Ferenc Adorjan 09/20/1997

Atomic Energy Research Institute, Budapest

H-1525 P.O.Box 49

Hungary

E-mail: fadorjan@freemail.hu

As a working hypothesis, let us assume that such closed curves exist that have two internal equichordal points. As a further assumption we assume that some of them have two symmetry axes: one determined by the two chor centers and the other perpendicular to the first. As soon as we know where the two equichordal points of the curve are and what the length of its "equichords" is, we have 6 trivial points of the curve:

Fig. 1 The 6 trivial points (A through F)

Of course, similar points also could also be obtained without assuming any symmetry. In Fig.1, let the chords (AB, CD, and EF) have the length of two units, and let us denote the distances OP and OQ by d. More generally, d can be defined as the ratio of the distance of the two equichordal centers to the specific chord length, where the specific chords are the ones pssing through one of the chord centers P or Q.

By using a simple recipe, four series of infinite number of points of the curve can be obtained:

1. start a chord with the same length as AB from one of the points C or D passing through the chordcenter Q, or from one of the points E or F passing through P,
1. the endpoint of the chord will be X₁, where X is one of the letters {C,D,E,F}
1. from X₁ start a new chord with the same length passing through the farther of the points P or Q,
1. the new endpoint is X₂,
1. analogously to points 3 and 4, from every X_i we can obtain X_i+1, ad infinitum...

Let us introduce a convenient coordinate system which is centered at O and its x axis is determined by the points P and Q. Each of the four point series have members winthin two opposite plane quadrants, so that the (++) quadrant contains every even indexed points from the series started from E and every odd indexed points from the series started from D. Note, that the points with even indices of the E series are centrally symmetrical counterparts to the even members of series D.

The above described recipe can easily be realized by an algorithm and a substantial number of points of the series can be calculated by a simple computer program. It is only necessary to set up a function that provides the coordinates of a point that is determined by the condition that it is at a fixed distance from a given first point, it is on the line determined by the first point and a second point, and it falls on the opposite side of the second point as the first point. By running this algorithm, one can obtain a "skeleton" of the curve of our interest, as it is shown in Fig. 3. Actually the vertices of the polylines determined by the specific chords - which are members of the four chord series - present an outline of something, that could easily be a curve. Zooming out, however, the region in the dashed rectangle, one can notice something surprising. This is shown in Fig. 4.

Figure 2 The point series starting from D

Figure 3 The "skeleton" of a 2-center equichordal curve with d=1/3 excentricity

The remarkable feature shown in Fig. 4 is that alternatingly every other acute vertice points are close to one or an other x coordinate. In other words, the odd numbered vertices seemingly converge to a different x coordinate than the even numbered ones. Anybody would suspect first that this effect appeared only due to some numerical error. According to what follows, this option seems to be excluded, however

The "skeleton" chords in Figs. 3 and 4 were calculated simply by solving directly the second order equation determining the meshpoints of a straight line and a circle. This method has, however, the drawback that the odd and the even numbered vertices form two separate series.

Figure 4 An enlarged fraction of the skeleton shown in Fig. 3.

The use of trigonometric functions offer, however, a method that unifies into a single series the points falling into one space quadrant. Let us denote its members as T_i. It can easily be shown that the following recursive formulae generate the [x_i,y_i] coordinates of this series of points:

Surprisingly, this different algorithm - even by using 16-bytes-long floating numbers, i.e. 34 decimal digits accuracy - has shown the same effect. I made some effort to evaluate the infinite limit of the odd and the even members of the point series, but did not succeed. Thus, I made some numerical experiments.

In these numerical experiments I interated the above equations until the two subsequent x co-ordinates of the points T_i broke the monotneity plus as many more iterations, to reach a 1% convergence of the difference of two subsequent x co-ordinates.

By playing with the parameter d, I noticed soon that the smaller the parameter d, the smaller the converged difference of the odd and the even x co-ordinates will be. In case of smaller d values it takes also a larger number of iterations to break the monotoneity if the subsequent x co-ordinates of the points T_i. These effects are shown in Figs. 5 and 6.

Figure 5 The logarithm of the magnitude of the first reversing step in the recursive sequence of x_i coordinates as function of d

It is clearly seen in Fig. 5 that only below d @ 0.03 becomes the algorithm numerically instable. This corresponds in Fig. 6 to over ~ 600 iterations of the recursive algorithm. This also means that we could not test the effect below d @ 0.03, thus we can not exclude numerically that there exists a threshold value of d, such that below that value the x coordinates of the T_i point series monotonously converge to the value of the half length of the specific chord. The result, however, does not suggest that, thus we anticipate that such threshold does not exist.

Figure 6 The first i index of the series T_i breaking the monotoneity of x_i-s
as function of parameter d.

If we accept that this bifurcated convergence of the skeleton points of the curve is being sought, the consequences are the followings:

• the cuve has infinite arc length, since it oscillates with an almost constant amplitude infinite times before it reaches the x axis,
• it has a discontinuity a y=0, since there is no such a small e region around this point that the curve would not oscillate between two different x coordinates.

Note, however that even in case of not very small d values, the amplitude of the oscillation is extremly small, e.g. in case of d=0.1, the converged distance between the two different x coordinates is 1.4 × 10^-11 times the length of the specific chord, according to the numerical results.

The next question is that: how can we add "flesh" to the "skeleton", i.e. how can we fill in the regions between the points T_i. To this end, the most important discernment is that the curve can be defined "freely" within the region determined by the points C and E (or D and F). The "free" choice of this section of the curve does not mean an absolute freedom, obviously some constraints need to be applied. This freedom is well illustrated, however, in Fig. 8.

Figure 7 depicts the method, how a full curve can be created from starting a pre-defined section between points C and E. Starting from the section marked by 1 we can map it point by point to section 2. Utilizing the symmetric nature of the expected curve we obtain the centrally symmetric counterpart of section 2, which is marked by 2’. By projecting this section passing the same chord center as previously, we obtain the section 3 and its symmetric counterpart 3’, etc. Note, that it can be shown rather easily that the derivative of the curve goes through the "skeleton" points continuously.

Figure 7 The method of generating a continuous curve meshing all the skeleton points and satisfying the two-center equichordal condition

In Fig. 8 we present a variety of such curves in "full" (i.e. up to the depth of iterating 50 skeleton points and the curve sections in between). The equations of the freely defined curve sections are as follows:

(1)

(2)

(3)

(4)

(5) 1

Figure 8 Five different "flesh" options fitting the same "skeleton"

In Figure 9 one can see, how the curves undulate (even the curve No. 5, which is the recursive map of a constant function) while they approach the x axis. This figure obviously raises the question: how could we generate the minimum length curve from among these infinite length curves?

Figure 9 The same five curves in the vicinity of the point of singularity

What is presented above only illustrates that the equichordal condition can not be satisfied by some Jordan curve. This has been proved, however, by Rychlik in 1997 [Rychlik, 1997 ] in a major paper. Perhaps, this paper presents some novelty by introducing the “flesh” and “skeleton” approach.

[Croft, 1991] H. Croft, K. Falconer, R. Guy: Unsolved Problems in Geometry, Springer 1991

[Rychlik, 1997 ] M. Rychlik: A Complete Solution to the Equichordal Point Problem of Fujiwara, Blaschke, Rothe and Weizenböck.
Inventiones Math., 129(1):141-212, 1997.