ELI5: Explain Like I'm 5

implicit curve

Okay, imagine you have a friend who really loves drawing, and they want to draw a really cool shape that goes all over the place. But they don't know exactly what they want the shape to look like yet, they just know some things they want it to do. For example, they might say "I want the shape to go up and then down again" or "I want the shape to have a loop in the middle".

So, they start drawing some lines and curves on a piece of paper, trying to make it match what they're picturing in their head. They might try drawing one line, and then see that it's not quite right, so they draw another line that gets closer to what they want. They keep doing this, adding and adjusting lines, until they have a drawing that they're happy with.

Now, imagine your friend wants to describe this shape they drew to someone else, but they don't actually want to show them the drawing. Maybe they want to use it for some kind of computer program, and need a way to tell the computer how to draw the shape without showing it a picture. This is where an implicit curve comes in.

An implicit curve is a way of describing a shape using a math equation. It's a bit like playing a game of "guess my number", but instead of thinking of a number between 1 and 10, your friend is thinking of a shape that goes up and down and has a loop in the middle. They can use an equation to describe that shape, even though it's not a perfect, specific shape like a circle or a square.

The equation might look something like this:

x^2 + y^2 = 1

This equation describes a circle: if you plug in different values for x and y, you'll get points that lie on the circle. But an implicit curve can be more complicated than just a circle. For example, your friend might use an equation like this:

x^2 + y^2 - 1.5x^3 - y^3 = 0

This equation describes a shape that kind of looks like a squished apple. If you plug in different values of x and y, you'll get points that lie on this shape.

So, an implicit curve is a way of using math to describe a shape that doesn't fit into a simple category like "square" or "triangle". It's a bit like drawing with equations instead of with a pencil!
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