GSpline formal definition

[rafaelrojasmiliani]

Let $(M, \mathcal{O}, \mathcal{A})$ be a smooth manifold, where $M$ is a set, $\mathcal{O}$ is a topology on $M$ and $\mathcal{A}$ is a smooth atlas.

Consider the interval $I=[a,b)$ and let $I_i=[t_i, t_{i+1}), i=1,.., N$ be a partition of $I$ with $t_0=a$ and $t_{N+1} =b$. We define the GSpline $\gamma: I \longrightarrow M$ with respect to the charts $(\psi_i, U_i)$ at the interval $I_i$ as the map
$$\gamma(t) = \psi_j^{-1}\Bigg(\sum \mathbf{y} B_i \circ s_j (t)\Bigg) \ \text{ if } t\in I_j$$

We define at each interval $I_i$ the coordinates of the GSplines as
$$q_i(t) = \sum \mathbf{y} B_i \circ s_j (t) \ \text{ if } t\in I_j$$
In other words, a GSpline with respect to the charts $(\psi_i, U_i)$ at the interval $I_i$ is a curve on $ I \longrightarrow M$ such that at the interval it is given at the interval $I_i$ by a GSpline in the codomain of the chart $\psi_i$.

Continuity condition

• $C^0$ continuity Unlike GSplines in $\mathbb{R}^n$ here the condition is not linear.
  $$\gamma(t_{i+1}^-) = \gamma(t_{i+1}^+)$$
  $$\psi_{i}^{-1}(q_i(t_{i_+1})) =\psi_{i+1}^{-1}(q_{i+1}(t_{i_+1})) $$
  $$q_i(t_{i_+1}) =\psi_i\Big(\psi_{i+1}^{-1}(q_{i+1}(t_{i_+1}))\Big) $$
  $$\psi_{i+1}\Big(\psi_{i}^{-1}(q_i(t_{i_+1}))\Big) =q_{i+1}(t_{i_+1}) $$

• $C^1$ continuity Here we use the concept of tangen vector to the curve which is chart-independent.
  $$\dot\gamma(t_{i+1}^-) = \dot\gamma(t_{i+1}^+)$$
  $$\mathsf{d}\psi_i^{-1} \frac{\mathsf{d} q_i}{\mathsf{d} t} = \mathsf{d}\psi_{i+1}^{-1} \frac{\mathsf{d} q_{i+1}}{\mathsf{d} t}$$

• $C^2$ continuity Were we look as the transport of the derivative given by the connection $\nabla$. Now we need a manifold with a connection.
  $$\Big(\nabla_{\dot\gamma} \dot\gamma\Big) (t_{i+1}^-) = \Big(\nabla_{\dot\gamma} \dot\gamma\Big) (t_{i+1}^+)$$
  $$\mathsf{d}\psi_i^{-1} \Big[\frac{\mathsf{d}^2 q_i}{\mathsf{d} t^2} + \Gamma_{\psi_i, k}^{jl} \frac{\mathsf{d} q_i}{\mathsf{d} t} \frac{\mathsf{d} q_i}{\mathsf{d} t} \Big] = \mathsf{d}\psi_{i+1}^{-1} \Big[\frac{\mathsf{d}^2 q_{i+1}}{\mathsf{d} t^2} + \Gamma_{\psi_{i+1}, k}^{jl} \frac{\mathsf{d} q_{i+1}}{\mathsf{d} t} \frac{\mathsf{d} q_{i+1}}{\mathsf{d} t} \Big]$$

GSpline representation

Any GSpline of on a manifold is completely define by

• 1 A Manifold of dimension $n$
• 2 A sequence of $n_c$ scalar basis $B_i:[-1,1]\longrightarrow \mathbb{R}$
• 3 An array $\boldsymbol{\tau}\in \mathbb{R}^N$ of interval lenghts
• 4 An array $\mathbf{y}\in \mathbb{R}^{n_c n N}$ of interval of coefficients of the basis at each interval
  $$\mathbf{y} = [\mathbf{y}_0^0 \mathbf{y}_0^1 \cdots \mathbf{y}_0^{n-1} \mathbf{y}_1^0 \mathbf{y}_1^1 \cdots \mathbf{y}^{n-1}_N \mathbf{y}_N^0 \mathbf{y}_N^1 \cdots]$$
• 5 An array of $N$ charts $(\psi_i, U_i)$

[rafaelrojasmiliani]

Desgin

classDiagram

GSpline --|>  LinearManifold
LinearManifold --|> Manifold
GSpline --|> Map

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