to the x-axis. As mentioned we can remove the restrictive assumptions one by one, but they facilitate modelling and agree with the formulation in classical spatial economics. The case is illustrated in Fig. 1. Let us denote a point on the boundary by cos 8, sin 19 and an interior point by x, y. The distance between them is ((cos 8 - x)~ + (sin 0 - Y)~)*/~ and we get transportation cost on land by multiplying the distance with the cost rate k. As for transportation on sea we only need to consider the part that is different for different points of the coastline. Normalizing it to zero at the rightmost point (8 = 0) the distance, and cost, on sea is 1 - cos 8. Accordingly, transportation cost from the point x, y through the boundary point, in direction 8 from the origin, is v=
0377-0427/88/$3.50
k((cos e-x)2
+ (sin e -y)2)1’2 + 1 - cos 8.
0 1988, Elsevier
Science Publishers
B.V. (North-Holland)
(1)
316
T. Puu / Catastrophe theory
Fig. 1.
The direction
8 can be chosen so an to minimize
dV
sin 0x - cos 8y
a=k
V, and this yields the condition + sin 9 = 0.
(4
((~0s 8 - x)’ + (sin 0 - y)*)l’* To see that this corresponds to the Palander-Stackelberg follows. Define the unit vectors
law of refraction
we can proceed
as
where y is the angle between the two unit vectors. However, y is also the angle of refraction shown in Fig. 1. On substituting from the last equation into (2) we obtain
as
A = (cos 8, sin 0) and B = (cos B - x, sin 8 - y)/((cos Obviously
the vector product
B - x)’ + (sin 8 - y)*)l’*.
of these equals
cos 8y - sin 8x
= sin y
((co, 8 - x)’ + (sin f3- y)*j”*
k sin y = sin 8
(3)
As 8 is the angle of incidence, (3) is nothing but the Palander-Stackelberg law. Figure 2 illustrates how the pencil of parallel transportation routes is refracted at the circular boundary and how they, in fact, form a caustic. For increased visibility the rays are not extended after they cross the horizon, but in reality they, of course, continue and are scattered again in all directions. If we differentiate (2) once more with respect to 9 and put the second derivative equal to zero we obtain the following condition for tangency between the set of confluent lines and their apparent envelope k*(sin
0x - cos 0y)(cos
0x + sin 8~)
= cos 6 sin 8(( cos 0 - x)’ + (sin 8 - y)‘) + sin28(sin
ex - cos
In principle the last equation could be used along with equation (2) to eliminate 8 obtaining an implicit relation between x and y, but this is computationally not se easy. Instead we can eliminate x, and y in turn to obtain the envelope by x and y given as functions of the parameter
8. x=cosB-(l-~)jcoso+
(k*-si;f~;+cose)
(4)
T. Puu / Catastrophe theory
317
Fig. 2. Refraction by a semicircular coastline.
and y = sin38/k2
(5)
valid for ) 8 1 < :T. A set of such curves for various values of k is illustrated in Fig. 3, where the caustics become more and more concentrated to the neighbourhood of the origin as k gets higher and higher. We can now relate the story to catastrophe theory. The envelopes or caustics are the bifurcation sets in control space (x, y). The state variable, subject to choice under the cost minimization assumption, is 0. The three-dimensional space 0 as a function of x and y is a typical surface related to the cusp catastrophe. It would not be easy to solve 0 explicitly from (2), but we can benefit from our knowledge that we deal with a ruled surface formed by lifting each
Fig. 3. The caustics for various k.
318
T. Puu / Catastrophe theory
Fig. 4. Catastrophe surface.
traffic ray to the level 0. It is then easy to make a computer graph of the whole surface. It is illustrated in Fig. 4. The Palander-Stackelberg model was generalized by Beckmann [l] to a case where transportation cost could vary continuously over space. The optimal flows of traffic then became general vector fields obeying certain optimality conditions equivalent to Huygens’ Principle. Denoting the vector field by + = (&(x, y), &(x, y)) and its euclidean norm by I$1 = (& + ~$22)~‘~the optimality condition reads k(x,
Y)WI+I
=grad
A
(6) where X is a potential function that can be interpreted as total transportation cost along a route. For (6) any kind of vector field can arise on sea and on land, but the law of refraction still holds on any boundary where there is sudden discontinuous change of transportation rates. Accordingly the formation of caustics still occurs in qualitatively the same manner as in the simplified example. And catastrophy theory ascertains that the phenomena we have treated are universal. They hold for a Beckmannian irregular flow field, and the exact shape of the convex boundary need not be specified.
References [I] [2] [3] [4] [5]
M.J. Beckmann, A continuous model of transportation, Econometrica 20 (1952) 643. T.F. Palander, BeitrLige zur Standortsthorie (Almqvist & Wiksell, Uppsala 1935). T. Poston and I. Stewart, Catastrophe Theory and its Applications (Pitman, London 1978). P.T. Saunders, An Introduction to Catastrophe Theory (Cambridge University Press, Cambridge 1980). H. von Stackelberg, Das Brechungsgesetz des Verkehrs, Jahrbiicher ftir Nationalijkonornie und Statistik 148 (1938) 680.
and Applied
Catastrophe of traffic
Mathematics
22 (1988) 315-318
315
theory applied to the refraction
Tijnu PUU Department
Received
Keywords:
of Economics,
University of Umecj, S-901 87 Ume& Sweden
1 April 1987
Transportation,
caustic, catastrophe
theory.
Fifty years ago Palander [2] and Von Stackelberg [5] discovered that transportation routes that cross two different media, like land and water, with different transportation costs obey the same refraction law at the boundary as do light rays passing two media with different refraction indices in optics. The idea of this article is to see what happens when a set of transportation routes from sea pass across for instance a circular coastline. Perhaps caustics, like those observed in a cup of coffee can occur? This might be an interesting observation as in the transportation case the counterpart to the highly illuminated caustic would be an extreme concentration of traffic and could be used to explain the formation of communication and settlement patterns. The problem is easy to treat analytically under the idealized assumptions that the coastline is perfectly circular and that transportation routes are straight lines. Catastrophe theory ensures that the results obtained in the exemplificatory case are of general validity. Textbooks on catastrophe theory that treat the formation of caustics include Poston and Stewart [3] and Saunders [4]. To make things precise suppose we deal with a semicircular costline of unit radius. Suppose, moreover, that transportation cost is unitary on sea and is k > 1 on land. (Transportration cost is thus uniform and isotropic in each medium, so that routes are straight lines, possibly broken at the coastline). Finally, suppose. that the points interior to the coastline communicate with an infinitely distant point to the right. Putting the origin in the centre of the circle, and denoting the euclidean coordinates by x, y we see the all the routes at sea are parallel to the x-axis. As mentioned we can remove the restrictive assumptions one by one, but they facilitate modelling and agree with the formulation in classical spatial economics. The case is illustrated in Fig. 1. Let us denote a point on the boundary by cos 8, sin 19 and an interior point by x, y. The distance between them is ((cos 8 - x)~ + (sin 0 - Y)~)*/~ and we get transportation cost on land by multiplying the distance with the cost rate k. As for transportation on sea we only need to consider the part that is different for different points of the coastline. Normalizing it to zero at the rightmost point (8 = 0) the distance, and cost, on sea is 1 - cos 8. Accordingly, transportation cost from the point x, y through the boundary point, in direction 8 from the origin, is v=
0377-0427/88/$3.50
k((cos e-x)2
+ (sin e -y)2)1’2 + 1 - cos 8.
0 1988, Elsevier
Science Publishers
B.V. (North-Holland)
(1)
316
T. Puu / Catastrophe theory
Fig. 1.
The direction
8 can be chosen so an to minimize
dV
sin 0x - cos 8y
a=k
V, and this yields the condition + sin 9 = 0.
(4
((~0s 8 - x)’ + (sin 0 - y)*)l’* To see that this corresponds to the Palander-Stackelberg follows. Define the unit vectors
law of refraction
we can proceed
as
where y is the angle between the two unit vectors. However, y is also the angle of refraction shown in Fig. 1. On substituting from the last equation into (2) we obtain
as
A = (cos 8, sin 0) and B = (cos B - x, sin 8 - y)/((cos Obviously
the vector product
B - x)’ + (sin 8 - y)*)l’*.
of these equals
cos 8y - sin 8x
= sin y
((co, 8 - x)’ + (sin f3- y)*j”*
k sin y = sin 8
(3)
As 8 is the angle of incidence, (3) is nothing but the Palander-Stackelberg law. Figure 2 illustrates how the pencil of parallel transportation routes is refracted at the circular boundary and how they, in fact, form a caustic. For increased visibility the rays are not extended after they cross the horizon, but in reality they, of course, continue and are scattered again in all directions. If we differentiate (2) once more with respect to 9 and put the second derivative equal to zero we obtain the following condition for tangency between the set of confluent lines and their apparent envelope k*(sin
0x - cos 0y)(cos
0x + sin 8~)
= cos 6 sin 8(( cos 0 - x)’ + (sin 8 - y)‘) + sin28(sin
ex - cos
In principle the last equation could be used along with equation (2) to eliminate 8 obtaining an implicit relation between x and y, but this is computationally not se easy. Instead we can eliminate x, and y in turn to obtain the envelope by x and y given as functions of the parameter
8. x=cosB-(l-~)jcoso+
(k*-si;f~;+cose)
(4)
T. Puu / Catastrophe theory
317
Fig. 2. Refraction by a semicircular coastline.
and y = sin38/k2
(5)
valid for ) 8 1 < :T. A set of such curves for various values of k is illustrated in Fig. 3, where the caustics become more and more concentrated to the neighbourhood of the origin as k gets higher and higher. We can now relate the story to catastrophe theory. The envelopes or caustics are the bifurcation sets in control space (x, y). The state variable, subject to choice under the cost minimization assumption, is 0. The three-dimensional space 0 as a function of x and y is a typical surface related to the cusp catastrophe. It would not be easy to solve 0 explicitly from (2), but we can benefit from our knowledge that we deal with a ruled surface formed by lifting each
Fig. 3. The caustics for various k.
318
T. Puu / Catastrophe theory
Fig. 4. Catastrophe surface.
traffic ray to the level 0. It is then easy to make a computer graph of the whole surface. It is illustrated in Fig. 4. The Palander-Stackelberg model was generalized by Beckmann [l] to a case where transportation cost could vary continuously over space. The optimal flows of traffic then became general vector fields obeying certain optimality conditions equivalent to Huygens’ Principle. Denoting the vector field by + = (&(x, y), &(x, y)) and its euclidean norm by I$1 = (& + ~$22)~‘~the optimality condition reads k(x,
Y)WI+I
=grad
A
(6) where X is a potential function that can be interpreted as total transportation cost along a route. For (6) any kind of vector field can arise on sea and on land, but the law of refraction still holds on any boundary where there is sudden discontinuous change of transportation rates. Accordingly the formation of caustics still occurs in qualitatively the same manner as in the simplified example. And catastrophy theory ascertains that the phenomena we have treated are universal. They hold for a Beckmannian irregular flow field, and the exact shape of the convex boundary need not be specified.
References [I] [2] [3] [4] [5]
M.J. Beckmann, A continuous model of transportation, Econometrica 20 (1952) 643. T.F. Palander, BeitrLige zur Standortsthorie (Almqvist & Wiksell, Uppsala 1935). T. Poston and I. Stewart, Catastrophe Theory and its Applications (Pitman, London 1978). P.T. Saunders, An Introduction to Catastrophe Theory (Cambridge University Press, Cambridge 1980). H. von Stackelberg, Das Brechungsgesetz des Verkehrs, Jahrbiicher ftir Nationalijkonornie und Statistik 148 (1938) 680.