Catenary and parabola are two types of plane curves —specifically, two classic shapes that appear in mathematics and physics when things hang or bend under force.

In geometry terms

  • Both are smooth, U‑shaped curves in the plane, but they arise from different underlying equations.
  • A parabola is the graph of a quadratic equation such as y=ax2+bx+cy=ax^2+bx+cy=ax2+bx+c.
  • A catenary is the graph of a hyperbolic‑cosine function y=acosh⁡(x/a)y=a\cosh(x/a)y=acosh(x/a), which comes from solving a hanging‑chain problem involving gravity and uniform tension.

In applied‑physics or engineering terms

  • In structural contexts, catenary and parabola are two types of funicular curves —shapes that trace how a flexible cable or arch naturally takes under certain loading patterns.
* A **catenary** is the shape of a **freely hanging chain or cable** under its own weight.

* A **parabola** appears when a cable **supports a uniform load along the horizontal direction** , as in many suspension‑bridge designs where the deck weight dominates.

So to answer your title directly:

Catenary and parabola are two types of plane curves (or, more specifically, two types of funicular curves in engineering).