Difference between revisions of "Regular curve"

Revision as of 21:29, 29 March 2015

Requires knowledge of Curve and Parametrisation

Definition

A curve $$\gamma:\mathbb{R}\rightarrow\mathbb{R}^3$$ usually (however $$\gamma:A\subseteq\mathbb{R}\rightarrow\mathbb{R}^n$$ more generally) is called regular if all points ( $$\in\text{Range}(\gamma)$$ ) are regular

Definition: Regular Point

A point $$\gamma(t)$$ is called regular of $$\dot\gamma\ne 0$$ otherwise it is a Singular point

Important point

The curve $$\gamma(t)\mapsto(t,t^2)$$ is regular however $$\gamma'(t)\mapsto(t^3,t^6)$$ is not - it is not technically a reparametrisation

Any reparametrisation of a regular curve is regular

TODO: reparametrisation theorem

See also

• Curve