Local $L^1$ sub-Finsler geometry in dimension 3: non-generic cases

Abstract

We study the local geometry of the sub-Finsler structure induced by a sub-Riemannian metric on a 3-dimensional manifold. We provide a description of the upper part of the cut locus of short geodesics, in some non generic cases.

Appendices

A: Permutations of the invariants

In Lemma 4, we saw that the geodesics of the sets $\Gamma _{-f}$ and $\Gamma _{\pm g}$ can be recovered from the $\Gamma _f$ ones, by applying a rotation around the z axis and a suitable permutation of the invariants. For the sake of clarity, in this section we provide these permutations, for the main invariants.

$\clubsuit$:

to obtain the coordinates (of the switching times) of a geodesics of the set $\Gamma _g$, starting from the corresponding expression associated with a geodesics belonging to the set $\Gamma _f$ one, we must perform a rotation of $\pi /2$ around the vertical axis (both in the space of the coordinates and in the momentum space) and apply the following transformations:

$$\begin{aligned} \begin{aligned}&A \mapsto -A\\&C_1 \mapsto C_2 \qquad C_2\mapsto C_1\\&D_1 \mapsto D_2 \qquad D_2\mapsto -D_1 \qquad E_1 \mapsto E_2 \qquad E_2\mapsto -E_1\\&\mathcal {c}_1\mapsto \mathcal {c}_2\qquad \mathcal {d}_1\mapsto \mathcal {d}_2 \qquad \mathcal {c}_2\mapsto -\mathcal {c}_1\qquad \mathcal {d}_2\mapsto -\mathcal {d}_1 \end{aligned} \end{aligned}$$

$\blacklozenge$:

analogously, to obtain the quantities associated with a geodesic of the set $\Gamma _{-f}$ from those associated with a geodesic of the set $\Gamma _f$, we must perform a rotation of $\pi$ around the vertical axis (in both spaces), and apply the following transformations:

$$\begin{aligned} \begin{aligned}&D_1 \mapsto -D_1 \qquad D_2\mapsto -D_2 \qquad E_1 \mapsto -E_1 \qquad E_2\mapsto -E_2\\&\mathcal {c}_1\mapsto -\mathcal {c}_1\qquad \mathcal {d}_1\mapsto -\mathcal {d}_1 \qquad \mathcal {c}_2\mapsto -\mathcal {c}_2\qquad \mathcal {d}_2\mapsto -\mathcal {d}_2 \end{aligned} \end{aligned}$$

$\spadesuit$:

finally, to obtain the quantities associated with a geodesic of the set $\Gamma _{-g}$ geodesic from a $\Gamma _f$ from those associated with a geodesic of the set $\Gamma _f$, we must perform a rotation of $3\pi /2$ around the vertical axis (in both spaces), and apply the following transformations:

$$\begin{aligned} \begin{aligned}&A \mapsto -A\\&C_1 \mapsto C_2 \qquad C_2\mapsto C_1\\&D_1 \mapsto -D_2 \qquad D_2\mapsto D_1 \qquad E_1 \mapsto -E_2 \qquad E_2\mapsto E_1\\&\mathcal {c}_1\mapsto -\mathcal {c}_2\qquad \mathcal {d}_1\mapsto -\mathcal {d}_2 \qquad \mathcal {c}_2\mapsto \mathcal {c}_1\qquad \mathcal {d}_2\mapsto \mathcal {d}_1 \end{aligned} \end{aligned}$$

Remark 5

We remark that $\spadesuit =\clubsuit \circ \blacklozenge =\blacklozenge \circ \clubsuit$.

B: Jets of the dynamics

1.1 Jets of the geodesics

Consider a geodesics of the set $\Gamma _f$, with initial adjoint covector $\varvec{\mu }_f(0)=(1,p_y^0,1/\rho _0)$. Let $\tau$ and $\mathscr {T}_3$ denote, respectively, the reparameterized time and the reparameterized third switching time, and set $\hat{\tau }=\tau -\mathscr {T}_3$. Then the jets of the geodesics are given by the following expressions:

$$\begin{aligned} \begin{aligned} x(\mathscr {T}_3+\hat{\tau })&= -(p_x^0+p_y^0)\rho _0+ \Big (4A p_x^0 p_y^0 -\frac{1}{6}C_1 (p_x^0+p_y^0)^3- 4C_2(p_x^0)^3\Big )\rho _0^3 \\&+\frac{1}{24} \bigg (-p_y^0 \big (24 (p_y^0)^2 \hat{\tau } \textrm{ax}_{11} + 12 p_y^0 \hat{\tau }^2 (\textrm{ax}_{12} + \textrm{ay}_{11}) + 8 \hat{\tau }^3 \textrm{ay}_{12} + (p_y^0)^3 (17 \textrm{ax}_{21} - 15 \omega _{xx31})\big ) \\&+ 4 p_x^0 \big (-6 (p_y^0)^2 \hat{\tau } (3 \textrm{ax}_{11} - 2 (\textrm{ax}_{12} + \textrm{ay}_{11})) - 6 p_y^0 \hat{\tau }^2 (\textrm{ax}_{12} + \textrm{ay}_{11} - 2 \textrm{ay}_{12}) - 2 \hat{\tau }^3 \textrm{ay}_{12} \\&+ (p_y^0)^3 (4 \textrm{ax}_{11} - 9 \textrm{ax}_{21} + 15 \omega _{xx31})\big ) - 6 (p_x^0)^2 \big (4 p_y^0 \hat{\tau } (3 \textrm{ax}_{11} - 4 (\textrm{ax}_{12} + \textrm{ay}_{11} - \textrm{ay}_{12})) \\&+ 2 \hat{\tau }^2 (\textrm{ax}_{12} + \textrm{ay}_{11} - 4 \textrm{ay}_{12}) + (p_y^0)^2 (16 \textrm{ax}_{11} + 8 \textrm{ax}_{12} + \textrm{ax}_{21} - 24 \textrm{ax}_{22} + 8 \textrm{ay}_{11} - 24 \textrm{ay}_{21}\\&- 15 \omega _{xx31} + 40 \omega _{xx32} + 40 \omega _{xy31} + 40 \omega _{yx31})\big ) + 4 (p_x^0)^3 \big (-6 \hat{\tau } (\textrm{ax}_{11} - 2 (\textrm{ax}_{12} + \textrm{ay}_{11} - 2 \textrm{ay}_{12})) \\&+ p_y^0 (12 \textrm{ax}_{11} + 48 \textrm{ax}_{12} + 7 \textrm{ax}_{21} + 48 \textrm{ay}_{11} + 16 \textrm{ay}_{12} - 72 \textrm{ay}_{22} + 15 \omega _{xx31} + 120 \omega _{xy32} \\&+ 120 \omega _{yx32} + 120 \omega _{yy31})\big ) - (p_x^0)^4 \big (32 \textrm{ax}_{11} + 48 \textrm{ax}_{12} - 15 \textrm{ax}_{21} + 80 \textrm{ax}_{22} + 48 \textrm{ay}_{11} + 256 \textrm{ay}_{12} \\&+ 80 \textrm{ay}_{21} - 15 \omega _{xx31} + 80 \omega _{xx32} + 80 \omega _{xy31} + 80 \omega _{yx31} + 960 \omega _{yy32}\big )\bigg )\rho _0^4+\mathcal {O}(\rho _0^5) \end{aligned} \end{aligned}$$

(38)

$$\begin{aligned} \begin{aligned} y(\mathscr {T}_3+\hat{\tau })&= (2p_x^0-\hat{\tau })\rho _0+ \Big (C_1 p_x^0(p_y^0-p_x^0)^2-2A (p_x^0)^2(p_y^0-p_x^0)+\frac{4}{3}C_2(p_x^0)^3\Big )\rho _0^3 \\&+\frac{1}{6} \bigg (-p_y^0 \hat{\tau } (6 (p_y^0)^2 \textrm{ax}_{21} + 3 p_y^0 \hat{\tau } (\textrm{ax}_{22} + \textrm{ay}_{21}) + 2 \hat{\tau }^2 \textrm{ay}_{22}) \\&+ 2 p_x^0 \big ((p_y^0)^2 \hat{\tau } (-9 \textrm{ax}_{21} + 6 (\textrm{ax}_{22} + \textrm{ay}_{21})) - 3 p_y^0 \hat{\tau }^2 (\textrm{ax}_{22} + \textrm{ay}_{21} - 2 \textrm{ay}_{22}) - \hat{\tau }^3 \textrm{ay}_{22} \\&+ 15 (p_y^0)^3 (\textrm{ax}_{21} - \omega _{xx31})\big ) + 3 (p_x^0)^2 \big (2 p_y^0 \hat{\tau } (-3 \textrm{ax}_{21} + 4 (\textrm{ax}_{22} + \textrm{ay}_{21} - \textrm{ay}_{22})) \\&- \hat{\tau }^2 (\textrm{ax}_{22} + \textrm{ay}_{21} - 4 \textrm{ay}_{22}) + 2 (p_y^0)^2 (3 \textrm{ax}_{11} - 11 \textrm{ax}_{21} + 5 (-\textrm{ax}_{22} - \textrm{ay}_{21} + 3 \omega _{xx31}\\&+ \omega _{xx32} + \omega _{xy3 1} + \omega _{yx31}))\big ) - 2 (p_x^0)^3 \big (3 \hat{\tau } (\textrm{ax}_{21} - 2 (\textrm{ax}_{22} + \textrm{ay}_{21} - 2 \textrm{ay}_{22})) \\&+ p_y^0 (18 \textrm{ax}_{11} + 4 \textrm{ax}_{12} + 3 \textrm{ax}_{21} - 30 \textrm{ax}_{22} + 4 \textrm{ay}_{11} - 30 \textrm{ay}_{21} - 20 \textrm{ay}_{22} + 45 \omega _{xx31} \\&+ 30 \omega _{xx32} + 30 \omega _{xy31} + 20 \omega _{xy32} + 30 \omega _{yx31} + 20 \omega _{yx32} + 20 \omega _{yy31})\big ) \\&+ 2 (p_x^0)^4 \big (9 \textrm{ax}_{11} + 4 \textrm{ax}_{12} + 5 \textrm{ax}_{21} + 9 \textrm{ax}_{22} + 4 \textrm{ay}_{11} + 2 \textrm{ay}_{12} + 9 \textrm{ay}_{21} - 36 \textrm{ay}_{22} \\&+ 15 \omega _{xx31} + 15 \omega _{xx32} + 15 \omega _{xy31} + 20 \omega _{xy32} + 15 \omega _{yx31} + 20 \omega _{yx32} + 20 \omega _{yy31} + 30 \omega _{yy32}\big )\bigg )\rho _0^4\\&+\mathcal {O}(\rho _0^5) \end{aligned} \end{aligned}$$

(39)

$$\begin{aligned} \begin{aligned} z(\mathscr {T}_3+\hat{\tau })&=\frac{1}{2}\big (6(p_x^0)^2+p_y^0(\hat{\tau }-2p_x^0)+\hat{\tau }p_x^0\big )\rho _0^2 \\&+ \frac{1}{6}\bigg (4 (p_x^0)^4 (19 \textrm{ax}_{31} + 15 \textrm{ax}_{32} + 15 \textrm{ay}_{31} + 44 \textrm{ay}_{32}) - p_y^0 \hat{\tau } \big (2 (p_y^0)^2 \textrm{ax}_{31} \\&+ 3 (p_y^0) \hat{\tau } (\textrm{ax}_{32} + \textrm{ay}_{31})\\&+ 2 \hat{\tau }^2 \textrm{ay}_{32}\big ) - 2 p_x^0 \big (10 (p_y^0)^3 \textrm{ax}_{31} + 3 (p_y^0)^2 \hat{\tau } (\textrm{ax}_{31} - 2 (\textrm{ax}_{32} + \textrm{ay}_{31})) \\&+ 3 p_y^0 \hat{\tau }^2 (\textrm{ax}_{32} + \textrm{ay}_{31} - 2 \textrm{ay}_{32}) + \hat{\tau }^3 \textrm{ay}_{32}\big ) + 3 (p_x^0)^2 \big (4 (p_y^0)^2 (11 \textrm{ax}_{31} + \textrm{ax}_{32} + \textrm{ay}_{31}) \\&- \hat{\tau }^2 (\textrm{ax}_{32} + \textrm{ay}_{31} - 4 \textrm{ay}_{32}) - 2 p_y^0 \hat{\tau } (\textrm{ax}_{31} + 4 (\textrm{ax}_{32} + \textrm{ay}_{31} + \textrm{ay}_{32}))\big ) \\&- 2 (p_x^0)^3 \big (p_y^0 (78 \textrm{ax}_{31} + 60 \textrm{ax}_{32} + 60 \textrm{ay}_{31} + 8 \textrm{ay}_{32}) + \hat{\tau } (\textrm{ax}_{31} - 6 (\textrm{ax}_{32} \\&+ \textrm{ay}_{31} + 6 \textrm{ay}_{32}))\big )\bigg ) \rho _0^4\\&+ \frac{1}{48} \bigg (2 (p_x^0)^5 \big (100 \textrm{ax}_{11} + 32 \textrm{ax}_{12} + 131 \textrm{ax}_{21} + 180 \textrm{ax}_{22} + 32 \textrm{ay}_{11} - 184 \textrm{ay}_{12} + 180 \textrm{ay}_{21} \\&- 16 \textrm{ay}_{22} + 171 \omega _{xx31} + 204 \omega _{xx32} + 204 \omega _{xy31} + 208 \omega _{xy32} + 204 \omega _{yx31} + 208 \omega _{yx32} \\&+ 208 \omega _{yy31} + 888 \omega _{yy32}\big ) + p_y^0 \hat{\tau } \big ((p_y^0)^3 (17 \textrm{ax}_{21} + 9 \omega _{xx31}) \\&+ 12 (p_y^0)^2 \hat{\tau } (\textrm{ax}_{11} + \omega _{xx32} + \omega _{xy31} + \omega _{yx31}) + 4 p_y^0 \hat{\tau }^2 (\textrm{ax}_{12} + \textrm{ay}_{11} + 2 (\omega _{xy32} + \omega _{yx32} + \omega _{yy31})) \\&+ 2 \hat{\tau }^3 (\textrm{ay}_{12} + 3 \omega _{yy32})\big ) - 2 p_x^0 ((p_y^0)^4 (53 \textrm{ax}_{21} - 51 \omega _{xx31}) + 2 (p_y^0)^3 \hat{\tau } \big (4 \textrm{ax}_{11} + 3 (-3 \textrm{ax}_{21} \\&- 3 \omega _{xx31} + 4 \omega _{xx32} + 4 \omega _{xy31} + 4 \omega _{yx31})\big ) - 6 (p_y^0)^2 \hat{\tau }^2 (3 \textrm{ax}_{11} - 2 \textrm{ax}_{12} - 2 \textrm{ay}_{11} + 3 \omega _{xx32} \\&+ 3 \omega _{xy31} - 4 \omega _{xy32} + 3 \omega _{yx31} - 4 \omega _{yx32} - 4 \omega _{yy31}) - 4 p_y^0 \hat{\tau }^3 \big (\textrm{ax}_{12} + \textrm{ay}_{11} + 2 (-\textrm{ay}_{12} \\&+ \omega _{xy32} + \omega _{yx32} + \omega _{yy31} - 3 \omega _{yy32})\big ) - \hat{\tau }^4 (\textrm{ay}_{12} + 3 \omega _{yy32})) + 2 (p_x^0)^2 \big (-4 (p_y^0)^3 (11 \textrm{ax}_{11} \\&- 69 \textrm{ax}_{21} + 9 (-\textrm{ax}_{22} - \textrm{ay}_{21} + 11 \omega _{xx31} + \omega _{xx32} + \omega _{xy31} + \omega _{yx31})) + 3 (p_y^0)^2 \hat{\tau } (16 \textrm{ax}_{11} \\&+ 8 \textrm{ax}_{12} + \textrm{ax}_{21} - 24 \textrm{ax}_{22} + 8 \textrm{ay}_{11} - 24 \textrm{ay}_{21} + 9 \omega _{xx31} + 16 \omega _{xx32} + 16 \omega _{xy31} + 16 \omega _{xy32} + 16 \omega _{yx31} \\&+ 16 \omega _{yx32} + 16 \omega _{yy31}) + 2 \hat{\tau }^3 (\textrm{ax}_{12} + \textrm{ay}_{11} + 2 (-2 \textrm{ay}_{12} + \omega _{xy32} + \omega _{yx32} + \omega _{yy31} - 6 \omega _{yy32})) \\&+ 6 p_y^0 \hat{\tau }^2 (3 \textrm{ax}_{11} - 4 \textrm{ax}_{12} - 4 \textrm{ay}_{11} + 4 \textrm{ay}_{12} + 3 \omega _{xx32} + 3 \omega _{xy31} - 8 \omega _{xy32} + 3 \omega _{yx31} \\&- 8 \omega _{yx32} - 8 \omega _{yy31} + 12 \omega _{yy32})\big ) + 4 (p_x^0)^3 \big (3 (p_y^0)^2 (26 \textrm{ax}_{11} - 25 \textrm{ax}_{21} - 46 \textrm{ax}_{22} - 46 \textrm{ay}_{21} - 8 \textrm{ay}_{22} \\&+ 111 \omega _{xx31} + 62 \omega _{xx32} + 62 \omega _{xy31} + 8 \omega _{xy32} + 62 \omega _{yx31} + 8 \omega _{yx32} + 8 \omega _{yy31}) \\&+ 3 \hat{\tau }^2 (\textrm{ax}_{11} - 2 \textrm{ax}_{12} - 2 \textrm{ay}_{11} + 4 \textrm{ay}_{12} + \omega _{xx32} + \omega _{xy31} - 4 \omega _{xy32} + \omega _{yx31} \\ \end{aligned} \end{aligned}$$

(40)

$$\begin{aligned} \begin{aligned}&- 4 \omega _{yx32} - 4 \omega _{yy31} + 12 \omega _{yy32}) - p_y^0 \hat{\tau } (12 \textrm{ax}_{11} + 48 \textrm{ax}_{12} + 7 \textrm{ax}_{21} + 48 \textrm{ay}_{11} + 16 \textrm{ay}_{12} \\&- 72 \textrm{ay}_{22} - 9 \omega _{xx31} + 36 \omega _{xx32} + 36 \omega _{xy31} + 72 \omega _{xy32} + 36 \omega _{yx31} + 72 \omega _{yx32} + 72 \omega _{yy31} + 48 \omega _{yy32})\big ) \\&+ (p_x^0)^4 \big (-8 (p_y^0) (69 \textrm{ax}_{11} - 16 \textrm{ax}_{12} + 55 \textrm{ax}_{21} - 15 \textrm{ax}_{22} - 16 \textrm{ay}_{11} - 2 \textrm{ay}_{12} - 15 \textrm{ay}_{21} - 104 \textrm{ay}_{22} \\&+ 159 \omega _{xx31} + 87 \omega _{xx32} + 87 \omega _{xy31} + 136 \omega _{xy32} + 87 \omega _{yx31} + 136 \omega _{yx32} + 136 \omega _{yy31} + 18 \omega _{yy32}) \\&+ \hat{\tau } (32 \textrm{ax}_{11} + 48 \textrm{ax}_{12} - 15 \textrm{ax}_{21} + 80 \textrm{ax}_{22} + 48 \textrm{ay}_{11} + 256 \textrm{ay}_{12} + 80 \textrm{ay}_{21} + 9 \omega _{xx31} + 32 \omega _{xx32} \\&+ 32 \omega _{xy31} + 96 \omega _{xy32} + 32 \omega _{yx31} + 96 \omega _{yx32} + 96 \omega _{yy31} + 768 \omega _{yy32})\big ) \bigg )\rho _0^5 \end{aligned} \end{aligned}$$

(41)

Consider a geodesics of the set $\Gamma _f$, with initial adjoint covector $(1,p_y^0,1/\rho _0)$. Set now $\hat{\tau }=\tau -\mathscr {T}_4$. Then

$$\begin{aligned} \begin{aligned} x(\mathscr {T}_4+\hat{\tau })&=x(\mathscr {T}_4)+\hat{\tau }\rho _0+\mathcal {O}(\rho _0^5) \end{aligned} \end{aligned}$$

(42)

$$\begin{aligned} \begin{aligned} y(\mathscr {T}_4+\hat{\tau })&=y(\mathscr {T}_4)+\mathcal {O}(\rho _0^5) \end{aligned} \end{aligned}$$

(43)

$$\begin{aligned} \begin{aligned} z(\mathscr {T}_4+\hat{\tau })&=z(\mathscr {T}_4) +2(p_x^0)^2\hat{\tau }(C_1p_y^0-Ap_x^0)\rho _0^4\\&+\hat{\tau }\big ((p_y^0)^2 D_1+2 E_2 p_y^0 p_x^0-\frac{1}{3}\mathcal {c}_1(p_x^0)^2\big )(p_x^0)^2\rho _0^5+\mathcal {O}(\rho _0^6) \end{aligned} \end{aligned}$$

(44)

1.2 Switching times of the geodesics of the set $\Gamma _f$

$$\begin{aligned} \begin{aligned} \mathscr {T}_1&=(p_x^0-p_y^0) +\frac{4}{3} \textrm{ax}_{31} (p_x^0-p_y^0)^3\rho _0^2\\&\quad + \frac{1}{24} \rho _0^3 \big (15 \textrm{ax}_{21} (p_x^0-p_y^0)^4+32 p_x^0 \textrm{ax}_{11}(p_x^0-p_y^0)^3+32 p_y^0 \textrm{ax}_{21} (p_x^0-p_y^0)^3\\&\quad +15 \omega _{xx31} (p_x^0-p_y^0)^4\big ) \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \mathscr {T}_2&=\mathscr {T}_1+2 p_x^0+\frac{2}{3} \big (12 (p_x^0)^2 \textrm{ax}_{32} (p_x^0-p_y^0)+12 p_x^0 \textrm{ax}_{31} (p_x^0-p_y^0)^2+16 (p_x^0)^3 \textrm{ay}_{32}\\&+12 (p_x^0)^2 \textrm{ay}_{31} (p_x^0-p_y^0)\big )\rho _0^2\\&+\frac{1}{24} \Big (32 (p_x^0)^3 \textrm{ax}_{12} (p_x^0-p_y^0)+72 (p_x^0)^2 \textrm{ax}_{11}(p_x^0-p_y^0)^2+120 (p_x^0)^2 \textrm{ax}_{22} (p_x^0-p_y^0)^2\\&+192 (p_x^0)^2 p_y^0 \textrm{ax}_{22} (p_x^0-p_y^0)+120 p_x^0 \textrm{ax}_{21} (p_x^0-p_y^0)^3+192 p_x^0 p_y^0 \textrm{ax}_{21} (p_x^0-p_y^0)^2\\&+16 (p_x^0)^4 \textrm{ay}_{12}\\&+32 (p_x^0)^3 \textrm{ay}_{11} (p_x^0-p_y^0)+160 (p_x^0)^3 \textrm{ay}_{22} (p_x^0-p_y^0)+256 (p_x^0)^3 p_y^0 \textrm{ay}_{22}\\&+120 (p_x^0)^2 \textrm{ay}_{21} (p_x^0-p_y^0)^2\\&+192 (p_x^0)^2 p_y^0 \textrm{ay}_{21} (p_x^0-p_y^0)+240 (p_x^0)^4 \omega _{yy32}+160 (p_x^0)^3\omega _{xy32} (p_x^0-p_y^0)\\&+160 (p_x^0)^3 \omega _{yx32} (p_x^0-p_y^0)+160 (p_x^0)^3 \omega _{yy31} (p_x^0-p_y^0)+120 (p_x^0)^2 \omega _{xx32} (p_x^0-p_y^0)^2\\&+120 (p_x^0)^2 \omega _{xy31} (p_x^0-p_y^0)^2\\&+120 (p_x^0)^2 \omega _{yx31} (p_x^0-p_y^0)^2+120 p_x^0 \omega _{xx31} (p_x^0-p_y^0)^3\Big ) \rho _0^3 \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \mathscr {T}_3&= \mathscr {T}_2+2 p_x^0+ \Bigg (\frac{8}{3} (p_x^0)^3 \textrm{ax}_{31}-16 (p_x^0)^2 p_y^0 \textrm{ax}_{32}+8 p_x^0 (p_y^0)^2 \textrm{ax}_{31}+32 (p_x^0)^3 \textrm{ay}_{32}\\&-16 (p_x^0)^2 p_y^0 \textrm{ay}_{31}\Bigg )\rho _0^2\\&+\frac{1}{3} \rho _0^3 \Big (-2 (p_x^0)^4 \textrm{ax}_{11}+10 (p_x^0)^4 \textrm{ax}_{22}+12 (p_x^0)^3 (p_y^0) \textrm{ax}_{12}-7 (p_x^0)^3 p_y^0 \textrm{ax}_{21}\\&-6 (p_x^0)^2 (p_y^0)^2 \textrm{ax}_{11}\\&-18 (p_x^0)^2 (p_y^0)^2 \textrm{ax}_{22}+9 (p_x^0) (p_y^0)^3 \textrm{ax}_{21}-24 (p_x^0)^4 \textrm{ay}_{12}+10 (p_x^0)^4 \textrm{ay}_{21}\\&+12 (p_x^0)^3 (p_y^0) \textrm{ay}_{11}+36 (p_x^0)^3 (p_y^0) \textrm{ay}_{22}\\&-18 (p_x^0)^2 (p_y^0)^2 \textrm{ay}_{21}+10 (p_x^0)^4 \omega _{xx32}+10 (p_x^0)^4 \omega _{xy31}+10 (p_x^0)^4 \omega _{yx31}+120 (p_x^0)^4 \omega _{yy32}\\&-15 (p_x^0)^3 p_y^0 \omega _{xx31}\\&-60 (p_x^0)^3 p_y^0\omega _{xy32}-60 (p_x^0)^3 p_y^0 \omega _{yx32}-60 (p_x^0)^3 p_y^0 \omega _{yy31}\\&+30 (p_x^0)^2 (p_y^0)^2 \omega _{xx32}+30 (p_x^0)^2 (p_y^0)^2 \omega _{xy31}\\&+30 (p_x^0)^2 (p_y^0)^2 \omega _{yx31}-15 p_x^0 (p_y^0)^3 \omega _{xx31}\Big ) \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \mathscr {T}_4&=\mathscr {T}_3+2 p_x^0+ \Big (8 (p_x^0)^3 \textrm{ax}_{31}-8 (p_x^0)^3 \textrm{ax}_{32}+16 (p_x^0)^2 p_y^0 \textrm{ax}_{31}-8 (p_x^0)^2 p_y^0 \textrm{ax}_{32}\\&+8 p_x^0 (p_y^0)^2 \textrm{ax}_{31}-8 (p_x^0)^3 \textrm{ay}_{31}\\&+\frac{32}{3} (p_x^0)^3 \textrm{ay}_{32}-8 (p_x^0)^2 p_y^0 \textrm{ay}_{31}\Big )\rho _0^2\\&+\frac{1}{3} \Big (9 (p_x^0)^4 \textrm{ax}_{11}-4 (p_x^0)^4 \textrm{ax}_{12}-15 (p_x^0)^4 \textrm{ax}_{21}+15 (p_x^0)^4 \textrm{ax}_{22}+18 (p_x^0)^3 (p_y^0) \textrm{ax}_{11}\\&-4 (p_x^0)^3 p_y^0 \textrm{ax}_{12}\\&-21 (p_x^0)^3 p_y^0 \textrm{ax}_{21}+6 (p_x^0)^3 p_y^0 \textrm{ax}_{22}+9 (p_x^0)^2 (p_y^0)^2 \textrm{ax}_{11}+3 (p_x^0)^2 (p_y^0)^2 \textrm{ax}_{21}\\&-9 (p_x^0)^2 (p_y^0)^2 \textrm{ax}_{22}\\&+9 p_x^0 (p_y^0)^3 \textrm{ax}_{21}-4 (p_x^0)^4 \textrm{ay}_{11}+2 (p_x^0)^4 \textrm{ay}_{12}+15 (p_x^0)^4 \textrm{ay}_{21}-20 (p_x^0)^4 \textrm{ay}_{22}\\&-4 (p_x^0)^3 p_y^0 \textrm{ay}_{11}\\&+6 (p_x^0)^3 (p_y^0) \textrm{ay}_{21}+12 (p_x^0)^3 p_y^0 \textrm{ay}_{22}-9 (p_x^0)^2 (p_y^0)^2 \textrm{ay}_{21}-15 (p_x^0)^4 \omega _{xx31}\\&+15 (p_x^0)^4 \omega _{xx32}+15 (p_x^0)^4 \omega _{xy31}\\&-20 (p_x^0)^4\omega _{xy32} +15 (p_x^0)^4 \omega _{yx31}-20 (p_x^0)^4 \omega _{yx32}-20 (p_x^0)^4 \omega _{yy31}\\&+30 (p_x^0)^4 \omega _{yy32}-45 (p_x^0)^3 (p_y^0) \omega _{xx31}+30 (p_x^0)^3 (p_y^0) \omega _{xx32}+30 (p_x^0)^3 p_y^0 \omega _{xy31}\\&-20 (p_x^0)^3 p_y^0\omega _{xy32}\\&+30 (p_x^0)^3 p_y^0 \omega _{yx31}-20 (p_x^0)^3 p_y^0 \omega _{yx32}-20 (p_x^0)^3 p_y^0 \omega _{yy31}-45 (p_x^0)^2 (p_y^0)^2 \omega _{xx31}\\&+15 (p_x^0)^2 (p_y^0)^2 \omega _{xx32}\\&+15 (p_x^0)^2 (p_y^0)^2 \omega _{xy31}+15 (p_x^0)^2 (p_y^0)^2 \omega _{yx31}-15 p_x^0 (p_y^0)^3 \omega _{xx31}\Big )\rho _0^3 \end{aligned} \end{aligned}$$

Analysis of the upper part of the cut locus

In this Section we provide the details of the analysis that leads to the main results of the article. For each of the three considered cases ($\varvec{A_-}$,$\varvec{B_+}$ and $\varvec{B_-}$), we consider separately the sub-cases $C_2<0$ (in which all geodesics of the sets $\Gamma _{\pm g}$ lose optimality before the fourth switching time) and $C_2>0$ (in which they may lose optimality after the fourth switching time, see Section 3.2).

Notation

For the sake of readability, in the following we will sometimes omit to specify that we are providing only the leading terms in the expansion with respect to $\zeta$ and $\rho _0$; in particular, this will be done in two cases: when we specify the intersection points (for instance, the value of the intersection between the fronts $\mathcal {F}_4$, $\mathcal {G}_4$ and $\bar{\mathcal {G}}_4$ is given by equation (45) up to fifth order terms in $\zeta$); when we describe the behavior of the geodesics according to the value of the initial adjoint vector (for instance, when we say that the geodesics with $p_y^0\in [-1,-1+c\rho _0^2]$ behave in some particular way, we mean $p_y^0\in [-1,-1+c\rho _0^2+\mathcal {O}(\rho _0^3)]$).

1.1 $E_1<-|D_1|$ (case ${A_-}$)

1.1.1 $C_2<0$

First of all, we focus on the Maxwell points of the geodesics belonging to the set $\Gamma _f$. In the nilpotent case, such geodesics lose optimality at the fourth swicthing time, when intersecting the front $\mathcal {G}_4$ and, in the case under concern, this intersection of fronts occurs (as sufficient conditions are satisfied); this suggest this intersection to cause global optimality loss for the geodesics of the class $\Gamma _f$, except maybe those with extreme values of the momentum, i.e. $|p_y^0|\sim 1$. To inspect more closely these ones, we look at the intersections $\mathcal {F}_4\cap \mathcal {G}_4$, $\mathcal {G}_4\cap \bar{\mathcal {F}}_4$ and $\bar{\mathcal {G}}_4\cap \mathcal {F}_4$ close to the origin, by taking the limits of equations (30)-(51) and as $\gamma \rightarrow -1$, $\upeta \rightarrow -1$ and $\upbeta \rightarrow 1$, respectively. We see that the reciprocal position of the intersections depend on the sign of the invariant A: indeed, the curve described by equation (30) crosses the one described by (52) if $A>0$, and the curve described by (51) if $A<0$, as can bee seen in Fig. 6. In the first case, the lines $\mathcal {F}_4\cap \mathcal {G}_4$ and $\bar{\mathcal {G}}_4\cap \mathcal {F}_4$ “bound” the front $\mathcal {F}_4$ (so, we guess that the cut locus of the geodesics of the set $\Gamma _f$ is contained in the union of these two lines); in the second one, it is the front $\mathcal {G}_4$ to be constrained (see Fig. 6).

For this reason, we must study the two cases separately.

$A>0$ The reasoning carried out few lines above suggests that, if $A>0$, the (suspension of the) cut locus of the geodesics of the set $\Gamma _f$ is contained in the union of the intersections $\mathcal {F}_4\cap \mathcal {G}_4$ and $\bar{\mathcal {G}}_4\cap \mathcal {F}_4$. In order to associate, with each geodesics, its cut point, we analyze the intersection among the suspension of three fronts $\mathcal {F}_4$, $\mathcal {G}_4$ and $\bar{\mathcal {G}}_4$. We find that, at time

$$\begin{aligned} \varvec{T}=8\zeta +\Big (\frac{8}{3}C_2 -4A\Big )\zeta ^3+\mathcal {O}(\zeta ^5) \end{aligned}$$

the three fronts are intersecting at the point

$$\begin{aligned} \Big (-4A\zeta ^3-\frac{4}{3}\big (2E_1+\mathcal {d}_1\big ) \zeta ^4,4A\zeta ^3-\frac{4}{3}\big (2E_2+\mathcal {d}_2\big ) \zeta ^4\Big ). \end{aligned}$$

(45)

The values of the corresponding adjoint vectors at time zero are respectively

$$\begin{aligned} \begin{aligned} \varvec{\mu }_f(0)=-1-4C_2\zeta ^2+\mathcal {O}(\zeta ^3)\\ \varvec{\mu }_g(0)=1+4\big (E_1+\frac{1}{3}D_1\big )\zeta ^3+\mathcal {O}(\zeta ^4)\\ \varvec{\mu }_{-g}(0)=-1+8A\zeta ^2+\mathcal {O}(\zeta ^3). \end{aligned} \end{aligned}$$

We can conclude that the geodesic associated with $\varvec{\mu }_f(0)=(1,p_y^0,1/ \rho _0)$ loses its global optimality by intersecting the geodesics belonging to the set $\Gamma _{- g}$ if $p_y^0\in [-1,-1-4C_2\rho _0^2]$ (we took indeed $\zeta =\rho _0+\mathcal {O}(\rho _0^2)$ here above), and those of the set $\Gamma _g$ otherwise.

We now consider the geodesics of the set $\Gamma _{-f}$. We recall that, since, for all of them, the conjugate time coincides with the fifth switching time, the cut time could be greater than the fourth switching time. However, as the geodesics of the set $\Gamma _{-f}$ cannot intersect neither those of the set $\Gamma _f$ (see Appendix 5-$\delta$-$\varepsilon$) nor those of the set $\Gamma _{-g}$ (because of (48)), we shall investigate their intersection with the trajectories of the set $\Gamma _g$.

From Appendix 5-$\upbeta$ we see that the intersection $\mathcal {G}_4\cap \bar{\mathcal {F}}_4$ involves only few geodesics of the set $\Gamma _{-f}$, that is, those associated with an adjoint vector $p_y^0\sim -1$; thus, we must consider also the intersection $\mathcal {G}_4\cap \bar{\mathcal {F}}_5$ (detailed in Appendix 5-$\iota$). We can then conclude that all geodesics of the set $\Gamma _{-f}$ lose their optimality by intersecting the trajectories from $\Gamma _{g}$; those with $p_y^0\in [1+4C_2\rho _0^2,1]$ lose optimality before their fourth switching times, the other ones after it.

We are left to describe the Maxwell set associated with the geodesics of the sets $\Gamma _{\pm g}$. Part of their Maxwell locus is already contained in the intersections pointed out above; for geodesics intersecting close to the origin, we are considering the only other intersection involving fourth bang fronts, that is, $\mathcal {G}_4\cap \bar{\mathcal {G}}_4$. Summing up, we can say that

• the geodesics of the set $\Gamma _{g}$ with initial momentum $(p_x^0,1,1/\rho _0)$ lose their optimality during their fourth bang arc, when intersecting the fourth arcs of $\Gamma _{f}$ (for $p_x^0\in [1+4\big (E_1+\frac{D_1}{3}\big )\rho _0^2,1]$), $\Gamma _{-g}$ (for $p_x^0\in [1-8A\rho _0^2,1+4\big (E_1+\frac{D_1}{3}\big )\rho _0^2,1]$) and the trajectories of the set $\Gamma _{-f}$.

• the geodesics of the set $\Gamma _{-g}$ with initial momentum $(p_x^0,-1,1/\rho _0)$ lose their optimality during their fourth bang arc, when intersecting the fourth arcs of $\Gamma _{f}$ (for $p_y^0\in [-1+8A\rho _0^2,1]$) and $\Gamma _{g}$, otherwise.

The suspension of the cut locus has three branches, each of that is at least $C^1$, at least up to the approximation in $\zeta$ that we considered. The graph of the suspension of the cut locus has been shown in Fig. 8 (left), and the cut locus itself in Fig. 10 (left).

$A<0$ Looking at Fig. 6, we guess that, when $A<0$, the (suspension of the) cut locus for the geodesics of the set $\Gamma _g$ is contained in the union of the intersections $\mathcal {F}_4\cap \mathcal {G}_4$ and $\mathcal {G}_4\cap \bar{\mathcal {F}}_4$. To determine exactly, for each geodesics, which intersections causes the loss of optimality, we proceed as above: we compare the suspension of the fronts (18), (20) and (22). By computations, we find that they meet at time $\varvec{T}=8\zeta +\big (\frac{8}{3}C_2+4A\big )\zeta ^3+\mathcal {O}(\zeta ^5)$ at the point

$$\begin{aligned} \Big (4A\zeta ^3-\frac{4}{3} (2E_1+\mathcal {d}_1)\zeta ^4,4A\zeta ^3-\frac{4}{3}\big (2E_2+\mathcal {d}_2\big )\zeta ^4\Big ). \end{aligned}$$

The values of the adjoint vectors at time zero are respectively $\varvec{\mu }_f(0)=(1,\gamma ,\rho _0)$, $\varvec{\mu }_f(0)=(\upbeta ,1,\widetilde{\rho }_0)$ and $\varvec{\mu }_{f-}(0)=(-1,\upnu ,\widetilde{\rho }_0)$, with $\rho _0,\widetilde{\rho }_0,\widehat{\rho }_0$ of order $\mathcal {O}(\zeta )$ and

$$\begin{aligned} \begin{aligned}&\gamma _1=\upbeta _1=\upnu _1=0\\&\gamma _2=-4C_2-8A\quad \upbeta _2=0 \quad \upnu _2=4C_2\\&\upbeta _3=4\big (E_1+\frac{1}{3}D_1\big ). \end{aligned} \end{aligned}$$

We can conclude that all geodesics of the set $\Gamma _g$ with $p_x^0\in [1+4(E_1+D_1/3)\rho _0^3,1]$ lose their optimality when intersecting the geodesics of the set $\Gamma _f$, whereas the others lose optimality by intersecting the geodesics of the set $\Gamma _{-f}$.

On the other hand, the geodesics of the set $\Gamma _{-g}$ meet (before the conjugate time) only the geodesics of the set $\Gamma _f$: then, these intersections constitute their cut point.

Let us now focus on the geodesics of the set $\Gamma _{\pm f}$. First of all, we notice that the front $\mathcal {F}_4$ intersects both the fronts $\bar{\mathcal {F}}_4$ and $\bar{\mathcal {F}}_5$ (see Appendices 5-$\delta$–5-$\varepsilon$), while $A<0$ forbids the intersection $\mathcal {G}_4\cap \bar{\mathcal {F}}_5$. Moreover, equating the jets of the expressions (18), (23) and (21), we can prove that the three fronts $\mathcal {F}_4$, $\bar{\mathcal {G}}_4$ and $\bar{\mathcal {F}}_{5}$ meet at the point

$$\begin{aligned} \Big (-4A\zeta ^3-\frac{4}{3} (2E_1+\mathcal {d}_1)\zeta ^4,-4A\zeta ^3-2\big (2E_1+D_1-\frac{1}{3}\mathcal {c}_1\big )\zeta ^4\Big ). \end{aligned}$$

Indeed, putting $p_y^0=-1$ in (21); $\gamma _1=0$, $\gamma _2=-4C_2$ and $\gamma _3=4E_2+\frac{4}{3}D_2$ in (18); $\upeta _1=\upeta _2=\upeta _3=0$ in (23); $\varvec{T}=8\zeta +\big (4A+\frac{8}{3}C_2\big )\zeta ^3$ in all of them, we see that the three expression are equal up to the third order in x, y and to the fourth order in $\zeta$. We conclude that

• the geodesics of the set $\Gamma _f$ with initial momentum $(1,p_y^0,\rho _0)$, with $p_y^0\in [-1,-1-4C_2\rho _0^2+4(E_2+D_2/3)\rho _0^3]$, lose optimality by intersecting the front $\bar{\mathcal {G}}_4$; those with $p_y^0\in [-1-4(C_2+2A)\rho _0^2+4(E_2+D_2/3)\rho _0^3,1]$, by intersecting the front $\mathcal {G}_4$; the others, when they meet the geodesics of the set $\Gamma _{-f}$.

• the geodesics of the set $\Gamma _{-f}$ with initial momentum $(-1,p_y^0,\rho _0)$ lose optimality in the following ways: for $p_y^0\in [1+4(E_1+\frac{1}{3}D_1)\rho _0^3,1]$, intersecting, before the fourth switching time, the geodesics of the set $\Gamma _g$. The others, by intersecting the geodesics of the set $\Gamma _f$, after their fourth switching time if $p_y^0\ge 1+(8A+4C_2)\rho _0^2$, before it otherwise.

As in the preceding case, the cut locus has three branches and each branch of the cut locus is at least $C^1$. Its suspension has been shown in Fig. 8 (right).

1.1.2 $C_2>0$

When $C_2>0$, the conjugate time of the trajectories of the sets $\Gamma _{\pm g}$ coincide with their fifth switching time. Further, intersections of the kind $\mathcal {G}_4\cap \bar{\mathcal {F}_4}$ and $\bar{\mathcal {G}}_4\cap \mathcal {F}_4$ do not occur. Thus, in order to describe the cut locus, we must also consider the wavefront made by the geodesics in $\Gamma _{\pm g}$ that have already passed the fourth switching time. From equation (24), we can see that the front $\mathcal {G}_5$ is constrained between the vertical lines $\{x=4(A-C_2)\zeta ^3\}$ and $\{x=4(A+C_2)\zeta ^3\}$ (up to fourth order terms in $\zeta$); for the fifth bang front $\bar{\mathcal {G}}_5$, an analogous bound holds. Then, to understand how the fronts coming from different strategies may intersect (and, in particular, if the intersection $\mathcal {G}_5\cap \bar{\mathcal {G}}_5$ may occur), we must look at the relative values of A and $C_2$; we have four cases, all shown in Fig. 8.

$A>C_2$ The suspension of the cut locus is a piecewise $C^1$ curve, composed by the concatenation of (pieces of) the following intersection between fronts:

$$\begin{aligned} \mathcal {F}_4\cap \mathcal {G}_4,\quad \mathcal {G}_4\cap \bar{\mathcal {G}}_5,\quad \mathcal {G}_4\cap \bar{\mathcal {G}}_4,\quad \bar{\mathcal {G}}_4\cap \mathcal {G}_5\quad \bar{\mathcal {F}}_5\cap \mathcal {G}_5. \end{aligned}$$

In order to prove this, first of all we take advantage of the results of Section 5, to conclude that all geodesics of the set $\Gamma _f$ lose their optimality during the fourth bang arc, by intersections with the geodesics of $\Gamma _g$ (as $C_2>0$ forbids the intersection with the front $\bar{\mathcal {G}}_4$ and $A>0$ forbids those with the fronts $\bar{\mathcal {F}}_4$ and $\bar{\mathcal {F}}_5$).

We then focus on the geodesics of the set $\Gamma _{-f}$. We recall that the front $\bar{\mathcal {F}}_4$ cannot intersect neither the front $\mathcal {G}_4$ nor the front $\bar{\mathcal {G}}_4$; we thus consider also the intersections between $\bar{\mathcal {F}}_5$ and $\mathcal {G}_5$ (all computations were detailed at page 32).

Let us now concentrate on the geodesics of the set $\Gamma _g$. From Fig. 18 (right), we see that the front arising from the geodesics of the set $\Gamma _g$ may cross the fronts $\mathcal {F}_4,\bar{\mathcal {G}}_4,\bar{\mathcal {G}}_4$ and $\bar{\mathcal {F}}_5$; on the other hand, equation (27) tells that the geodesics that lose their optimality by intersecting the front $\mathcal {F}_4$ are those whose initial momentum $\varvec{\mu }_g(0)=(p_x^0,1,1/\rho _0)$ satisfies $p_x^0\in [1-4(E_1+D_1/3)\rho _0^3,1]$. To see what happens for $p_x^0<1-4(E_1+D_1/3)\rho _0^3$ we now look at the intersection between the front $\mathcal {G}_4$ and the fronts $\bar{\mathcal {G}}_4$ and $\bar{\mathcal {G}}_5$; the detailed computations are provided, respectively, at page 31 and in the Appendix 5-$\zeta$.

From (55), we can see that the suspension of the first intersection describes, up to higher order terms in $\zeta$, an arc of parabola (with concavity $-C_2$) connecting the points $(-4(A+C_2)\zeta ^3,4A\zeta ^3)$ and $(-4(A-C_2)\zeta ^3,4(A-C_2)\zeta ^3)$. From equation (54), we see that the geodesics of the set $\Gamma _g$ involved in this intersection are those such that $p_x^0\ge 1-4C_2\rho _0^2$.

Analogously, the suspension of the intersection $\bar{\mathcal {G}}_4\cap \mathcal {G}_5$ (that can be recovered by applying to the intersection $\mathcal {G}_{4}\cap \bar{\mathcal {G}}_5$ a rotation of $\pi$ around the z axis and the corresponding permutation of the invariants $\blacklozenge$), describes an arc of parabola of concavity $C_2$ connecting the points $(4(A-C_2)\zeta ^3,-4(A-C_2)\zeta ^3)$ and $(4(A+C_2)\zeta ^3,-4A\zeta ^3)$.

As we have already pointed out, the front $\bar{\mathcal {G}}_5$ is confined in the region $x<-4(A-C_2)\zeta ^3$, and the front $\mathcal {G}_5$ in the region $x>4(A-C_2)\zeta ^3$; then, as $A>C_2$, to describe the part of the cut locus for $|x|<4(A-C_2)\zeta ^3$, we must also consider the intersection $\mathcal {G}_4\cap \bar{\mathcal {G}}_4$. Equation (31) tells us that the suspension of this intersection is indeed a segment that joins the points $(-4(A+C_2)\zeta ^3,4A\zeta ^3)$ and $(-4(A-C_2)\zeta ^3,4(A-C_2)\zeta ^3)$. We thus conclude that the geodesics of the set $\Gamma _g$ (with $\varvec{\mu }_g(0)=(p_x^0,1,1/\rho _0)$) lose optimality in the following ways:

• if $p_x^0\ge 1+4(E_1+1/3D_1)\rho _0^3+\mathcal {O}(\rho _0^4)$, the geodesic loses its optimality by intersecting $\mathcal {F}_4$.

• if $1-4C_2\rho _0^2\le p_x^0\le 1+4(E_1+1/3D_1)\rho _0^3$, then the geodesic loses its optimality by intersecting $\bar{\mathcal {G}}_5$ (see equation (55)).

• for $1-8A\rho _0^2\le p_x^0\le 1-4C_2\rho _0^2$, then the geodesic loses its optimality during its fourth bang arc, by intersecting $\bar{\mathcal {G}}_4$ (see equation (31)).

• all geodesics with $p_x^0\le -1+2\sqrt{\zeta }\sqrt{\frac{-D_1-3E_1}{3C_2}}$ lose their optimality during the fifth bang arc, by intersecting the front $\bar{\mathcal {F}}_5$ (see page 32);

• finally, all geodesics of the set $\Gamma _g$ with $-1+2\sqrt{\zeta }\sqrt{\frac{-D_1-3E_1}{3C_2}}\le p_x^0\le 1-8A\rho _0^2+\mathcal {O}(\rho _0^3)$ lose their optimality during the fifth bang arc, by intersecting the front $\bar{\mathcal {G}}_4$.

Last of all, the loss of optimality of the geodesics of the set $\Gamma _{-g}$ is analogous to the one of the trajectories $\Gamma _{g}$ (with the difference that they do not meet other trajectories than those of the set $\Gamma _g$).

The cut locus is shown in Fig. 10 (right).

$0<A<C_2$ This case is very similar to the precedent one, with one major exception: if $A<C_2$, then the intersection between the fourth arcs of the geodesics of the sets $\Gamma _g$ and $\Gamma _{-g}$ does not occur, so that we look at the intersection between the fronts $\mathcal {G}_5$ and $\bar{\mathcal {G}}_5$. Such intersections involve only those geodesics of the set $\Gamma _{-g}$ that are associated with an initial covector $\varvec{\mu }_{-g}(0)=(\upeta ,-1,1/\rho _0)$ satisfying $\upeta \ge 1-\frac{2A}{C_2}$; besides, the intersection $\mathcal {G}_5\cap \bar{\mathcal {G}}_5$ requires $\upeta \ge \frac{2A}{C_2}-1$ to exist. Then, the cut locus contains: the part of $\mathcal {G}_4\cap \bar{\mathcal {G}}_5$ between the points $(-4(A+C_2)\zeta ^3,4A\zeta ^3)$ and $(4(A-C_2)\zeta ^3,(-4A^2/C_2+4A)\zeta ^3)$; the part of the intersection $\bar{\mathcal {G}}_4\cap \mathcal {G}_5$ between the points $(4(A+C_2)\zeta ^3,-4A\zeta ^3)$ and $(-4(A-C_2)\zeta ^3,(4A^2/C_2-4A)\zeta ^3)$; the portion of the intersection between $\mathcal {G}_5\cap \bar{\mathcal {G}}_5$ joining the points $(4(A-C_2)\zeta ^3,(-4A^2/C_2+4A)\zeta ^3)$ and $(-4(A-C_2)\zeta ^3,(4A^2/C_2-4A)\zeta ^3)$; this last part is a segment of slope $-A/C_2$.

Thus, the cut locus has one branch and is given by the concatenation of (pieces of) the following intersections:

$$\begin{aligned} \mathcal {F}_4\cap \mathcal {G}_4,\quad \mathcal {G}_4\cap \bar{\mathcal {G}}_5,\quad \mathcal {G}_5\cap \bar{\mathcal {G}}_5,\quad \bar{\mathcal {G}}_4\cap \mathcal {G}_5,\quad \bar{\mathcal {F}}_5\cap \mathcal {G}_5. \end{aligned}$$

It can be easily verified that all junctions are $C^1$, except the one involving $\bar{\mathcal {F}}_5\cap \mathcal {G}_5$.

$-C_2<A<0$ We are showing that the cut locus has one branch and is given by the concatenation of (pieces of) the following intersections:

$$\begin{aligned} \mathcal {F}_{4}\cap \mathcal {G}_{4},\quad \mathcal {F}_{4}\cap \mathcal {G}_5,\quad \mathcal {G}_5\cap \bar{\mathcal {G}}_5,\quad \bar{\mathcal {F}}_{4}\cap \bar{\mathcal {G}}_5,\quad \bar{\mathcal {F}}_{5}\cap \bar{\mathcal {G}}_5. \end{aligned}$$

We start by considering the geodesics of the set $\Gamma _g$; first of all, we notice that the intersections with the geodesics of the set $\Gamma _{-f}$ are forbidden (see equation (50) and Appendix-5-$\iota$); on the other hand, the only allowed intersection between the geodesics of the sets $\Gamma _{g}$ and $\Gamma _{-g}$ is $\mathcal {G}_5\cap \bar{\mathcal {G}}_5$ that involves only some of the geodesics of the set $\Gamma _g$ (see Appendix-5-$\upeta$). Then, to detect the loss of optimality of the geodesics of the set $\Gamma _g$, we study their intersections with those of the set $\Gamma _f$. Consider the geodesics of the set $\Gamma _g$ associated with the initial covector $\varvec{\mu }_g(0)=(p_x^0,1,1/\rho _0)$. We already saw that those with $p_x^0\ge 1+4\big (E_1+D_1/3\big )\rho _0^3+\mathcal {O}(\rho _0^4)$ are intersecting the front $\mathcal {F}_4$, before their fourth switching time; we now evaluate the intersection that may occur after the forth switching time (the details can be found in Section 5-$\kappa$). We find that these intersections occur, and cause a loss of global optimality, if $p_x^0\in [ 1+\frac{2A}{C_2},1+4\big (E_1+D_1/3\big )\rho ^3]$; equation (57) shows that the suspension (on the plane $\{z=4\zeta ^2\}$, with $\zeta =\rho _0+\mathcal {O}(\rho _0^2)$) of this intersection describes an arc of parabola; in particular, the arc joining the points $(4(A-C_2)\zeta ^3,4A\zeta ^3)$ and $(-4(A+C_2)\zeta ^3,4(A+A^2/C_2)\zeta ^3)$ (corresponding respectively to $p_x^0=1+\mathcal {O}(\rho _0^3)$ and to $p_x^0=1+\frac{2A}{C_2}$) belongs to the cut locus.

We can repeat the same reasoning for the fronts $\bar{\mathcal {F}}_4$ and $\bar{\mathcal {G}}_5$ and see that the symmetric arc of parabola joining the points $(-4(A-C_2)\zeta ^3,-4A\zeta ^3)$ and $(4(A+C_2)\zeta ^3,-4(A+A^2/C_2)\zeta ^3)$ belongs to the cut locus.

Finally, all trajectories of the set $\Gamma _g$ with $p_x^0\in [-1,1+\frac{2A}{C_2}]$ lose their optimality along the fifth bang arc, by intersecting the fifth bang front $\bar{\mathcal {G}}_5$.

It is left to prove how the trajectories of the set $\Gamma _{-f}$ with initial covector $(-1,p_y^0,\rho _0)$ and $p_y^0\le 1+8A\rho _0^3$ lose optimality. By geometric considerations on the position of the fronts, the only possibility is an intersection with the front $\bar{\mathcal {G}}_5$. To find such intersection, we proceed exactly as we did when studying the intersection $\bar{\mathcal {F}}_5\cap \mathcal {G}_5$ (see page 32). As in that case, we obtain expressions for the intersection time and the momentum depending on powers of $\sqrt{\zeta }$. Thus, the suspension of the intersection is the parameterized curve

$$\begin{aligned} {\left\{ \begin{array}{ll} x= -4(A-C_2)\zeta ^3-4 \sqrt{\frac{(2D_1p_y^0-D_1-3E_1)C_2}{3}} (1+p_y^0) \zeta ^{7/2}+\mathcal {O}(\zeta ^4)\\ y=-4A\zeta ^3+\mathcal {O}(\zeta ^4) \end{array}\right. } \end{aligned}$$

(46)

As above, we can prove that all the junctions, except the one with (46), are at least $C^1$, up to higher order terms.

$A<-C_2<0$ This case looks alike the precedent one; with respect to it, we spot two main differences: first of all, the intersection between the fronts $\mathcal {F}_4$ and $\mathcal {G}_5$ (as well as its symmetric) is allowed for every $p_x^0\in [-1,1]$ ($p_x^0$ denotes the first component of the adjoint covector associated at time 0 with the geodesic of the set $\Gamma _g$); also, the intersection between the fifth fronts of the geodesics of the sets $\Gamma _g$ and $\Gamma _{-g}$ does not occur: the part of the cut locus closer to the origin of the plane $\{z=4\zeta ^2\}$ is given by the intersection $\mathcal {F}_4\cap \bar{\mathcal {F}}_4$, which is studied in Section 5-$\delta$.

The cut locus has one branch and is given by the concatenation of (pieces of) the following intersections:

$$\begin{aligned} \mathcal {F}_4\cap \mathcal {G}_4,\quad \mathcal {F}_4\cap \mathcal {G}_5,\quad \mathcal {F}_4\cap \bar{\mathcal {F}}_4,\quad \bar{\mathcal {F}}_4\cap \bar{\mathcal {G}}_5,\quad \mathcal {F}_5\cap \bar{\mathcal {G}}_5. \end{aligned}$$

Again, we can prove that all the junctions, except the one with (46), are $C^1$.

Fig. 15

The formation of the cut locus in the cases $A_-$ with $C_2<0$ and $A>0$ (on the left) and $A_-$ with $A>C_2>0$ (on the right). The arrows show the direction in which the front evolves in time; when the front self-intersects, it gives rise to the cut locus (shown in black). In red (respectively, blue, orange, purple) the front $\mathcal {F}$ (respectively $\mathcal {G}, \bar{\mathcal {F}},\bar{\mathcal {G}}$)

1.2 $0<E_1<D_1$ (case $\varvec{B_+}$)

When $\frac{|E_1|}{|D_1|}<1$, the conjugate locus of the geodesics belonging to the sets $\Gamma _{\pm f}$ does not depend only on the values of the invariants, but also on the value of the associated adjoint vector at time 0. Moreover, the cut locus does not depend on the values of the two invariants $C_2$ and A only, but also on the ratio between $E_1$ and $D_1$, as it will be seen in the following. In particular, the last invariant that determines the shape of the cut locus is the ratio $\frac{3E_1}{D_1}$. We will study all the cases in details.

1.2.1 $C_2<0$

There are four subcases, depending on the values of $\frac{3E_1}{D_1}$ and of A. The suspensions of the cut loci have been illustrated in Fig. 11.

Fig. 16

The formation of the cut locus in the case $B_+$ with $C_2<0$, $A>0$ and $\frac{3E_1}{D_1}<1$. The arrows show how the fronts deplace as time increases

$A>0$ For these values of the invariants, the evolution of the suspensions of the front as time increases is sketched in Fig. 19. To understand how the whole front self-intersects, we first concentrate on the geodesics of the set $\Gamma _f$. The “natural” intersection $\mathcal {F}_4\cap \mathcal {G}_4$ does not occur, as condition (27) is never satisfied, but the front $\mathcal {F}_4$ does intersect the front $\mathcal {F}_5$. Moreover, the geodesics of the set $\Gamma _f$ with $p_y^0$ close to -1 are also intersecting, before the fourth switching time, the front $\bar{\mathcal {G}}_4$ (as already seen in Section 5); on the other hand, the intersection $\mathcal {F}_5\cap \bar{\mathcal {G}}_4$ is allowed (see Appendix 5-$\theta$). To understand which parts of these intersections actually belong to the cut locus, we search for an intersection of the three fronts: we fix some $\zeta >0$ and a time $\varvec{T}=8\zeta +\mathcal {O}(\zeta ^2)$, and we consider these three geodesics: the two geodesics (of the set $\Gamma _f$) with initial covector given respectively by $(1,\gamma ,1/\rho _0)$ and $(1,\widetilde{\gamma },1/\widetilde{\rho }_0)$ and the geodesic (of the set $\Gamma _{-g}$) geodesic with initial covector $(\upeta ,-1,1/\widehat{\rho }_0)$, where $\rho _0,\widetilde{\rho }_0$ and $\widehat{\rho }_0$ are chosen in such a way that the third coordinate of each of the three trajectories equals $4\zeta ^2$, up to sixth order powers of $\zeta$. Imposing the equality (at each order of the jets) for the other coordinates, we find that the intersection occurs at time

$$\begin{aligned} \varvec{T}=8\zeta +\Big (\frac{8}{3}C_2-4A\Big )\zeta ^3-\frac{3(E_1+D_1/3)^2}{2D_1}\zeta ^4+\mathcal {O}(\zeta ^5) \end{aligned}$$

and for

$$\begin{aligned} \begin{aligned}&\gamma =-1-4C_2\zeta ^2+4(E_2+D_2/3)\zeta ^3+\mathcal {O}(\zeta ^4)\\&\widetilde{\gamma }=\frac{1}{2}-\frac{3E_1}{2D_1}+\mathcal {O}(\zeta )\\&\upeta =-1+8A\zeta ^2+\big (5E_1+\frac{3E_1^2}{2D_1} +\frac{3D_1}{2} \big )\zeta ^3+\mathcal {O}(\zeta ^4) . \end{aligned} \end{aligned}$$

Summing up these elements, we can describe the cut points of the geodesics belonging to the set $\Gamma _f$. Let $\varvec{\mu }_f(0)=(1,p_y^0,1/\rho _0)$ be the associated adjoint vector; then:

• the geodesics with $p_y^0\in [-1,-1-4C_2\rho _0^2)$ lose global optimality because of the intersection $\mathcal {F}_4\cap \bar{\mathcal {G}}_4$.

• the geodesics with $p_y^0\in [-1-4C_2\rho _0^2,\frac{1}{2}-\frac{3E_1}{2D_1})$ lose global optimality due to the intersection of $\mathcal {F}_4$ with $\mathcal {F}_5$, as described in Section 4.1.

• the geodesics with $p_y^0\in (\frac{1}{2}-\frac{3E_1}{2D_1},1]$ lose global optimality during their fifth bang arc, by intersecting with the fourth bang of geodesics of the set $\Gamma _{-g}$.

For studying the geodesics of the set $\Gamma _{-f}$, we must distinguish the two cases in which $\frac{3E_1}{D_1}\ge 1$ or $\frac{3E_1}{D_1}<1$. Indeed, in the first case, every geodesic in $\Gamma _{-f}$ intersects, at (reparameterized) time $\mathcal {T}=7+p_y^0+\mathcal {O}(\rho _0^2)$, the front $\bar{\mathcal {G}}_4$; in the second one, equation (48) is satisfied only for $p_y^0\le \frac{3E_1}{2D_1}+\frac{1}{2}<1$.

Then, if $\frac{3E_1}{D_1}\ge 1$, all geodesics in the set $\Gamma _{-f}$ lose optimality during their fourth bang arc: those with $p_y^0\in [-1,1+4C_2\rho _0^2]$ by intersecting $\bar{\mathcal {G}}_4$, those with $p_y^0\in [1+4C_2\rho _0^2,1]$ by intersecting $\mathcal {G}_4$. For what concerns the geodesics of the sets $\Gamma _{\pm g}$, we can adapt the demonstration done in Section 5: more precisely, the geodesics of the set $\Gamma _{g}$ lose their global optimality when intersecting the fourth front of geodesics of the sets $\Gamma _{-f}$ or $\Gamma _{-g}$, the suspension of such intersections being respectively provided in equations (51) and (31). The geodesic of the set $\Gamma _{-g}$ lose their global optimality when intersecting the geodesics of the sets $\Gamma _{f}$ (equations (51) and (56)), $\Gamma _g$ and $\Gamma _{-f}$ (equation (30)). By direct computation of the tangents to the curves, we see that junction between the intersections $\mathcal {F}_5\cap \bar{\mathcal {G}}_4$ and $\mathcal {G}_4\cap \bar{\mathcal {G}}_4$ is $C^1$.

Let us now assume that $\frac{3E_1}{D_1}<1$, which implies that equation (48) is satisfied only for $p_y^0\le \frac{3E_1}{2D_1}+\frac{1}{2}<1$; in other words, a geodesic with initial adjoint vector $(-1,p_y^0,1/\rho _0)$ may lose its global optimality before its fourth switching time only if $p_y^0\le \frac{3E_1}{2D_1}+\frac{1}{2}$ or $p_y^0\ge 1+4C_2\rho _0^2$: indeed, the front $\bar{\mathcal {F}}_4$ cannot intersect $\mathcal {F}_4$ or $\mathcal {F}_5$, because $A>0$, and we can neglect eventual intersections with the fronts $\mathcal {G}_5$ and $\bar{\mathcal {G}}_5$, which are not optimal. So, to figure out what happens if $p_y^0 \in [\frac{1}{2}+\frac{3E_1}{2D_1},1+4C_2\zeta ^2]$, we must study the front $\bar{\mathcal {F}}_5$.

Equation (21) shows that the suspension of the front $\bar{\mathcal {F}}_5$ is contained in a horizontal strip of width $\mathcal {O}(\zeta ^4)$ centered about $\{y=-4A\zeta ^3\}$. This suggests to study its intersection with the fourth bang front of the trajectories $\Gamma _g$; from Appendix 5-$\iota$, we see that it is an arc of curve of length $\mathcal {O}(\zeta ^4)$ that connects the parts of the intersections $\mathcal {G}_4\cap \bar{\mathcal {G}}_4$ and $\bar{\mathcal {F}}_5\cap \mathcal {G}_4$.

Computing the slope of the tangent to the curve (56), we can see the suspension of the cut locus is at least $C^1$ (up to the third order in $\zeta$) at the junction between $\bar{\mathcal {F}}_5\cap \mathcal {G}_4$ and $\mathcal {G}_4\cap \bar{\mathcal {G}}_4$.

Summing up, differently from the case $\frac{3E_1}{D_1}\ge 1$, the suspension of the cut locus is disconnected, with one connected component made by only one branch, and the other one made by three branches, one of which is not $C^1$.

$A<0$ In this case, the existence condition for the intersection of the fronts $\mathcal {F}_5$ and $\bar{\mathcal {G}}_4$ is violated; our first concern is then to understand how the geodesics of the set $\Gamma _f$ with $p_y^0\ge \frac{1}{2}-\frac{3E_1}{2D_1}$ lose optimality. Inspired by the case $\varvec{A_-}$ with both $C_2$ and A negative, where a piece of the intersection between the (suspension of the) fronts $\mathcal {F}_4$ and $\bar{\mathcal {F}}_4$ participates to the cut locus, we look at the intersection between the fronts $\mathcal {F}_5$ and $\bar{\mathcal {F}}_4$; from (53), we see that this intersection occurs at time $\varvec{T}=8\rho _0+\mathcal {O}(\rho _0^2)$, while the self-intersection between the fronts $\mathcal {F}_4$ and $\mathcal {F}_5$ at time $\varvec{T}=(7-\gamma _0)\rho _0+\mathcal {O}(\rho _0^2)$ (equation (36)), with $\gamma _0\ge -1$; then, for $p_y^0\in [-\frac{E_1}{D_1},\frac{1}{2}-\frac{3E_1}{2D_1}]$, the intersection with the front $\mathcal {F}_4$ occurs before, and thus belongs to the cut locus.

Summing up, we can distinguish two cases; if $\frac{3E_1}{D_1}\ge 1$, then the cut locus is connected and made of five $C^1$ branches. More precisely:

• as it occurs in the case $A_-$, with $C_2<0$ and $A<0$, the geodesics of the set $\Gamma _f$ with initial adjoint vector $(1,p_y^0,1/\rho _0)$ and $p_y^0\in [-1,-1-4(2A+C_2)\zeta ^2)$ lose their optimality before the fourth switching time, by intersecting the front $\bar{\mathcal {F}}_4$; if $p_y^0\in (-1-(4C_2+8A)\zeta ^2,\frac{1}{2}-\frac{3E_1}{2D_1})$, then the geodesics lose optimality because of the self intersection $\mathcal {F}_4\cap \mathcal {F}_5$; finally, if $p_y^0\in (\frac{1}{2}-\frac{3E_1}{2D_1},1]$, they lose optimality after the fourth switching time, intersecting $\bar{\mathcal {F}}_4$.

• the geodesics of the set $\Gamma _{-f}$ associated with an adjoint vector at time zero equal to $\varvec{\mu }_{-f}(0)=(-1,p_y^0,1/\rho _0)$ and $p_y^0\in (1+4C_2\zeta ^2+4(E_2+D_2/3)\zeta ^3,1]$ lose optimality by intersecting $\mathcal {G}_4$; those with $p_y^0\in (1+4C_2\zeta ^2+4(E_2+D_2/3)\zeta ^3,1]$, by intersecting $\mathcal {F}_5$; those with $p_y^0\in (1+4(2A+C_2)\zeta ^2,1+4C_2\zeta ^2)$, by intersecting $\mathcal {F}_4$; finally, for $p_y^0\in [-1,1+4(2A+C_2)\zeta ^2]$, intersecting the front $\bar{\mathcal {G}}_4$.

• the geodesics of the kind $\Gamma _{\pm g}$ lose optimality during the fourth bang arc, by intersecting, respectively, the front $\bar{\mathcal {F}}_4$ or both fronts $\bar{\mathcal {F}}_4$ and $\mathcal {F}_4$.

If $\frac{3E_1}{D_1}< 1$, the suspension of the cut locus is disconnected and made by two connected components. The main difference with respect to the preceding case ($\frac{3E_1}{D_1}\ge 1$) is that the geodesics of the set $\Gamma _{-f}$ associated with an initial covector $(-1,p_y^0,1/\rho _0)$ with $p_y^0\in (\frac{1}{2}+\frac{3E_1}{2D_1},1+4(2A+C_2)\zeta ^2)$ are not intersecting the front $\bar{\mathcal {G}}_4$. As the intersections of the front $\bar{\mathcal {F}}_4$ with the fronts $\mathcal {G}_4$, $\mathcal {F}_4$ and $\mathcal {F}_5$ occur for $p_y^0$ close to $-1$, the only option left is the intersection $\bar{\mathcal {F}}_5\cap \mathcal {F}_4$, which is studied in Appendix 5-$\varepsilon$. In particular, its tangent for $p_y^0=\frac{1}{2}+\frac{3E_1}{2D_1}$ has slope $-\frac{2D_1}{3(E_1+D_1)}$, which shows that the junction with $\mathcal {F}_4\cap \bar{\mathcal {F}}_4$ is not $C^1$.

1.2.2 $C_2>0$

To describe the cut locus when $C_1=0$, $D_1>E_1>0$ and $C_2>0$, we can rely on the analysis carried out up to now; as could be expected, the shape of the cut locus depends on the relative values of the invariants A and $C_2$ and on the value of $\frac{3E_1}{D_1}$. We are thus studying them separately. The main differences with the cases studied in Section 5 involve the cut locus of the geodesics of the sets $\Gamma _{\pm f}$.

The suspensions of the cut loci have been illustrated in Fig. 13 (cases with $\frac{3E_1}{D_1}\ge 1$) and 14 (cases with $\frac{3E_1}{D_1}< 1$).

$A>C_2$ As already proved, the geodesics of the set $\Gamma _f$ with initial covector $(1,p_y^0,1/\rho _0)$ and $p_y^0\in \big [-1,\frac{1}{2}-\frac{3}{2}\frac{E_1}{D_1}\big ]$ lose their optimality because of the intersection of the fronts $\mathcal {F}_4$ and $\mathcal {F}_5$; what happens for the geodesics with $p_y^0\ge \frac{1}{2}-\frac{3}{2}\frac{E_1}{D_1}$ ? To answer this question, we recall that the intersection $\bar{\mathcal {G}}_4\cap \mathcal {F}_4$ does not occur (because $C_2>0$), we observe that the front $\bar{\mathcal {G}}_5$ remains “on the left” of the front $\mathcal {G}_5$ (as $A>0$) and, relying on the computations made in Section 5, we remark that the intersection $\bar{\mathcal {G}}_5\cap \mathcal {G}_4$ occurs (and will likely participate to the cut locus). This suggests to investigate the intersection between $\bar{\mathcal {G}}_5$ and $\mathcal {F}_5$. To do it, we can repeat the procedure already carried out for the analysis of the intersections $\mathcal {G}_5\cap \bar{\mathcal {F}}_5$ and $\bar{\mathcal {F}}_5\cap \bar{\mathcal {G}}_5$; we obtain that the suspension of this intersection is the parameterized curve

$$\begin{aligned} {\left\{ \begin{array}{ll} x=-4(A+C_2)\zeta ^3+4(1+p_y^0)C_2\sqrt{\frac{3E_1-D_1+2D_1p_y^0}{3C_2}}\zeta ^{7/2}+\mathcal {O}(\zeta ^4)\\ y=4A\zeta ^3+\mathcal {O}(\zeta ^4), \end{array}\right. } \end{aligned}$$

for $p_y^0\in \big (\frac{1}{2}-\frac{3E_1}{2D_1},1\big ]$. Summing up, we can conclude that the geodesics of the set $\Gamma _f$ with $p_y^0\in \big [-1,\frac{1}{2}-\frac{3E_1}{2D_1}\big ]$ lose their optimality because of the self-intersection between $\mathcal {F}_4$ and $\mathcal {F}_5$, and those with $p_y^0\in \big (\frac{1}{2}-\frac{3E_1}{2D_1},1\big ]$ lose optimality after their fourth switching time, by intersecting with the (fifth front) of the geodesics of the set $\Gamma _{-g}$.

For what concerns the cut locus of the geodesics of the set $\Gamma _{-f}$, it depends on the value of $\frac{3E_1}{D_1}$; indeed, if $\frac{3E_1}{D_1}\ge 1$, then all $\Gamma _{-f}$ geodesics lose their optimality before the fourth switching time, by intersecting with (the fourth arc of) the geodesics with of the set $\Gamma _{-g}$; if instead $\frac{3E_1}{D_1}<1$, then only the geodesics with $p_y^0\in \big [-1,\frac{1}{2}+\frac{3E_1}{2D_1}\big ]$ lose their optimality by intersecting with (the fourth arc of) the geodesics of the set $\Gamma _{-g}$, whereas those with $p_y^0\in \big (\frac{1}{2}+\frac{3E_1}{2D_1},1\big ]$ lose their optimality after the fourth switching time, by intersecting the geodesics of the set $\Gamma _g$, as already seen in Section 5.

The remaining part of the cut locus is completely analogous of the one studied in Section 5, therefore we do not repeat its description. We can conclude that, if $\frac{3E_1}{D_1}\ge 1$, the suspension of the cut locus is connected and composed by a single branch, whereas if $\frac{3E_1}{D_1}< 1$, the suspension of the cut locus is composed by two connected components. In both cases, they are piecewise $C^1$.

$0<A<C_2$ This case is very similar to the precedent one, with the sole exception that, as already remarked in Section 5, the fronts $\mathcal {G}_4$ and $\bar{\mathcal {G}}_4$ do not intersect, while $\mathcal {G}_5$ and $\bar{\mathcal {G}}_5$ do.

Thus, if $\frac{3E_1}{D_1}\ge 1$, the cut locus is made by the concatenation of pieces of the following intersections

$$\begin{aligned} \mathcal {F}_4\cap \mathcal {F}_5\quad \mathcal {F}_5\cap \bar{\mathcal {G}}_5\quad \mathcal {G}_4\cap \bar{\mathcal {G}}_5\quad \mathcal {G}_5\cap \bar{\mathcal {G}}_5\quad \bar{\mathcal {G}}_4\cap \mathcal {G}_5\quad \bar{\mathcal {F}}_4\cap \bar{\mathcal {G}}_4 \end{aligned}$$

and it is piecewise $C^1$.

If $\frac{3E_1}{D_1}< 1$, the suspension of the cut locus is disconnected; one connected component is made by the concatenation of the intersections

$$\begin{aligned} \mathcal {F}_4\cap \mathcal {F}_5\quad \mathcal {F}_5\cap \bar{\mathcal {G}}_5\quad \mathcal {G}_4\cap \bar{\mathcal {G}}_5\quad \mathcal {G}_5\cap \bar{\mathcal {G}}_5\quad \bar{\mathcal {G}}_4\cap \mathcal {G}_5\quad \bar{\mathcal {F}}_5\cap \mathcal {G}_5 \end{aligned}$$

and is piecewise smooth; the other connected component is the intersection $\bar{\mathcal {F}}_4\cap \bar{\mathcal {G}}_4$.

$-C_2<A<0$ We start by recalling that, for these values of the invariants, the front $\mathcal {F}_4$ is intersecting the front $\mathcal {G}_5$ (see Appendix 5-$\kappa$); this intersection involves the geodesics belonging to the set $\Gamma _f$ with $p_y^0$ close to -1. On the other hand, the geodesics with $p_y^0\in [-1,\frac{1}{2}-\frac{3E_1}{2D_1}]$ are involved in the intersection $\mathcal {F}_4\cap \mathcal {F}_5$, analyzed in the previous sections.

To detect the cut points of the geodesics of the set $\Gamma _f$ with $p_y^0\ge \frac{1}{2}-\frac{3E_1}{2D_1}$, we investigate the intersection between the fronts $\mathcal {F}_5$ and $\mathcal {G}_5$. Following the same approach adopted, for instance, when studying the intersection $\bar{\mathcal {F}}_5\cap \mathcal {G}_5$ (page 32), we can describe the suspension of such intersection as the curve

$$\begin{aligned} {\left\{ \begin{array}{ll} x=4(A-C_2)\zeta ^3-4(1-p_y^0)\sqrt{\frac{(3E_1+D_1+2D_1p_y^0)C_2}{3}}\zeta ^{7/2}+\mathcal {O}(\zeta ^4)\\ y=4A\zeta ^3+\mathcal {O}(\zeta ^4), \end{array}\right. } \end{aligned}$$

parameterized by $p_y^0\in \big [\frac{1}{2}+\frac{3E_1}{2D_1},1\big ]$.

Concerning the geodesics of the set $\Gamma _{-f}$, we still must distinguish the two cases $\frac{3E_1}{D_1}\ge 1$ and $\frac{3E_1}{D_1}< 1$. In the first one, all geodesics lose optimality before the fourth switching time: those with $p_y^0\in [-1,1+8A\rho _0^2]$ by intersecting $\bar{\mathcal {G}}_4$, the ones with $p_y^0\in [1+8A\rho _0^2,1]$ by intersecting $\bar{\mathcal {G}}_5$ (see Appendix 5-$\upbeta$).

If $\frac{3E_1}{D_1}< 1$, then only the geodesics with $p_y^0 \le \frac{1}{2}+\frac{3E_1}{2D_1}$ lose optimality before the fourth switching time, intersecting $\bar{\mathcal {G}}_4$; on the other hand, those with $p_y^0\in [1+8A\rho _0^2,1]$ still intersect the front $\bar{\mathcal {G}}_5$, before their fourth switching time. The geodesics with $p_y^0\in \big [\frac{1}{2}+\frac{3E_1}{2D_1},1+8A\rho _0^2\big ]$lose optimality because of the intersection between the fronts $\bar{\mathcal {F}}_5$ and $\bar{\mathcal {G}}_5$, as already seen in Section 5.

The rest of the cut locus is, as in the case $A_-$, $-C_2<A<0$, given by the intersection of $\mathcal {G}_4\cap \bar{\mathcal {G}}_4$.

Summing up, if $\frac{3E_1}{D_1}\ge 1$, the suspension of the cut locus is connected and given by three $C^1$ branches; if $\frac{3E_1}{D_1}< 1$, the cut locus is composed by two connected components.

$A<-C_2<0$ This last case can be deduced by gathering the arguments used to study the precedent cases. In particular, the only difference with the case just analyzed ($-C_2<A<0$) involves the “central part” of the (suspension of the) cut locus, which is made by the concatenation of the intersections $\mathcal {F}_4\cap \mathcal {G}_5$, $\mathcal {F}_4\cap \bar{\mathcal {F}}_4$ and $\bar{\mathcal {F}}_4\cap \bar{\mathcal {G}}_5$. More precisely:

• the geodesics of the set $\Gamma _f$ associated with the adjoint vector $\varvec{\mu }_f(0)=(1,p_y^0,1/\rho _0)$, with $p_y^0\in [-1,-1-8(A+C_2)\rho _0^2]$, lose optimality by intersection with the fourth bang arc $\bar{\mathcal {F}}_4$; those with $p_y^0\in [-1-8(A+C_2)\rho _0^2,-1-8A\rho _0^2]$, by intersection with the fifth bang arc $\mathcal {G}_5$; for $p_y^0\in \big [-1-8A\rho _0^2,\frac{1}{2}-\frac{3E_1}{2D_1}\big ]$, because of the self intersection between $\mathcal {F}_4$ and $\mathcal {F}_5$; finally, those with $p_y^0\in \big [\frac{1}{2}-\frac{3E_1}{2D_1},1\big ]$ lose optimality after the fourth switching time, by intersection with $\mathcal {G}_5$, as described above.

• analogously, the geodesics of the set $\Gamma _{-f}$ with initial adjoint vector $\varvec{\mu }_{-f}(0)=(-1,p_y^0,1/\rho _0)$, with $p_y^0\in [1+8(A+C_2)\rho _0^2,1]$, lose optimality by intersection with the fourth bang arc $\mathcal {F}_4$, and those with $p_y^0\in [1+8A\rho _0^2,1+8(A+C_2)\rho _0^2]$ by intersection with the fifth bang arc $\bar{\mathcal {G}}_5$. For $p_y^0\le 1+8A\rho _0^2$, we must distinguish the cases: if $\frac{3E_1}{D_1}\ge 1$, then all geodesics lose optimality by intersecting $\bar{\mathcal {G}}_4$, whereas, if $\frac{3E_1}{D_1} < 1$, only the geodesics with $p_y^0\le \frac{1}{2}+\frac{3E_1}{2D_1}$ lose optimality in this way; those with $p_y^0\in \big [\frac{1}{2}+\frac{3E_1}{2D_1},1+8A\rho _0^2\big ]$ lose optimality after the fifth bang arc, by intersecting $\bar{\mathcal {G}}_5$ (see equation 46).

1.3 $-D_1<E_1<0$ (case ${B_-}$)

The last case we consider is the one in which $|D_1|>|E_1|$, but the two invariants have opposite sign; in particular, we assume that $D_1$ is positive and $E_1$ negative.

A coarse analysis reveals that the major differences between the case $B_-$ and the case $B_+$ involve the geodesics of the set $\Gamma _f$, as, if $\frac{3E_1}{D_1}<-1$, then a part of the front $\mathcal {F}_4$ intersects the front $\mathcal {G}_4$. Assume indeed that $\frac{3E_1}{D_1}<-1$, and consider the self intersection of the front of the geodesics of the set $\Gamma _f$ (equation (35)); from the fact that $\gamma _0\le \upeta$, we see that $\gamma _0$ must be greater than or equal to $-\frac{3E_1}{D_1}-2$ (which is strictly greater than $-1$): the geodesics with initial covector $p_y^0\le -\frac{3E_1}{D_1}-2$ are not involved in this intersection. On the other hand, (28) states that the intersection between the fourth bang fronts of $\Gamma _f$ and $\Gamma _g$ trajectories may occur only if $p_y^0\le -\frac{3E_1}{D_1}-2$, and the suspension of this intersection is a horizontal segment joining the points $\big ((\frac{3E_1}{D_1}+1)\zeta ,4A\zeta ^3\big )$ and $\big (-4(A+C_2)\zeta ^3,4A\zeta ^3\big )$. This suggests that the only geodesics of the set $\Gamma _f$ losing optimality after the fourth switching time are those with $p_y^0\in [-\frac{E_1}{D_1},\frac{1}{2}-\frac{3E_1}{2D_1}]$, which are involved in the intersection $\mathcal {F}_4\cap \mathcal {F}_5$ and, to describe the cut locus close to the origin of the plane $\{z=4\zeta ^2\}$ (that is, in a ball of radius $\mathcal {O}(\zeta ^3)$), we must concentrate on the front $\mathcal {F}_4$ only.

In instead $\frac{3E_1}{D_1}\ge -1$, then the behavior of the geodesics of the set $\Gamma _f$ is not much different from the one described in Section 5.

The other difference with the precedent cases is that $\frac{3E_1}{2D_1}+\frac{1}{2}<1$, which implies that the intersection of a geodesics of the set $\Gamma _{-f}$, with initial covector $(-1,p_y^0,1/\rho _0)$, with one of the set $\Gamma _{-g}$ is possible only if $p_y^0\in \big [-1,\frac{3E_1}{2D_1}+\frac{1}{2}\big ]$; as already seen in Section 5, the suspension of this intersection is, up to higher order terms in $\zeta$, a segment of length $\frac{3}{2}(1+\frac{E_1}{D_1})\zeta$ and constitutes a connected component of the cut locus. Then, in the case $\varvec{B_-}$ the suspension of the cut locus is always disconnected.

A detailed analysis of the cut locus in the 12 different sub-cases of the case $\varvec{B_-}$ is not necessary, as it can easily be deduced from the preceding ones. For the sake of completeness, we just give, here below, a brief description.

1.3.1 $C_2<0$

When $C_2$ is negative, the cut locus has two connected components, one composed by three piecewise-smooth branches, the other one constituted by one smooth branch. We have four possible cases, depending on the value of A and $\frac{3E_1}{D_1}\ge - 1$. The suspension of the cut locus is plot in Fig. 15.

$A>0$ As already anticipated at the beginning of the section, if $\frac{3E_1}{D_1}\ge - 1$ the cut locus is very similar to the one shown in Fig. 11 (right). Indeed, all geodesics of the set $\Gamma _f$ with initial covector $p_y^0\in [-1,-\frac{3E_1}{2D_1}+\frac{1}{2}]$ lose optimality because of the self-intersection $\mathcal {F}_4\cap \mathcal {F}_5$, as already seen before; those with $p_y^0\in [-\frac{3E_1}{2D_1}+\frac{1}{2},1]$ intersect the fourth bang arc of the geodesics of the set $\Gamma _{-g}$.

The geodesics of the set $\Gamma _{-f}$ associated with the covector $\varvec{\mu }_{-f}(0)=(-1,p_y^0,1/\rho _0)$, with $p_y^0\in [-1,\frac{3E_1}{2D_1}+\frac{1}{2}]$, lose optimality when they intersect the fourth bang front $\bar{\mathcal {G}}_4$ (equation (49)); the suspension of this intersection is, up to higher order terms in $\zeta$, a segment of length $\frac{3}{2}(1+\frac{E_1}{D_1})\zeta$ and constitute one connected component of the cut locus. On the other hand, the geodesics of the set $\Gamma _{-f}$ with $p_y^0\ge \frac{3E_1}{2D_1}+\frac{1}{2}$ lose optimality after the fourth switching time, by intersecting the front $\mathcal {G}_4$. Finally, the geodesics of the sets $\Gamma _{\pm g}$ lose optimality during their fourth bang arc.

When $\frac{3E_1}{D_1}< - 1$, the geodesics of the set $\Gamma _f$ involved in the intersection $\mathcal {F}_4\cap \mathcal {F}_5$ are only those that with initial covector $p_y^0\in [-\frac{3E_1}{D_1}-2,1]$; those with $p_y^0\in [-1+8A\rho _0^2,-\frac{3E_1}{D_1}-2]$ lose optimality when intersecting the front $\mathcal {G}_4$.

$A<0$ As when $\frac{3E_1}{D_1}\ge -1$ the cut locus is almost identical to the one shown in Fig. 11 (down), we describe in details only the case in which $\frac{3E_1}{D_1}< -1$.

Consider a geodesic of the set $\Gamma _{f}$, associated with the adjoint vector $\varvec{\mu }_f(0)=(1,p_y^0,1/\rho _0)$, and assume that $\frac{3E_1}{D_1}< -1$; we have that

• if $p_y^0\in \big [-\frac{3E_1}{2D_1}+\frac{1}{2},1\big ]$, then the geodesic loses optimality during its fifth bang arc, by intersection with the front $\bar{\mathcal {F}}_4$ (see Appendix 5-$\varepsilon$);

• if $p_y^0\in \big (-1,\frac{3E_1}{2D_1}+\frac{1}{2}\big ]$, the geodesics loses optimality because of the self intersection of the fronts $\mathcal {F}_4$ and $\mathcal {F}_5$.

For what concerns the geodesics of the set $\Gamma _{-f}$, they lose optimality in the following way, according to the value of the initial covector $(-1,p_y^0,1/ \rho _0)$:

• if $p_y^0\in \big [-1,\frac{3E_1}{2D_1}+\frac{1}{2}\big ]$, during their fifth bang arc, by intersection with the front $\bar{\mathcal {G}}_4$;

• for $p_y^0\in \big [\frac{3E_1}{2D_1}+\frac{1}{2},1+4(2A+C_2)\rho _0^2\big ]$, during their fifth arc, by intersection with the front $\mathcal {F}_4$;

• for $p_y^0\ge 1+4(2A+C_2)\rho _0^2$, again by intersection with the front $\mathcal {F}_4$, but before their fourth switching time.

1.3.2 $C_2>0$

If $\frac{3E_1}{D_1}\ge -1$, the suspension of the cut locus is very similar to the one, corresponding to the same relative values of A and $C_2$, shown in Fig. 14. More difference with the precedent cases arise when $\frac{3E_1}{D_1}< -1$; these last cases are described in more details and shown in Fig. 17.

$A>C_2$ If $\frac{3E_1}{D_1}\ge -1$ the suspension of the cut locus is given by the concatenation of (pieces of) the following intersections:

$$\begin{aligned} \mathcal {F}_4\cap \mathcal {F}_5,\quad \mathcal {F}_5\cap \bar{\mathcal {G}}_5,\quad \mathcal {G}_4\cap \bar{\mathcal {G}}_5,\quad \mathcal {G}_4\cap \bar{\mathcal {G}}_4,\quad \bar{\mathcal {G}}_4\cap \bar{\mathcal {G}}_5,\quad \bar{\mathcal {F}}_5\cap \mathcal {G}_5 \end{aligned}$$

plus the connected component $\bar{\mathcal {F}}_4\cap \bar{\mathcal {G}}_4$.

On the other hand, if $\frac{3E_1}{D_1}<-1$, then the intersection between the fronts $\mathcal {F}_5$ and $\bar{\mathcal {G}}_5$ does not occur; then, the geodesics of the set $\Gamma _f$ associated with $p_y^0\in [-\frac{3E_1}{D_1}-2,1]$ lose their optimality at the self intersection $\mathcal {F}_4\cap \mathcal {F}_5$, while those associated with $p_y^0\in [-1,-\frac{3E_1}{D_1}-2]$ lose optimality when intersecting the front $\mathcal {G}_4$. The cut locus is thus given by the concatenation

$$\begin{aligned} \mathcal {F}_4\cap \bar{\mathcal {F}}_5,\quad \mathcal {F}_4\cap \mathcal {G}_4,\quad \mathcal {G}_4\cap \bar{\mathcal {G}}_5,\quad \mathcal {G}_4\cap \bar{\mathcal {G}}_4,\quad \bar{\mathcal {G}}_4\cap \mathcal {G}_5,\quad \bar{\mathcal {F}}_5\cap \mathcal {G}_5, \end{aligned}$$

plus the connected component $\bar{\mathcal {F}}_4\cap \bar{\mathcal {G}}_4$.

$0<A<C_2$ The only difference between this case and the precedent one is that the intersection $\mathcal {G}_4\cap \bar{\mathcal {G}}_4$ does not participate to the cut locus, and that the intersection of the fronts $\mathcal {G}_4$ and $\bar{\mathcal {G}}_5$ is optimal only if the first component $p_x^0$ of the initial covector $\varvec{\mu }_g(0)$ associated with the geodesic belonging to the set $\Gamma _g$ satisfies $p_x^0\le \frac{2A}{C_2}-1$ (as already discussed at page 60).

Then, if $\frac{3E_1}{D_1}\ge -1$, the suspension of the cut locus is given by the concatenation of (pieces of) the intersections

$$\begin{aligned} \mathcal {F}_4\cap \mathcal {F}_5,\quad \mathcal {F}_5\cap \bar{\mathcal {G}}_5,\quad \mathcal {G}_4\cap \bar{\mathcal {G}}_5,\quad \mathcal {G}_5\cap \bar{\mathcal {G}}_5,\quad \bar{\mathcal {G}}_4 \cap \mathcal {G}_5,\qquad \mathcal {G}_5\cap \bar{\mathcal {F}}_5 \end{aligned}$$

plus the connected component $\bar{\mathcal {F}}_4\cap \bar{\mathcal {G}}_4$.

If instead $\frac{3E_1}{D_1}<-1$, the geodesics of the set $\Gamma _f$ with initial momentum $p_y^0\in [-\frac{3E_1}{D_1}-2,1]$ lose their optimality at the self intersection $\mathcal {F}_4\cap \mathcal {F}_5$, while those with $p_y^0\in [-1,-\frac{3E_1}{D_1}-2]$ lose optimality when intersecting the front $\mathcal {G}_4$. The cut locus is thus given by the concatenation of (pieces of) the intersections

$$\begin{aligned} \mathcal {F}_4\cap \mathcal {F}_5,\quad \mathcal {F}_4\cap \mathcal {G}_4,\quad \mathcal {G}_4\cap \bar{\mathcal {G}}_5,\quad \mathcal {G}_5\cap \bar{\mathcal {G}}_5,\quad \bar{\mathcal {G}}_4\cap \mathcal {G}_5,\quad \bar{\mathcal {F}}_5\cap \mathcal {G}_5, \end{aligned}$$

and the connected component $\bar{\mathcal {F}}_4\cap \bar{\mathcal {G}}_4$.

$-C_2<A<0$ If $\frac{3E_1}{D_1} \ge -1$ the suspension of the cut locus is disconnected: one connected component is made by the intersection of the geodesics of the set $\Gamma _{-f}$ with initial momentum $p_y^0\in \big [-1,-\frac{1}{2}-\frac{3E_1}{2D_1}\big ]$ with the front $\bar{\mathcal {G}}_4$; the other connected component is made by 3 branches, meeting (in the plane $\{z=4\zeta ^2\}$) at the point $(4(A-C_2)\zeta ^3,4A\zeta ^3)$: the self intersection $\mathcal {F}_4\cap \mathcal {F}_5$, the intersection $\mathcal {F}_5\cap \mathcal {G}_5$ and the concatenation of the intersections

$$\begin{aligned} \mathcal {F}_4\cap \mathcal {G}_5,\quad \mathcal {G}_5\cap \bar{\mathcal {G}}_5,\quad \bar{\mathcal {F}}_5\cap \bar{\mathcal {G}}_5. \end{aligned}$$

If $\frac{3E_1}{D_1}<- 1$, as seen above, then the self intersection $\mathcal {F}_4\cap \mathcal {F}_5$ involves only the geodesics with initial momentum $p_y^0\in \big [-\frac{3E_1}{D_1}-2,1\big ]$; moreover, the intersection between the fifth front of the geodesics of the sets $\Gamma _{f}$ and $\Gamma _g$ does not occur. The cut locus is thus made by two connected components: one is the intersection $\bar{\mathcal {F}}_4\cap \bar{\mathcal {G}}_4$; the other one, the concatenation of the intersections

$$\begin{aligned} \mathcal {F}_4\cap \bar{\mathcal {F}}_5,\quad \mathcal {F}_4\cap \mathcal {G}_4,\quad \mathcal {F}_4\cap \mathcal {G}_5,\quad \mathcal {G}_5\cap \bar{\mathcal {G}}_5,\quad \bar{\mathcal {F}}_4\cap \bar{\mathcal {G}}_5\quad \bar{\mathcal {F}}_5\cap \bar{\mathcal {G}}_5. \end{aligned}$$

$-A<-C_2<0$ If $\frac{3E_1}{D_1} \ge - 1$ the cut locus is completely analogous to the corresponding one depicted in Fig. 14.

If instead $\frac{3E_1}{D_1}<- 1$, then the cut locus is thus made by two connected components: the intersection $\bar{\mathcal {F}}_4\cap \bar{\mathcal {G}}_4$ and the concatenation of the intersections

$$\begin{aligned} \mathcal {F}_4\cap \bar{\mathcal {F}}_5,\quad \mathcal {F}_4\cap \mathcal {G}_4,\quad \mathcal {F}_4\cap \mathcal {G}_5,\quad \mathcal {F}_4\cap \bar{\mathcal {F}}_4,\quad \bar{\mathcal {F}}_4\cap \bar{\mathcal {G}}_5\quad \bar{\mathcal {F}}_5\cap \bar{\mathcal {G}}_5. \end{aligned}$$

Intersections between regular bang-bang geodesics with different initial control

In this Section, we provide the existence conditions, the intersection times and the expression of the suspensions on the plane $\{z=4\zeta ^2\}$ of the intersections between bang-bang geodesics. These results are obtained following the same procedure described in Section 4.1.

$\alpha$:

$\bar{\mathcal {F}}_4\cap \bar{\mathcal {G}}_4$ may be easily recovered from the computation at page 30, by applying suitably Lemmas 4-5. In particular, we deduce that such intersections may occur only if $C_1\le 0$ and, if $C_1=0$, if

$$\begin{aligned} E_1+\frac{2-p_y^0}{3}D_1\ge 0 \end{aligned}$$

(47)

$$\begin{aligned} E_1+\frac{1-2p_y^0}{3}D_1\ge 0, \end{aligned}$$

(48)

where the geodesics belonging to the set $\Gamma _{-f}$ is associated with the initial adjoint vector $\varvec{\mu }_{-f}(0)=(-1,p_y^0,1/\rho _0)$. The intersection occurs at (reparameterized) time $\mathcal {T}$ such that

$$\begin{aligned} \mathscr {T}_4-\mathcal {T}=-2(1+p_y^0)C_1\rho _0^2-2(1+p_y^0)\Big (E_1+\frac{1-2p_y^0}{3}D_1\Big )\rho _0^3\mathcal {O}(\rho _0^4), \end{aligned}$$

and the suspension of the intersection at $z=4\zeta ^2$ is given by

$$\begin{aligned} {\left\{ \begin{array}{ll} x=(1-p_y^0)\zeta -\big ( (1-12p_y^0-5(p_y^0)^2)\frac{A}{4}-(p_y^0+3)C_2\big )\zeta ^3 +\mathcal {O}(\zeta ^4)\\ y=-4A\zeta ^3-\frac{2}{3}\big ( D_1(p_y^0)^2+(3E_1+D_1)(1-p_y^0) - \mathcal {c}_1 \big ) \zeta ^4 +\mathcal {O}(\zeta ^5). \end{array}\right. } \end{aligned}$$

(49)

$\upbeta$:

$\mathcal {G}_4\cap \bar{\mathcal {F}}_4$ This case can be obtained from the precedent one, by a rotation of $\pi /2$ around the $z-$axis and the permutation $\clubsuit$. We consider a $\Gamma _g$ geodesic with initial covector $(\upbeta ,1,1/\rho _0)$, and a $\Gamma _{-f}$ geodesic with initial covector $(-1,\upnu ,1/\widetilde{\rho }_0)$, where $\upnu$ and $\widetilde{\rho }_0$ are power series in $\rho _0$. We fix $\varvec{T}=(7+\upbeta )\rho _0+\mathcal {O}(\rho _0^2)$ and look at the intersection of the two geodesics at time $\varvec{T}$. As we find

$$\begin{aligned} \upnu = 1+2(\upbeta +1)C_2\rho _0^2+\mathcal {O}(\rho _0^3), \end{aligned}$$

(50)

this intersection occurs only if $C_2\le 0$. The intersection occurs at (reparameterized) time $\mathcal {T}= \mathscr {T}_4+2(1-\upbeta )C_2\rho _0^2+\mathcal {O}(\rho _0^3)$ and its suspension is the curve, parameterized by $\upbeta \in [-1,1],$

$$\begin{aligned} {\left\{ \begin{array}{ll} x= (4A +2(1-\upbeta ) C_2)\zeta ^3+\mathcal {O}(\zeta ^4)\\ y=(\upbeta -1)\zeta -\big ( (1-12\upbeta -5\upbeta ^2)\frac{A}{4}-\frac{1}{24}(5+9\upbeta -9\upbeta ^2-5\upbeta ^3)C_2\big )\zeta ^3 +\mathcal {O}(\zeta ^4). \end{array}\right. } \end{aligned}$$

(51)

$\gamma$:

$\bar{\mathcal {G}}_4\cap \mathcal {F}_4$ We describe here the last intersection between fourth bang front, that is, the one between the fronts $\bar{\mathcal {G}}_4$ and $\mathcal {F}_4$. This one too yields from the computation at page 30, after a suitable application of Lemmas 4-5. In particular, we find that this intersections occur only for $C_2\le 0$ at time

$$\begin{aligned} \mathcal {T}=\varvec{\mathscr {T}}_4+ 2(1-\upeta )C_2\rho _0^2+\mathcal {O}(\rho _0^3), \end{aligned}$$

where $\varvec{\mathscr {T}}_4$ is the fourth switching time of the geodesic of the set $\Gamma _{-g}$ with initial momentum $\varvec{\mu }_{-g}(0)=(\upeta ,-1,1/\rho _0)$. The suspension of such intersection is described by the following curve, parameterized by $\upeta \in [-1,1]$:

$$\begin{aligned} {\left\{ \begin{array}{ll} x=- (4A +2(1+\upeta ) C_2)\zeta ^3+\mathcal {O}(\zeta ^4)\\ y=(1+\upeta )\zeta +\big ( (1+12\upeta -5\upeta ^2)\frac{A}{4}-\frac{1}{24}(5-9\upeta -9\upeta ^2+5\upeta ^3)C_2\big )\zeta ^3 \\ \qquad +\mathcal {O}(\zeta ^4). \end{array}\right. } \end{aligned}$$

(52)

Up to third order terms in $\zeta$, this curve is a segment of length $\sim 2\zeta$.

$\delta$:

$\mathcal {F}_4\cap \bar{\mathcal {F}}_4$ In general, the intersections among geodesics with opposite initial velocity occur only close to the the vertical axis, that is, at a time $\varvec{T}\sim 8/p_z(0)$; since we are considering fourth arcs, this means that the second component of the initial momentum must be close to $\pm 1$. We then set $p_y^0=-1+\sum _{k\ge 1} \gamma _k \rho _0^k$ for the second component of the adjoint covector (at time zero) associated with any trajectory of the set $\Gamma _f$, and $p_y^0=1+\sum _{k\ge 1} \upnu _k \rho _0^k$ for the second component of the adjoint covector (at time zero) associated with any trajectory of the set $\Gamma _{-f}$. Equating the jets as usual, we find that the intersection occurs if $\upnu _1=\gamma _1=0$ and $\gamma _2-\upnu _2=-8(A+C_2)$, which is admissible only if $A+C_2\le 0$. The suspension of the intersection is given by the curve, parameterized by $\gamma _2\in [0,-8(A+C_2)]$, is given by

$$\begin{aligned} {\left\{ \begin{array}{ll} x=-(4A + 4C_2+ \gamma _2)\zeta ^3+\mathcal {O}(\zeta ^4)\\ y=-(4A + 4C_2+ \gamma _2)\zeta ^3+\mathcal {O}(\zeta ^4). \end{array}\right. } \end{aligned}$$

$\varepsilon$:

$\bar{\mathcal {F}}_5\cap \mathcal {F}_4$ For $\rho _0$ and $\varvec{T}$ fixed, we consider a trajectory of the set $\Gamma _{-f}$ with initial covector $(-1,p_y^0,1/\rho _0)$ and a trajectory of the set $\Gamma _{f}$ with initial covector $(1,\gamma ,1/\widetilde{\rho }_0)$, with $\gamma =\sum _{k\ge 0}\gamma _k\rho _0^k$ and $\widetilde{\rho }_0=\rho _0+\sum _{k\ge 2} \alpha _k\rho _0^k$. We assume that $\varvec{T}$ is greater than the fourth switching time of the first geodesics, but smaller than the fourth switching time of the second one. Imposing the equality of the jets at each order, we find that

$$\begin{aligned} \begin{aligned}&\gamma _0=-1,\quad \gamma _1=0,\quad \gamma _2=-4C_2+(p_y^0+1)^2C_1,\\&\mathbb {T}_4-\varvec{T}=-\big (4(p_y^0+1)C_1+8A\big )\rho _0^3+\mathcal {O}(\rho _0^4), \end{aligned} \end{aligned}$$

where $\mathbb {T}_4$ denotes the fourth switching time of the trajectory belonging to the set $\Gamma _f$. This implies that, if $C_1=0$, the intersection occurs only if $C_2\le 0$ and $A\le 0$. If $C_1=0$, the suspension of the intersections to the plane $\{z=4\zeta ^2\}$ is given by the curve

$$\begin{aligned} {\left\{ \begin{array}{ll}x=-4A\zeta ^3+\big (-\frac{2}{3} D_1 (p_y^0)^3 + (E_1 - D_1) (p_y^0)^2 + 2 E_1 p_y^0 \\ \qquad +\frac{1}{3}(D_1-5E_1-4\mathcal {d}_1) \big )\zeta ^4 +\mathcal {O}(\zeta ^5)\\ y=-4A\zeta ^3-\big ( 2D_1(p_y^0)^2-4E_1p_y^0- \frac{2}{3}\mathcal {c}_1\big ) \zeta ^4 +\mathcal {O}(\zeta ^5). \end{array}\right. } \end{aligned}$$

Applying a rotation of $\pi$ around the axis z (and $p_z$ in the adjoint space) and the transformation $\blacklozenge$, we obtain the expression of the suspension of the intersection $\mathcal {F}_5\cap \bar{\mathcal {F}}_4$:

$$\begin{aligned} {\left\{ \begin{array}{ll}x=4A\zeta ^3+\big (\frac{2}{3} D_1 (p_y^0)^3 + (E_1 - D_1) (p_y^0)^2 - 2 E_1 p_y^0 \\ \qquad +\frac{1}{3}(D_1-5E_1-4\mathcal {d}_1) \big )\zeta ^4 +\mathcal {O}(\zeta ^5)\\ y=4A\zeta ^3-\big ( 2D_1(p_y^0)^2+4E_1p_y^0- \frac{2}{3}\mathcal {c}_1\big ) \zeta ^4 +\mathcal {O}(\zeta ^5), \end{array}\right. } \end{aligned}$$

where $p_y^0$ denotes the second component of the initial momentum of the geodesic belonging to the set $\Gamma _f$. The existence conditions for such intersection are the same, and the intersection time is

$$\begin{aligned} \mathbb {T}_4-\varvec{T}=-\big (4(1-p_y^0)C+8A\big )\rho _0^3+\mathcal {O}(\rho _0^4), \end{aligned}$$

(53)

where $\mathbb {T}_4$ denotes the fourth switching time of the trajectory belonging to the set $\Gamma _{-f}$.

$\zeta$:

$\mathcal {G}_4\cap \bar{\mathcal {G}}_5$ For $\rho _0$ and $\varvec{T}$ fixed, we consider a trajectory of the set $\Gamma _{-g}$ with initial covector $\varvec{\mu }_{-g}(0)=(p_x^0,-1,1/\rho _0)$ and a trajectory of the set $\Gamma _{g}$ with initial covector $(\upbeta ,1,1/\widehat{\rho }_0)$, with $\upbeta =\sum _{k\ge 0}\upbeta _k\rho _0^k$. We assume that $\varvec{T}$ is greater than the fourth switching time of the first geodesics, but smaller than the fourth switching time of the second one (denoted with $\mathbb {T}_4$ in the following). Imposing the equality of the jets at each order, we find the constraints

$$\begin{aligned} \upbeta _0=1,\quad \upbeta _1=0,\quad \upbeta _2=4C_1-(1-p_x^0)^2C_2, \end{aligned}$$

(54)

so that, for $C_1=0$, this intersection occurs only if $C_2\ge 0$. As $\mathbb {T}_4-\varvec{T}=\big (8A+4(p_x^0-1)C_2\big )\rho _0^3+\mathcal {O}(\rho _0^4)$, then this intersections occurs only for

$$\begin{aligned} p_x^0\ge 1-\frac{2A}{C_2}, \end{aligned}$$

which also imposes $A\ge 0$. When $C_1=0$, the suspension of the intersections to the plane $\{z=4\zeta ^2\}$ is given by the following parameterized curve

$$\begin{aligned} {\left\{ \begin{array}{ll}x=-4(A+p_x^0 C_2)\zeta ^3+\big (2 D_2(p_x^0)^2+ 4E_2p_x^0+\frac{2}{3} \mathcal {c}_2 \big )\zeta ^4 +\mathcal {O}(\zeta ^5)\\ y=(4A-(1-p_x^0)^2C_2)\zeta ^3+\big ( \frac{2}{3}D_2(p_x^0)^3+(E_2-D_2)(p_x^0)^2\\ \quad -2E_2p_x^0+ \frac{1}{3}D_2+\frac{5}{12}\mathcal {c}_1-\frac{1}{2}\mathcal {d}_2-\frac{5}{12}D_1\big ) \zeta ^4 +\mathcal {O}(\zeta ^5), \end{array}\right. } \end{aligned}$$

(55)

The third order term describes is an arc of parabola with vertex for $p_x^0=1$. The slope of the tangent to the curve is given by

$$\begin{aligned} \frac{y\prime (p_x^0)}{x\prime (p_x^0)}=\frac{p_x^0-1}{2}+\mathcal {O}(\rho _0). \end{aligned}$$

The intersection $\bar{\mathcal {G}}_4\cap \mathcal {G}_5$ can be easily recovered from the one just studied, by changing all signs in the suspension and applying a suitable permutation of the invariants. As, under such permutations, A and $C_2$ are unchanged, at the third order term the suspension is the symmetric of (55) with respect to the origin.

$\upeta$:

$\mathcal {G}_5\cap \bar{\mathcal {G}}_5$ For $\rho _0$ and $\varvec{T}$ fixed, we consider a trajectory of the set $\Gamma _{g}$ with initial covector $\varvec{\mu }_{g}(0)=(\upbeta ,1,1/\rho _0)$ and a trajectory of the set $\Gamma _{-g}$ with initial covector $\varvec{\mu }_{-g}(0)=(\upeta ,-1,1/\widehat{\rho }_0)$. These two trajectories intersects only if $\upbeta -\upeta =\frac{2A}{C_2}$; in particular, this imposes $|A|<C_2$. The intersection occurs at time $\varvec{T}=8\rho _0+\frac{4}{C_2}\Big (C_2^2\upbeta ^2-2AC_2 \upbeta -A^2+\frac{2}{3}C_2^2\Big )\rho _0^3+\mathcal {O}(\rho _0^4).$ The suspension of the intersections to the plane $\{z=4\zeta ^2\}$ is given by

$$\begin{aligned} {\left\{ \begin{array}{ll} x=4(A-C_2\upbeta )\zeta ^3+\mathcal {O}(\zeta ^4)\\ y= -4\frac{A}{C_2}\big (A-C_2\upbeta \big )\zeta ^3 +\mathcal {O}(\zeta ^4), \end{array}\right. } \end{aligned}$$

where $\upbeta \in \big [\max \{-1,\frac{2A}{C_2}-1\},\min \{1-\frac{2A}{C_1},1\}\big ]$. The leading term is a segment of slope $-A/C_2$.

$\theta$:

$\mathcal {F}_5\cap \bar{\mathcal {G}}_4$ For $\rho _0$ and $\varvec{T}$ fixed, we consider a trajectory of the set $\Gamma _{f}$ with initial covector $(1,p_y^0,1/\rho _0)$ and a trajectory of the set $\Gamma _{-g}$ with initial covector $(\upbeta ,-1,1/\widetilde{\rho }_0)$, with $\upbeta =\sum _{k\ge 0}\upbeta _k\rho _0^k$ and $\widetilde{\rho }_0=\rho _0+\sum _{k\ge 2}\alpha _k\rho _0^k$. We assume that $\varvec{T}$ is greater than the fourth switching time of the first geodesics, but smaller than the fourth switching time of the second one. Equating as usual the jets of the two geodesics, we find

$$\begin{aligned} \begin{aligned}&\upbeta _0=-1,\quad \upbeta _1=0,\quad \upbeta _2=8A-4(1+p_y^0)C_1,\\&\mathbb {T}_4-\varvec{T}=\big (2A(1+p_y^0)+(1+p_y^0)C_1+((p_y^0)^2+2p_y^0-3 )C_2\big )\rho _0^3+\mathcal {O}(\rho _0^4), \end{aligned} \end{aligned}$$

where $\mathbb {T}_4$ denotes the fourth switching time of the geodesics belonging the set $\Gamma _{-g}$. We can verify that, for $C_1=0$, $A > 0$ and $C_2 < 0$, this expression is always negative. Indeed, it is a concave parabola in $p_y^0$, which is positive for $p_y^0=1$ and $p_y^0=-1$, so positive on the whole interval. This implies that, for $C_1=0$, the intersection occurs only if $C_2\le 0$ and $A\ge 0$. For $C_1=0$, the suspension of the intersections to the plane $\{z=4\zeta ^2\}$ is the curve

$$\begin{aligned} {\left\{ \begin{array}{ll}x=-4A\zeta ^3+\big (\frac{2}{3} D_1 (p_y^0)^3 + (E_1 + D_1) (p_y^0)^2 + 2 E_1 p_y^0 \\ \quad - \frac{1}{3}(D_1+5E_1+4\mathcal {d}_1) \big )\zeta ^4 +\mathcal {O}(\zeta ^5)\\ y=4A\zeta ^3-\big ( 2D_1(p_y^0)^2+4E_1p_y^0- \frac{2}{3}\mathcal {c}_1\big ) \zeta ^4 +\mathcal {O}(\zeta ^5). \end{array}\right. } \end{aligned}$$

(56)

$\iota$:

$\bar{\mathcal {F}}_5\cap \mathcal {G}_4$ The analysis of this intersection can be derived from the precedent one, just by applying the suitable permutations of invariants and rotations. In particular, we get that, when $C_1=0$, this intersection may occur only if $A>0$ and $C_2<0$. The suspension of this intersection is given by the curve

$$\begin{aligned} {\left\{ \begin{array}{ll} x=4A\zeta ^3+\big (-\frac{2}{3}D_1 (p_y^0)^3+(E_1+D_1)(p_y^0)^2-2 E_1 p_y^0\\ \quad - \frac{1}{3}(D_1+5E_1+4\mathcal {d}_1)\big )\zeta ^4+\mathcal {O}(\zeta ^5)\\ y=-4A\zeta ^3-\big (2D_1(p_y^0)^2-4 E_1 p_y^0-\frac{2}{3}\mathcal {c}_1\big )\zeta ^4+\mathcal {O}(\zeta ^5), \end{array}\right. } \end{aligned}$$

where $p_y^0\in [-1,1]$ denotes the second component of the initial covector of the geodesics belonging to the set $\Gamma _{-f}$.

$\kappa$:

$\mathcal {F}_4\cap \mathcal {G}_5$ These case can be recovered by applying to the results of $\theta$ a rotation of $\pi /2$ around the z and the $p_z$ axes and the permutation of the invariants $\clubsuit$. We consider a trajectory of the set $\Gamma _g$ with initial covector $(p_x^0,1,1/\rho _0)$ and we intersect it with a geodesic of the set $\Gamma _f$. We obtain that such intersections occur only for $p_x^0\ge 1+\frac{2A}{C_2}$. If $C_1=0$, the suspension on the plane $\{z=4\zeta ^2\}$ reads

$$\begin{aligned} {\left\{ \begin{array}{ll} x=4(A-p_x^0C_2)\zeta ^3+\big (2D_2(p_x^0)^2-4E_2 p_x^0 -\frac{2}{3}\mathcal {d}_2 \big )\zeta ^4+\mathcal {O}(\zeta ^5)\\ y=(4A+(1-p_x^0)^2C_2)\zeta ^3+\big (-\frac{2}{3}D_2(p_x^0)^3+(E_2-D_2) (p_x^0)^2\\ -2E_2p_x^0 +\frac{1}{3}(D_2+5E_2+4\mathcal {d}_2\big )\zeta ^4+\mathcal {O}(\zeta ^5) \end{array}\right. } \end{aligned}$$

(57)