# Triangle pursuit

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 select:

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 .

**Formula for **

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 ].

**Adherent points**

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.*

**References**

- Code is available on my github. Many examples are provided, and code contains some generalization with more initial points, higher dimension, etc.
- I wrote a question in math.stackexchange asking about the asymptotic behavior of . Thanks for achille hui for his comment.