# curvature (space curve)

Let $I\subset \mathbb{R}$ be an interval^{}, and let $\gamma :I\to {\mathbb{R}}^{3}$ be
an arclength parameterization of an oriented space curve, assumed to
be regular^{}, and free of points of inflection. We interpret $\gamma (t)$ as
the trajectory of a particle moving through 3-dimensional space. Let
$T(t),N(t),B(t)$ denote the corresponding moving trihedron. The
speed of this particle is given by

$$v(t)=\parallel {\gamma}^{\prime}(t)\parallel .$$ |

The quantity

$$\kappa (t)=\frac{\parallel {T}^{\prime}(t)\parallel}{v(t)}=\frac{\parallel {\gamma}^{\prime}(t)\times {\gamma}^{\prime \prime}(t)\parallel}{{\parallel {\gamma}^{\prime}(t)\parallel}^{3}}$$ |

is called the
*curvature ^{}* of the space curve. It is invariant with respect to
reparameterization, and is therefore a measure of an intrinsic property
of the curve, a real number geometrically assigned to the point
$\gamma (t)$. If one parameterizes the curve with respect to the arclength $s$, one gets the more concise relation

^{}that

$$\kappa (s)=\frac{1\cdot \parallel {\gamma}^{\prime \prime}(s)\parallel \cdot \mathrm{sin}\frac{\pi}{2}}{{1}^{3}}=\parallel {\gamma}^{\prime \prime}(s)\parallel .$$ |

Physically, curvature may be conceived as the ratio of the normal acceleration of a particle to the particle’s speed. This ratio measures the degree to which the curve deviates from the straight line at a particular point. Indeed, one can show that of all the circles passing through $\gamma (t)$ and lying on the osculating plane, the one of radius $1/\kappa (t)$ serves as the best approximation to the space curve at the point $\gamma (t)$.

To treat curvature analytically, we take the derivative^{} of the relation

$${\gamma}^{\prime}(t)=v(t)T(t).$$ |

This yields the following decomposition of the acceleration vector:

$${\gamma}^{\prime \prime}(t)={v}^{\prime}(t)T(t)+v(t){T}^{\prime}(t)=v(t)\left\{{(\mathrm{log}v)}^{\prime}(t)T(t)+\kappa (t)N(t)\right\}.$$ |

Thus, to change speed,
one needs to apply acceleration along the tangent vector^{}; to change
heading the acceleration must be applied along the normal.

Title | curvature (space curve) |
---|---|

Canonical name | CurvaturespaceCurve |

Date of creation | 2013-03-22 12:14:58 |

Last modified on | 2013-03-22 12:14:58 |

Owner | slider142 (78) |

Last modified by | slider142 (78) |

Numerical id | 13 |

Author | slider142 (78) |

Entry type | Definition |

Classification | msc 53A04 |

Synonym | curvature |

Related topic | SpaceCurve |

Related topic | Torsion |

Related topic | ExpressionsForCurvatureAndTorsion |

Related topic | SerretFrenetFormulas |