Parametric curves are really important in robotics and motion planning. They help describe complicated paths that robots need to follow.

In robotics, these curves help us understand how robotic arms and mobile robots can move around. They make it easier for robots to get around safely in their surroundings. By using equations, like $x(t)$ and $y(t)$ for a 2D path, we can program robots to follow these paths smoothly and avoid obstacles in their way.

One big use of parametric curves is for planning how a robot will move, called trajectory planning. For example, when a robot wants to pick up an object, it can use these equations to create a smooth path. This path takes into account where the obstacles are. Following a smooth path helps robots move better, which is very important for tasks that need careful movements.

Also, using polar coordinates with curves can be helpful when robots work in round or circular areas. This makes it easier to figure out angles and distances.

Parametric curves also help when testing robotic movements virtually. By simulating these paths before actually using them, developers can find problems, make things work better, and keep things safe.

In short, using parametric curves helps make motion planning easier and plays a key role in making robotic systems work better in the real world.