The curved line is one of the most basic and important shapes in mathematics, around which endless structures and relationships of great importance are established. We could describe the curved line as a straight line that progressively takes some kind of straightness deviation, not suddenly or violently, because in that case we would be talking about the union of two perpendicular straight curves at a point. The curved line can form, if it is closed, different shapes and structures that vary depending on the angle with which that line is built on space and on the plane.
The curved line is an interesting phenomenon in mathematics since its morphology makes it difficult to describe in comparison with many other phenomena more adjustable to logical or formulaic definitions. The curved line has been classified in many different ways and in some cases the traditionally accepted definitions have required updates because mathematics itself has proven useless to explain the phenomenon so simple but at the same time so complex of the curved line.
In simple terms, we could say that the curved line can be open or closed. When we talk about open curved lines, we are referring to the parabola (the line that is projected when a conical shape is cut by the plane parallel to its generatrix), to the hyperbola (the one that is generated when a cone is cut through a plane oblique to its axis of symmetry) and to the catenary (the curve that an element such as a chain obtains when exposed to gravity).
Closed curved lines can form different surfaces that vary depending on the angle of your space. Thus, we talk about the ellipse (a closed symmetrical curved line) and the circumference (a line that establishes that all the points that start from its radius or center are at the same distance from the line, which is why it is a perfect curved line). . On the other hand, there is also the flat curved line, which is one that only exists in a plane or space, for which we speak of a representation of a curved line.