EDB — 1NV

[1NV] Let $I\subseteq ℝ$ be an interval.

• A continuous function $𝛾:I\to X$ is called parametric curve, or more simply in the following curve.

• If $𝛾$ is injective, the curve is said to be simple.

• If $𝛾$ is a homeomorphism onto its image, the curve is said to be embedded.

• If $X={ℝ}^n$ and $𝛾$ is of class $C^1$ and ${𝛾}^{\prime }(t)\ne 0$ for every $t\in I$, then $𝛾$ is called an immersed curve or regular curve.

We will call support or trace the image $𝛾(I)$ of a curve.

The term arc is also used as a synonym for curve; ¹ this term is mainly used when the curve is not (necessarily) closed.