Lissajous curves

Lissajous curves are shapes that are made by connecting a moving point to two axes that move at different speeds. It's like drawing on a piece of paper while the paper is moving in different directions.

Imagine you are playing a game with two spinning tops. One top spins horizontally and the other spins vertically. You put a pen on the top of one of the spinning tops and follow the movement of the two tops on a piece of paper below. The resulting shape is a lissajous curve.

These curves can come in many different shapes and patterns, depending on how the two tops are spinning and how fast they are spinning. Some lissajous curves look like circles, while others look like squares or even intricate patterns.

Scientists and engineers use lissajous curves to study the motion of things that move in two dimensions, like sound waves or electrical signals. By analyzing the shape of the curve, they can learn more about how these things move and how they can be controlled.