space curve (original) (raw)
Kinematic definition.
Regularity hypotheses.
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
γ′(t) never vanishes. Also, we say that γ(t) is a point of inflection if the first and second derivatives γ′(t),γ′′(t) are linearly dependent. Space curves with points of inflection are beyond the scope of this entry. Henceforth we make the assumption
that γ(t) is both and lacks points of inflection.
Geometric definition.
Arclength parameterization.
We say that γ:I→ℝ3 is an arclength parameterization of an oriented space curve if
With this hypothesis the length of the space curve between points γ(t2) and γ(t1) is just |t2-t1|. In other words, the parameter in such a parameterization measures the relative distance along the curve.
Starting with an arbitrary parameterization γ:I→ℝ3, one can obtain an arclength parameterization by fixing a t0∈I, setting
and using theinverse function σ-1 to reparameterize the curve. In other words,
is an arclength parameterization. Thus, every space curve possesses an arclength parameterization, unique up to a choice of additive constant in the arclength parameter.
| Title | space curve |
|---|---|
| Canonical name | SpaceCurve |
| Date of creation | 2013-03-22 12:15:03 |
| Last modified on | 2013-03-22 12:15:03 |
| Owner | Mathprof (13753) |
| Last modified by | Mathprof (13753) |
| Numerical id | 15 |
| Author | Mathprof (13753) |
| Entry type | Definition |
| Classification | msc 53A04 |
| Synonym | oriented space curve |
| Synonym | parameterized space curve |
| Related topic | Torsion |
| Related topic | CurvatureOfACurve |
| Related topic | MovingFrame |
| Related topic | SerretFrenetFormulas |
| Related topic | Helix |
| Defines | point of inflection |
| Defines | arclength parameterization |
| Defines | reparameterization |