# Triangle pursuit

Let \(x_1, x_2, x_3\) be three points in a plane. We define \(x_4\) the point on the ray \([x_3 x_1)\) located at a distance \(1\) of \(x_3\). It is as \(x_1\) has been attracted to \(x_3\) but kept at distance. We continue by defining \(x_5\) the point on the ray \([x_4 x_2)\) located at a distance \(1\) of \(x_4\).

On the whole, we define from \((x_1, x_2, x_3)\) a recurrent sequence taking values in \(\mathbb{R}^2\) and such that, for \(n \geq 4\) (with \(N: x \mapsto \frac{x}{\| x \|}\) where \(\|.\|\) is a norm):

\[x_{n} = x_{n-1} - N(x_{n-1} - x_{n-3}).\]We would like to study how behaves this sequence for large \(n\).

*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 \(x_1=(0,0)\), \(x_2=(1,0)\), and \(x_3=(1,1)\). The construction of \(x_4\) is shown in Fig. 2. This can be computed from the formula:

\[x_{4} = x_{3} - N(x_{3} - x_{1}) = (1,1) - N((1,1)) = (1 - 1/\sqrt{2}, 1 - 1/\sqrt{2}).\]

*Fig. 2. Construction of \(x_4\) from \(x_1\) and \(x_3\)*

Next points \(x_5\) and \(x_6\) are more difficult to calculate from the formula, and we only provide the construction (Fig. 3 and 4).

*Fig. 3. Construction of \(x_5\) from \(x_2\) and \(x_4\)*

*Fig. 4. Construction of \(x_6\) from \(x_3\) and \(x_5\)*

After some steps, we obtain \(3\) 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 \(x_1 = x_2 = x_3\)).

**Reducing dimension of the problem**

Each triplet contains \(6\) real parameters. We will show that we can reduce the *triangle pursuit* problem to \(1\) parameter without loss of generality. Explicitly, our final parameter will be \(t \in (0, 2 \pi) \setminus \lbrace \pi \rbrace\), related with triplet \((x_1, x_2, x_3) = (0, 1, e^{it})\).
Calculations are tedious, so you can skip them at first reading.

*Applying rotation and translation*

Suppose that \((x_n)_n\) is well-defined from triplet \((x_1, x_2, x_3)\). Let \(\theta \in [0, 2 \pi)\) and \(b \in \mathbb{C}\). Let for \(k \in \lbrace 1, 2, 3 \rbrace\): \(x'_k := e^{-i \theta} (x_k - b).\)

Then, for \(k \in \lbrace 1, 2, 3 \rbrace\), \(x_k = e^{i \theta} x'_k + b.\) We rewrite \(x_4\) as follows:

\[x_4 = x_3 - N(x_3 - x_1) = e^{i \theta} x'_3 + b - N(e^{i \theta} x'_3 - e^{i \theta} x'_1).\]Because \(N(.)\) is defined with the Euclidian norm, we obtain:

\[x_4 = e^{i \theta} x'_3 + b - e^{i \theta} N( x'_3 - x'_1) = e^{i \theta} \left(x'_3 - N( x'_3 - x'_1) \right) + b.\]Since \(x_4\) exists, \(x'_3 - N( x'_3 - x'_1)\) exists and we define: \(x'_4 := x'_3 - N( x'_3 - x'_1).\)

We can continue and define \((x'_n)\) such that for all \(n\): \(x'_n := e^{i \theta} x'_n + b.\)

*From 6 to 3 parameters*

Suppose as before that \((x_n)_n\) is well-defined from triplet \((x_1, x_2, x_3)\).

Rotation and translation have released \(3\) degree of freedom. In this paragraph, we select \(\theta\) and \(b\) to obtain a triplet \((x'_1, x'_2, x'_3)\) verifying those \(3\) conditions:

\[x'_1 \text{ on the ray } ]x'_3 0)~~~;~~~x'_2 \in \mathbb{R}^{+}~~~;~~~\| x'_3 \| = 1.\]Positions of \(x'_1, x'_2, x'_3\) are illustrated in Fig. 6.

*Fig. 6. Typical positions of \(x'_1, x'_2\) and \(x'_3\) after transformation*

First, we have \(x_3 \neq x_1\), otherwise \(x_4\) cannot be defined. Then, we let:

\[s:= \text{Arg}(x_3 - x_1) \in [0, 2 \pi),\] \[r:= 1 - \| x_3 - x_1 \| \in (-\infty, 1),\] \[A \geq 0 \text{ and } t \in [0, 2 \pi) \text{ such that } A e^{-it} := 1 + (x_2 - x_3)e^{-is}.\]We select:

\[\theta := s - t,\] \[b:= x_3 - e^{is}.\]We compute \(x'_1, x'_2, x'_3\):

\[x'_1 = e^{-i \theta}(x_1 - b) = e^{-i s}e^{i t}(x_1 - x_3 + e^{is}) = e^{-i s}e^{i t}(-(1-r) e^{is} + e^{is}) = r e^{it}.\] \[x'_2 = e^{-i \theta}(x_2 - b) = e^{-is}e^{it}(x_2 - x_3) + e^{it} = e^{it} (1 + (x_2 - x_3) e^{-is}) = A.\] \[x'_3 = e^{-i \theta}(x_3 - b) = e^{-is}e^{it}e^{is} = e^{it}.\]\(x'_1, x'_2, x'_3\) verify the \(3\) conditions, so the conclusion.

*From 3 to 1 parameters*

We consider \((x'_1, x'_2, x'_3)\) as is the last paragraph.

From the recurrence relation, we obtain: \(x'_4 = 0\) regardless of \(r\). The term \(x'_1\) is not used for subsequent terms, so we can let \(r = 0\) and consider \((x''_1, x''_2, x''_3) = (0, x'_2, x'_3).\)

Then, we observe that \(A \neq 0\), otherwise \(x''_5\) cannot be defined. From the recurrence relation, we obtain: \(x''_5 = 0\) regardless of remaining \(A\). The term \(x''_2\) is not used for subsequent terms, so we can let \(A = 1\) and consider \((x'''_1, x'''_2, x'''_3) = (0, 1, e^{it}).\)

This construction is illustrated in Fig. 7.

*Fig. 7. From \(3\) parameters to \(1\) parameter*

We have reduced the problem to \(1\) dimension without loss of generality. We observe that parameters \(t = 0 \text{ mod } \pi\) are impossible. We are now interested to understand the behavior of the sequence as a function of \(t\).

**Formula for \(x_n\)**

In this section, we let \(t \in (0, \pi)\) and \((x_1, x_2, x_3) := (0, 1, e^{it})\).

First terms are easy to compute:

\[x_4 = 0,~~~x_5 = 1.\]After that, I follow indication provided by achille hui here.

Let \(u_n = x_{n} - x_{n-1}\) for \(n \geq 2\). \(u_n\) represents the vector from \(x_{n-1}\) to \(x_n\). For \(n \geq 4\), we have \(\|u_n\| = 1\). So for \(n \geq 5\), there exists \(\theta_n\) such that \(e^{i \theta_n} = u_n / u_{n-1}\). \(\theta_n\) represents the angle between \(u_{n-1}\) and \(u_{n}\).

For \(n \geq 5\), we observe that triangle \((x_{n-2}, x_{n-1}, x_n)\) is isocele with angles \(\pi - \theta_n\), \(\pi - \theta_{n+1}\) and \(\pi - \theta_{n+1}\) (see Fig. 8 for details). It follows \(\pi = \pi - \theta_n + 2(\pi - \theta_{n+1})\) i.e. \(\theta_{n+1} = \pi - (1/2) \theta_n.\)

*Fig. 8. Angles in triangle \((x_{n-2}, x_{n-1}, x_n)\)*

After some calculations, we get for \(n \geq 5\):

\[x_n = \sum_{k=0}^{n-5} e^{\frac{2ik \pi}{3}} e^{\frac{i}{3}\left( t - \frac{\pi}{3} \right) \left[ 1 - \left(-\frac{1}{2} \right)^k \right]}.\][this formula in only valid in the interval \((0, \pi)\)].

**Adherent points**

For each \(t \in (0, 2 \pi) \setminus \lbrace \pi \rbrace\) and \((x_1, x_2, x_3) := (0, 1, e^{it})\), we observe that \((x_n)_n\) has \(3\) adherent points forming an equilateral triangle.

We map each initial triplet to the corresponding adherent points.

We show in Fig. 9 the mapping from \((0, 2 \pi) \setminus \lbrace \pi \rbrace\) to the corresponding adherent points. Images of components \((0, \pi)\) and \((\pi, 2 \pi)\) are symmetric with respect to the x-axis.

*Fig. 9. Mapping from \((0, 2 \pi) \setminus \lbrace \pi \rbrace\) (left) to the corresponding adherent points (right). Bright colors correspond to small values of \(t\), and faded colors to larger values.*

We restrict the mapping on the interval \((0, \pi)\) and show a more detailed plot in Fig. 10. Notice that triangle corresponding to \(t = \pi / 3 \approx 1.05\) remains unchanged by the mapping.

*Fig. 10. Mapping from \((0, \pi)\) (left) to the corresponding adherent points (right).*

**Illustration with other norms**

Let \(x_1=(0,0)\), \(x_2=(1,0)\), and \(x_3=(1,1)\).

*Map of a rotation*

We are interested to see mapping of \(e^{i \theta}(x_1, x_2, x_3)\) for \(\theta \in (-\pi, \pi)\).

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 \(e^{i \theta}(x_1, x_2, x_3)\) 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 \(\theta \times (x_1, x_2, x_3)\) for \(\theta \in (-\pi, \pi)\).

The mappings are depicted in Fig. 12.

*Fig. 12. Mapping from \(\theta \times (x_1, x_2, x_3)\) 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 \((x_n)_n\). Thanks for achille hui for his comment.