can we say that any 2D shapes or plane figures are also a curve in geometry

Question

Definition of curve is "A curve is a shape or a line which is smoothly and continuously drawn in a plane having a bent or turns in it". So according to the definition of curve, can we say that any 2D shapes or plane figures are also a curve in geometry?

Answer 1

A disc, for example the set in an xy-coordinate system defined by $x^2+ y^2\le 1$, is a "plane figure" but is NOT a "curve".

Answer 2

As Wikipedia defines it:

A 'topological curve' can be specified by a continuous function $\gamma \colon I \rightarrow X$ from an interval $I\subset$ $\mathbb{R}$ into a topological space $X$. Properly speaking, the ''curve'' is the image of $\gamma$.

We can actually simplify this in our case to just:

A curve is the image of a continuous function $\gamma \colon [0,1] \rightarrow \mathbb{R}^2$.

If we assume this definition of a curve, then a '2d plane figure' is just a special case, for which $\gamma(0)=\gamma(1)$. This is called a closed curve. Examples would be circles, triangles, any polygons.

But as @george-ivey pointed out solid 2d shapes are not curves, only the boundary is a curve.