Parametric surface

Okay, so imagine you have some Play-Doh and you want to make a cool shape, like a butterfly. But instead of just smooshing the Play-Doh into a butterfly shape randomly, you decide to follow some rules. You figure out that if you make a long snake out of Play-Doh, then bend it in the middle, and curve the ends up, you can make a butterfly shape!

Now let's take it a step further. Imagine instead of just making one butterfly, you want to make a whole bunch of butterflies with different patterns and sizes. You could use the Play-Doh snake method for each one, but that would take a looong time. Instead, you come up with a set of rules called parametric equations that can create the shape of a butterfly.

These rules tell you things like how much to bend the Play-Doh snake, how much to curve the ends, and where to place the head and wings. You can even adjust the rules to make different sizes and shapes of butterflies!

In math, a parametric surface works kind of like this. Instead of using Play-Doh to make shapes, mathematicians use equations to create surfaces (like a blobby, 3D shape). Parametric equations tell you things like how to move a point around in space, how fast to move it, and in which direction. These rules can be used to create all kinds of fun shapes and patterns!