Differential geometry and terms

The solution of a continuous dynamical system is a trajectory $\mathcal{T}_\mathbf{s}(t)$ as defined by Eq. 3.1 [40,69]. Any point on the trajectory is given by its parameter t and an initial state s of the system. Parameter t can be interpreted as the time passed since the system evolved from s. Note, that Eq. 3.1 is a ``recursive'' definition (integral equation) that cannot be expressed explicitly in most cases.

$$\mathcal{T}_\mathbf{s}(t) = \mathbf{s} + \int_{u=0}^{t} \mathbf{f}_\mathbf{p}(\mathcal{T}_\mathbf{s}(u))\,\mathrm{d}u$$ (3.1)

Differential geometry includes the analysis of curves and surfaces in higher dimensions. The construction of a local coordinate system (Frenét-Frame) helps to get insight into local characteristics of a spatial curve, e.g., curvature and torsion [10,30]. Local analysis of trajectories requires a good working knowledge of various terms of differential geometry. They are shortly discussed in the following.

Given a parameterized curve $\mathcal{C}(t)$ in three-space a re-parameterization is possible such that the curve's new parameter s is equal to the arc length of curve $\mathcal{C}$ in the parameter interval [0,s). In respect to these distinct parameters derivations of curve $\mathcal{C}$ are written differently:

$$\dot{\mathcal{C}} = \frac{\mathrm{d}\mathcal{C}}{\mathrm{d}t... ...hrm{d}^2\mathcal{C}}{\mathrm{d}s^2},\ \textrm{etc.}\nonumber$$

By the use of these derivations a local coordinate system (Frenét-Frame) can be built at a curve point by the curve's tangent vector $\mathbf{t}_\mathcal{C}=\mathcal{C}'$, its principal normal $\mathbf{n}_\mathcal{C}=\mathcal{C}''\!/\!\left\vert\mathcal{C}''\right\vert$, and its binormal $\mathbf{b}_\mathcal{C}=\mathbf{t}_\mathcal{C}\!\times\!\mathbf{n}_\mathcal{C}$. These three vectors span an orthonormal basis at a curve point. Note, that $\mathbf{n}_\mathcal{C}$ and $\mathbf{b}_\mathcal{C}$ are ambiguous when the curve is locally equal to a straight line.

By building the Frenét-Frame at a point on the curve the curvature $\kappa$ and the torsion $\tau$ of curve $\mathcal{C}$ at this point can be derived in a straightforward way from the orthonormal basis [10]:

$$\frac{\mathrm{d}}{\mathrm{d}s}\left( \begin{array}{c} \math... ...ac{\mathrm{d}\mathbf{b}_\mathcal{C}}{\mathrm{d}s} \right\vert$$

Curvature $\kappa$ and torsion $\tau$ of curve $\mathcal{C}$ can be described in other terms as well. For example, the curvature of a curve can be written as 1/r, when r is the radius of the osculating circle [13]. As a third possibility, $\kappa$ can be derived by the following procedure: assuming $\alpha$ to be the angle enclosed by the curve's tangent and the line running through $\mathcal{C}(s)$ and some point $\mathcal{C}(s+\Delta{}s)$, slightly ahead on the curve, the curvature $\kappa$ can be calculated as $\kappa=\lim_{\Delta{}s\to0}\alpha/\Delta{}s$.

Torsion can be similarly derived by a differential quotient. Assuming $\beta$ to be the angle enclosed by a line through $\mathcal{C}(s)$ and $\mathcal{C}(s+\Delta{}s)$ the rectifying plane (spanned by $\mathbf{t}_\mathcal{C}$ and $\mathbf{b}_\mathcal{C}$), the torsion $\tau$ can be calculated as $\tau=\lim_{\Delta{}s\to0}\beta/\Delta{}s$ [13].

mailto:helwig@cg.tuwien.ac.at.