Let be three points in a plane. We define the point on the ray located at a distance of . It is as has been attracted to but kept at distance. We continue by defining the point on the ray located at a distance of .
On the whole, we define from a recurrent sequence taking values in and such that, for (with where is a norm):
We would like to study how behaves this sequence for large .
Fig. 1. Illustration (see Fig. 11 for details)
Except for the last part of this post, we select the Euclidian norm and identify the plane with the complex plane.
Understanding the recurrence on an example
We illustrate the first steps of the sequence when , , and . The construction of is shown in Fig. 2. This can be computed from the formula:
Fig. 2. Construction of from and
Next points and are more difficult to calculate from the formula, and we only provide the construction (Fig. 3 and 4).
Fig. 3. Construction of from and
Fig. 4. Construction of from and
After some steps, we obtain adherent points forming an equilateral triangle. Initial and final steps are shown in Fig. 5.
Fig. 5. Initial and final steps
Note that the sequence may be undefined for some initial triplets (for example when ).
Reducing dimension of the problem
Each triplet contains real parameters. We will show that we can reduce the triangle pursuit problem to parameter without loss of generality. Explicitly, our final parameter will be , related with triplet . Calculations are tedious, so you can skip them at first reading.
Applying rotation and translation
Suppose that is well-defined from triplet . Let and . Let for :
Then, for , We rewrite as follows:
Because is defined with the Euclidian norm, we obtain:
Since exists, exists and we define:
We can continue and define such that for all :
From 6 to 3 parameters
Suppose as before that is well-defined from triplet .
Rotation and translation have released degree of freedom. In this paragraph, we select and to obtain a triplet verifying those conditions:
Positions of are illustrated in Fig. 6.
Fig. 6. Typical positions of and after transformation
First, we have , otherwise cannot be defined. Then, we let:
We compute :
verify the conditions, so the conclusion.
From 3 to 1 parameters
We consider as is the last paragraph.
From the recurrence relation, we obtain: regardless of . The term is not used for subsequent terms, so we can let and consider
Then, we observe that , otherwise cannot be defined. From the recurrence relation, we obtain: regardless of remaining . The term is not used for subsequent terms, so we can let and consider
This construction is illustrated in Fig. 7.
Fig. 7. From parameters to parameter
We have reduced the problem to dimension without loss of generality. We observe that parameters are impossible. We are now interested to understand the behavior of the sequence as a function of .
In this section, we let and .
First terms are easy to compute:
After that, I follow indication provided by achille hui here.
Let for . represents the vector from to . For , we have . So for , there exists such that . represents the angle between and .
For , we observe that triangle is isocele with angles , and (see Fig. 8 for details). It follows i.e.
Fig. 8. Angles in triangle
After some calculations, we get for :
[this formula in only valid in the interval ].
For each and , we observe that has adherent points forming an equilateral triangle.
We map each initial triplet to the corresponding adherent points.
We show in Fig. 9 the mapping from to the corresponding adherent points. Images of components and are symmetric with respect to the x-axis.
Fig. 9. Mapping from (left) to the corresponding adherent points (right). Bright colors correspond to small values of , and faded colors to larger values.
We restrict the mapping on the interval and show a more detailed plot in Fig. 10. Notice that triangle corresponding to remains unchanged by the mapping.
Fig. 10. Mapping from (left) to the corresponding adherent points (right).
Illustration with other norms
Let , , and .
Map of a rotation
We are interested to see mapping of for .
When is the Euclidian norm, we already know the global behavior. But with one-norm or maximum norm, strange figures are obtained. Those mappings are depicted in Fig. 11.
Fig. 11. Mapping from to the corresponding adherent points for one-norm, Euclidian norm and maximum norm respectively.
Map of an homothety
We are interested to see mapping of for .
The mappings are depicted in Fig. 12.
Fig. 12. Mapping from to the corresponding adherent points for one-norm, Euclidian norm and maximum norm respectively.