Editing Differentiable curve (section)

{{short description|Study of curves from a differential point of view}}
{{About|curves in Euclidean space|curves in an arbitrary topological space|Curve}}

'''Differential geometry of curves''' is the branch of [[geometry]] that deals with [[smoothness (mathematics)|smooth]] [[curve]]s in the [[Euclidean plane|plane]] and the [[Euclidean space]] by methods of [[differential calculus|differential]] and [[integral calculus]].

Many [[list of curves|specific curves]] have been thoroughly investigated using the [[Synthetic geometry|synthetic approach]]. [[Differential geometry]] takes another path: curves are represented in a [[parametric equation|parametrized form]], and their geometric properties and various quantities associated with them, such as the [[curvature]] and the [[arc length]], are expressed via [[derivative]]s and [[integral]]s using [[vector calculus]]. One of the most important tools used to analyze a curve is the [[Frenet frame]], a moving frame that provides a coordinate system at each point of the curve that is "best adapted" to the curve near that point.

The theory of curves is much simpler and narrower in scope than the [[differential geometry of surfaces|theory of surfaces]] and its higher-dimensional generalizations because a regular curve in a Euclidean space has no intrinsic geometry. Any regular curve may be parametrized by the arc length (the ''natural parametrization''). From the point of view of a [[test particle|theoretical point particle]] on the curve that does not know anything about the ambient space, all curves would appear the same. Different space curves are only distinguished by how they bend and twist. Quantitatively, this is measured by the differential-geometric invariants called the ''[[curvature]]'' and the ''[[torsion of curves|torsion]]'' of a curve. The [[fundamental theorem of curves]] asserts that the knowledge of these invariants completely determines the curve.