catenary and parabola are two types of what?

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).