Triangular city: the green wave
by
Professor Ezio Marchi

Abstract: In this paper we obtain the neccesary and sufficient condition for having a
general green wave in a triangular grid city.
 Director del Instituto de Matemática Aplicada San Luis,
CONICET – Universidad Nacional de San Luis.
Ejército de los Andes 950 – C.P. 5700
San Luis
Argentina.
1) Introduction
In a previous paper we have studied the problem of the coordination of lights, or green
wave, for a rectangular city. This study stimulated our interest to study other cities with
different shapes and topologies. This is a complement of studies made by R. Allshop and
N. Improta.
For this reason we present in this paper the study of the “green wave” in a city with a
different topology, namely a city where all the blocks are triangular and its decomposition
in very well order. Under general conditions, considering that all the vehicles go in a
straight line, we obtain a neccesary and sufficient condition for having a green wave. This
generalizes the problem studied by the Transportation Research Laboratory group.
We would like to stress that the most important point for obtaining the results of this paper
is choosing a suitable system of coordinates. If this is not the right system the problem
might be almost impossible or very hard to solve.
2) Description of the coordination topology
Consider a city where the graph describing its distribution is triangular. That is to say that
all the blocks are triangular, with all equilateral triangles arranged in a ordered form. Thus
the basic block is an equilateral triangle. The city looks like
Where there are three different directions of straight lines, namely the first one west-south
to east-north, the second one west-north to east-south and the remaining direction west to
east. Only for simplicity in the computations we assume that the streets have width zero.
At this point it is neccesary to choose a system of coordinates; if cartesian coordinates are
used, all the following problems of solving the green wave in such a graph become
extremely hard or almost impossible. On the other hand if the choice is what we call
simplicial coordinates, the solution becomes simpler to formulate.
What we call simplicial coordinates is the following attachment of the nodes (m,n) as
follows
(8,8)
(7,7) (8,7)
(6,6) (7,6) (8,6)
(5,5) (6,5) (7,5) (8,5)
(4,4) (5,4) (6,4) (7,4) (8,4)
(3,3) (4,3) (5,3) (6,3) (7,3) (8,3)
(2,2) (3,2) (4,2) (5,2) (6,2) (7,2) (8,2)
(1,1) (2,1) (3,1) (4,1) (5,1) (6,1) (7,1) (8,1)
(0,0) (1,0) (2,0) (3,0) (4,0) (5,0) (6,0) (7,0) (8,0)
Therefore, in this system of coordinates the already mentioned straight lines have the
following equations
m  k , n  k and m  n  k
m , n and k positive integers.
A point moving in one of these straight lines has the equation
t  n0  v1n , t  m0  v2 m , t  n0  v3 n
where t is the time when the point is at n  R, v1 , v2 , v3 are the respective inverses of the
velocities.
Now consider a triangular block
If we have a light at each intersection, that is to say at its vertices, this is a mecanism telling
the vehicles the right of passing. Therefore if an arrow in the corresponding direction marks
this right of way we could have the following situation
In this case there will be a problem at the right corner since the vehicles coming from above
and from the left will join. Therefore the only two possible arrangements for easy flow are
SOME RESEARCH IN THIS DIRECTION WAS PERFORMED BY??
Since we have three phases the system of lights will have to work in three phase way.
In order to have in the grid the possibility of the existence of a green wave/way we consider
intuitively that for each index l , the times t m ,n ,l  when the “green” marked by the
corresponding arrow in the intersection m, n fulfills that there are times t l such that
t l t  m ,n ,l  t l 1
for each m, n
therefore the green lights are introduced by a vectorial field which indicates in the
respective intervals of time the right of way in the direction marked by the corresponding
arrow.
From t 3l 1 to t 3l  2 we have
that analytically might be written as if
m  n  1 (mod 3)
that we also write m  n   1 then the vector assigned is e
m  n  2
2
then
e
e 2  m  n    / 3 i .
Now from t 3l  2 and t 3l 3
3i
and finally for
m  n  0
2
3i
as a complex number.
then e 0i or the function is
which analytically might be written as if
m  n  2 the vector is
2
e
3i
m  n  1
the vector assigned is e 0i , if
and if m  n  3 the vector is e
e
4
3i
or concisely
2 m n  1 / 3i
Finally in the interval t 3l to t 3l 1
And here analytically for m  n   1 the corresponding vector is e
e 0i and if m  n  3 , e
e
2
3i
4
3i
, if m  n  2 ,
or concisely
2  m n  2  / 3i
from the above material we have in a schematical way
t  m ,n ,3l 1
t  m , n , 3l  2 
t  m , n , 3l  3 
t  m , n , 3l  4 
m  n  1
m  n  2
m  n  3
3 Coordination of the way
In the previous section we have described the topology and the introduction of the vector
field and its dynamics.
Now here in this paragraph we will explain the coordination or green wave.
If the equation of a straight line expressing the time as function of the position is
 m, n 
 m, n 
t m,n   n0
 v1 n
for m  n   1 and then if from t m,n,3l 1 to t m,n,3l 2  we have
t m,n,3l 1  t m,n   n0
 m, n 
 v1
 m, n 
n  t m,n,3l 2 
then in the next block we must have
t m,n,3l 11  t m,n   n0
 m, n 
 v1
 m, n 
n  1  t m,n,3l 21
and in general
t m,n,3l 1 r   t m,n   n0
 m, n 
n  r   t m,n,3l 2r 
m  n  2 passing from
 v1
For another example with
 m, n 
t m,n,3l 1 to t m,n,3l 2  for m, n we
have for the concise inequality
t m,n,3l 1  t m,n   n0
 m, n 
 v3
 m, n 
n  t m,n,3l 2 
and since the straight line is m  n  k , in the next block we must have
 m, n 
 m, n 
 v3 n  1  tk  m 1, n 1,3l  2 1
t k m1,n1,3l 11  n0
and in general
 m, n 
 m, n 
 v3 n  r   tk  m  r , n  r ,3l  2  r 
tk  m  r , n  r ,3l 1 r   n0
and finally for the remaining cases with similar inequalities it is possible to obtain the
neccesary and sufficient conditions for having the green wave in each case.
We remark that as in [1] it is possible to perform the study of having platoons.
We would like to point out that for the other orientation that we had, a similar study might
be performed.
In three dimensions it is possible to study similar situations with 3! possible cases or
models. This might be of of interest for telecommunications.
As a generalization for n-dimension it seems possible to generalize it with the possibility of
n  1! way.
4 Bibliography
[1] E. Marchi: On the green wave for a rectangular city. Proceedings of the 30th
International Symposium on Automatic Technology & Automation. Vol. “The Motor
Vehicle and the Environment – Entering a new century” pp. 61 – 68. Florence, Italy, 16th19th June 1997.
[2) R. Allshop: Network models in traffic management and control. Transport Review
1983, Number 2, pp. 157-182.
[3] N. Improta:
[4] J. Cesco and E. Marchi