The first order space by time relation

x˙2+y˙2=R˙2+R2a˙2 \dot{x}^2 + \dot{y}^2 = \dot{R}^2 + R^2 \dot{a}^2

The second order space by time relation

x¨2+y¨2=(R¨−R(a˙)2)2+(2R˙a˙+Ra¨)2 \ddot{x}^2 + \ddot{y}^2 = \left( \ddot{R} - R (\dot{a})^2 \right)^2 + \left( 2 \dot{R} \dot{a} + R \ddot{a} \right)^2

The third order space by time relation

∥ r ( 3 ) ∥ 2 = ( R ( 3 ) − 3 R α ˙ α ¨ - 3 R ˙ α ˙ 2 ) 2 + ( R α ( 3 ) − R α ˙ 3 + 3 R ˙ α ¨ + 3 R ¨ α ˙ ) 2 \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2

The 4th order space by time relation

∥ r ( 4 ) ∥ 2 = ( R ( 4 ) − 6 R ¨ α ˙ 2 - 12 R ˙ α ˙ α ¨ + R α ˙ 4 -3 R α ¨ 2 -4 R α ˙ α ( 3 ) ) 2 + ( -4 R ( 3 ) α ˙ −6 R ¨ α ¨ +4 R ˙ α ˙ 3 -4 R ˙ α ( 3 ) + 6 R α ˙ 2 α ¨ - R α ( 4 ) ) 2 \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2

The 5th order space by time relation

∥ r ( 5 ) ∥ 2 \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2 = \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2 ( R ( 5 ) - 10 R ( 3 ) α ˙ 2 − 30 R ¨ α ˙ α ¨ + 5 R ˙ α ˙ 4 - 15 R ˙ α ¨ 2 - 20 R ˙ α ˙ α ( 3 ) + 10 R α ˙ 3 α ¨ - 10 R α ¨ α ( 3 ) - 5 R α ˙ α ( 4 ) ) 2 \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2

+ ( −5 R ( 4 ) α ˙ −10 R ( 3 ) α ¨ + 10 R ¨ α ˙ 3 - 10 R ¨ α ( 3 ) + 30 R ˙ α ˙ 2 α ¨ - 5 R ˙ α ( 4 ) - 1 R α ˙ 5 + 15 R α ˙ α ¨ 2 + 10 R α ˙ 2 α ( 3 ) - R α ( 5 ) ) 2 \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2

The 6th order space by time relation

∥ r ( 6 ) ∥ 2 \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2 = \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2 ( R ( 6 ) − 15 R ( 4 ) α ˙ 2 − 60 R ( 3 ) α ˙ α ¨ + 15 R ¨ α ˙ 4 − 45 R ¨ α ¨ 2 − 60 R ¨ α ˙ α ( 3 ) + 60 R ˙ α ˙ 3 α ¨ − 60 R ˙ α ¨ α ( 3 ) − 30 R ˙ α ˙ α ( 4 ) − 1 R α ˙ 6 + 45 R α ˙ 2 α ¨ 2 + 20 R α ˙ 3 α ( 3 ) − 10 R α ( 3 ) α ( 3 ) - 15 R α ¨ α ( 4 ) - 6 R α ˙ α ( 5 ) ) 2 \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2

+ ( −6 R ( 5 ) α ˙ −15 R ( 4 ) α ¨ + 20 R ( 3 ) α ˙ 3 - 20 R ( 3 ) α ( 3 ) + 90 R ¨ α ˙ 2 α ¨ - 15 R ¨ α ( 4 ) - 6 R ˙ α ˙ 5 + 90 R ˙ α ˙ α ¨ 2 + 60 R ˙ α ˙ 2 α ( 3 ) - 6 R ˙ α ( 5 ) - 15 R α ˙ 4 α ¨ + 15 R α ¨ 3 + 60 R α ˙ α ¨ α ( 3 ) + 15 R α ˙ 2 α ( 4 ) - R α ( 6 ) ) 2 \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2 \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2 \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2 \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2 \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2 \|\mathbf{r}^{(3)}\|^2 = \left(R^{(3)} - 3 R \dot{\alpha}\ddot{\alpha} - 3\dot{R}\dot{\alpha}^2\right)^2 + \left(R\alpha^{(3)} - R\dot{\alpha}^3 + 3\dot{R}\ddot{\alpha} + 3\ddot{R}\dot{\alpha}\right)^2

Radial snap:

s r = j ˙ r − α ˙ j θ s_r = \dot{j}_r - \dot{\alpha} j_\theta

Transverse snap:

sθ=j˙θ+α˙jrs_\theta = \dot{j}_\theta + \dot{\alpha} j_r