A parameterized space curve is a parameterized curve taking values in 3-dimensional Euclidean space. It may be interpreted as the trajectory of a particle moving through space. Analytically, a smooth space curve is represented by a sufficiently differentiable mapping of an interval into 3-dimensional Euclidean space . Equivalently, a parameterized space curve can be considered a 3-vector of functions:
To preclude the possibility of kinks and corners, it is necessary to add the hypothesis that the mapping be regular (http://planetmath.org/Curve), that is to say that the derivative never vanishes. Also, we say that is a point of inflection if the first and second derivatives are linearly dependent. Space curves with points of inflection are beyond the scope of this entry. Henceforth we make the assumption that is both and lacks points of inflection.
A space curve, per se, needs to be conceived of as a subset of rather than a mapping. Formally, we could define a space curve to be the image of some parameterization . A more useful concept, however, is the notion of an oriented space curve, a space curve with a specified direction of motion. Formally, an oriented space curve is an equivalence class of parameterized space curves; with and being judged equivalent if there exists a smooth, monotonically increasing reparameterization function such that
We say that is an arclength parameterization of an oriented space curve if
With this hypothesis the length of the space curve between points and is just . In other words, the parameter in such a parameterization measures the relative distance along the curve.
Starting with an arbitrary parameterization , one can obtain an arclength parameterization by fixing a , setting
and using the inverse function to reparameterize the curve. In other words,
|Date of creation||2013-03-22 12:15:03|
|Last modified on||2013-03-22 12:15:03|
|Last modified by||Mathprof (13753)|
|Synonym||oriented space curve|
|Synonym||parameterized space curve|
|Defines||point of inflection|