1. Project Final Report, Fall 2015
University of Illinois at Urbana-Champaign
December 19, 2015
1 Introduction
The interaction of particles on a given curve, like an ellipse or rectangle, has
been studied by mathematicians for a long period. Particularly, problems, such
as how to discover the position of particles with maximum product of distances
between each two particles, and what is the trajectory of center of mass when
particles moving along the curve, are drawing attention of many mathemati-
cians.
2 Summary of Outcomes
Mainly using numerical methods, our group made some breakthroughs in anal-
ysis of problems stated above, on an ellipse and rectangle. We found that the
center of mass of three points on an ellipse that divide the perimeter into equal
arc lengths is a smaller ellipse with the same shape as the initial one. In addi-
tion, we then reached proof that the same phenomenon occurs for the square.
Also, we have made some progress in finding the maximum product of distances
between four particles on rectangles, and three particles on an ellipse.
2.1 Optimization of Particles on Ellipses
Some analytical results for the problem of maximization of the product of dis-
tances were obtained for three points on an ellipse. Assuming symmetry, the
optimal positions are
(a, 0)
(x, y)
(x, −y)
Where
x2
a2
+ y2
= 1
x = a
a2
+ 3 −
√
25a4 − 18a2 + 9
6 (a2 − 1)
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2. We were able to narrow down the number of possible results in certain cases.
For instance, we proved that
x1, y1
x2, y2
−x3, −y3
has a lower product of distances than
x1, y1
x2, −y2
−x3, −y3
Where
1 ≥ a ≥ x3 ≥ x2 ≥ x1 ≥ 0
1 ≥ y1 ≥ y2 ≥ y3 ≥ 0
There are two other cases similar to this which would need to be similarly com-
pared for optimality. From there, it may be possible to show that x3 must be
a, and that the other two points must be symmetric.
Varying the minor axis, the following numerical solution was found:
For comparison, this is the analytic solution from above:
They are identical within computer precision.
Interestingly enough, for even number of points, there appears to be a kind
of non-linear behavior in the optimal positions of the points:
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3. 2.2 Arc Length Approximation
One particularly important scenario that we take into consideration is three
particles with equal arc length between each two on some specific ellipse. If we
are given any point – may be called starting point – on an ellipse, where one
of the three particles falls on, we can find the location of other two particles
satisfying the requirement of equal arc length, by numerical methods with com-
puter programming. Via moving the starting point around whole ellipse, we can
acquire all possible distribution of three particles with equal arc length on ellipse.
With these distributions, we mainly analyzed two subjects: the distribution
giving maximum product of distances between each two particles, and the tra-
jectory of center of mass of three particles as the starting point moving along
ellipse. On the ellipse with function
x2
52
+
y2
32
= 1,
we found the distribution with maximum product of distances is symmetric ac-
cording to y-axis, with one of three particles falling on the very top (or bottom),
as shown in picture below. This finding can be expanded to other ellipse with
major radius falls on x-axis.
As to trajectory of center of mass, the results of numerical simulation tells us
that, it is an ellipse with exactly same shape, i.e the eccentricity, as the ellipse
that three particles fall on, except for a extremely size. By repeating same
procedure on other ellipse with different functions, we found that this finding is
of generality for all ellipse.
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4. We then focused on the relationship between the ellipse which particles fall on,
and trajectory of center of mass of three particles, particularly attempting to
figure out determinants of size of trajectory. By intuition, we conjectured that
the ratio of trajectory to ellipse, which can be expressed as
r =
a
A
where a is the half length of major radius of trajectory, and A is the half length
of major radius of ellipse, would be closely related to the eccentricity of ellipse,
which is
e =
√
A2 − B2
A
where A, B are half length of major and minor radius of ellipse respectively. By
running programming on numerous ellipses with different eccentricity, collecting
the data of ratio described above, and then conducting regression on these
data, we find there exits linear relationship between logrithm of eccentricity
and logrithm of ratio,
log e = α log r
which is plausible since this equation satisfies properties of ellipse. According
to the result of regression, we found that the value of α is
α = 21.5260903...
2.3 Optimization of Particles on Rectangles
It is a known fact that on squares, 4 particles that give the maximum product
of distances are located at the four vertexes of the squares. However, where
should the 4 particles be on the rectangles? We suspect that the 4 particles will
still locate on the 4 vertexes of rectangles with the length of the rectangles small
enough. But then, when will the 4 particles no longer be on the four vertexes?
By mapping particles from the real-axis to the boundary of rectangles, and
implementing Steepest Descent Method and Bisection Method, we found the
combination of four particles which gives the maximum product of distances on
rectangles. Moreover, we got some interesting observations about the position
of the optimized particles:
Suppose l = length of rectangle, w = width of rectangle, r = l
w .
For x ∈ range(5.75, 5.8) and y ∈ range(6.5, 6.75):
• When 1 ≤ r ≤ x, the four points which produce the maximum product of
distance between four particles are located on the four vertexes.
• When x < r ≤ y, three points are on the vertexes and one point is a little
bit of the vertex (As shown in Figure 1).
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5. Figure 1: When r = 6.
Figure 2: When r = 7.
• When y < r, two points are on the vertexes and two points are off the
vertexes. The four points are symmetric (As shown in Figure 2).
These surprising observations show that r can be actually very large for the 4
particles to be located on the the 4 vertexes of the rectangles.
2.4 Center of Mass of Particles on Square and Ellipses
From the computer simulation results, we concluded that the trajectory of the
Center of Mass (COM) of three points on an ellipse is a smaller ellipse. How-
ever,the analytic equation of the trajectory of COM in ellipse remains to be
determined since it involves ellipse function and integration.
We found that the product of distances of three particles on square that equally
divide its perimeter is also maximized when one point is on the vertex and a
symmetry is obtained. This is a similar phenomenon as in the ellipse case. So
we started investigating similar phenomena in the square.
Suppose we have three points A,B,C on the edge of a unit square that equally
divide the perimeter of square. As A travels on the edges of the square in a
full cycle, the CoM of ABC moves around the center of the square on a small
rectangle for 3 cycles. The vertices of the small rectangle is given by:
(
4
9
,
4
9
), (
4
9
,
5
9
), (
5
9
,
5
9
), and(
5
9
,
4
9
)
.
Figure 3: Motion of COM as Point A moves from (0,0) to (1,0)
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6. • In this case, the product of the distance is maximized when A is on any
vertex with the other 2 points in positions that is symmetric with respect
to the diagonal.
• Progress in determining the trajectory of COM in ellipse: the same phe-
nomenon occurs in ellipse as in square due to the symmetry: As one point
moves from (a, 0) to (b, 0), where a is the major axis and b is the minor
axis, the COM traces out a 3
4 ellipse. So as this point moves around the
ellipse in a full circle, the COM traces out a smaller ellipse 3 cycles.
We then examined the pattern as the number of points on the square goes larger.
With the same argument, the center of mass of five points that equally divide
the perimeter of square trace out a smaller square for 5 cycles with vertices
(
12
25
,
12
25
), (
13
25
,
12
25
), (
13
25
,
13
25
), and(
12
25
,
13
25
)
.
We saw patterns after finding the trajectories of 7 points and 8 points. And
then we found the generalized results to n points on squares. For an arbitrary
number of points that equally divide the perimeter of the square, we consider
two cases:
• when n is odd,i.e. n = 2k + 1, k = 1, 2, 3..., the COM traces out a smaller
square n times with vertices being
(
n2
− 1
2n2
,
n2
− 1
2n2
), (
n2
+ 1
2n2
,
n2
− 1
2n2
), (
n2
+ 1
2n2
,
n2
+ 1
2n2
), and(
n2
− 1
2n2
,
n2
− 1
2n2
)
.
• When n is even,i.e. n = 2k, k = 1, 2, 3...., the COM is always the COM of
the Square i.e.
(
1
2
,
1
2
)
3 Future Directions
• Find the relationship between the length of rectangles and the position of
the off-vertex optimized particles.
• Find particles that give the maximum product of distances on other poly-
gons, and discover a rule to determine whether the interested points are
located at the vertexes of the polygons
• Use the elliptical functions and elliptical integration to find the Analytic
equation of the trajectory of the center of mass of three points on ellipse.
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