Curve (parametrized curve)

A (parametrized) curve in $\mathbb{R}^k$ is a function

usually assumed at least continuous (and often differentiable or $C^1$ depending on context). The set $\gamma([a,b])\subseteq\mathbb{R}^k$ is the trace (or image) of the curve.

Curves provide a way to describe paths, boundaries, and parameterized geometry. In analysis, curves are used to build paths in spaces and to study continuous images of intervals.

Examples:

• The line segment from $x$ to $y$ is $\gamma(t)=(1-t)x+ty$ on $[0,1]$.
• The circle can be parametrized by $\gamma(t)=(\cos t,\sin t)$ on $[0,2\pi]$.
• The graph of a function $f:[a,b]\to\mathbb{R}$ is the curve $\gamma(t)=(t,f(t))$ in $\mathbb{R}^2$.