Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

For questions about or involving curves.

Let $X$ be a topological space and $I$ an interval in $\mathbb{R}$. A continuous curve in $X$ is a continuous map $\gamma : I \to X$.

Let $X$ be a smooth manifold and again, let $I$ be an interval in $\mathbb{R}$. A smooth curve in $X$ is a smooth map $\gamma : I \to X$.

Note, it both cases, a curve is more than its image. That is, given two curves $\gamma_1 : I_1 \to X$ and $\gamma_2 : I_2 \to X$, it may be the case that $\gamma_1(I_1) = \gamma_2(I_2)$. A particular instance of this occurs when there is a map $\sigma : I_2 \to I_1$ which is a homeomorphism in the case of continuous curves or a diffeomorphism in the case of smooth curves, such that $\gamma_2 = \gamma_1\circ\sigma$. In this case, we say that $\gamma_2$ is a reparameterisation of $\gamma_1$.

history | excerpt history