Is the curve open or closed?
To answer this question, we need to understand what the word curve means. In everyday life, we would say that a curved line or surface is any line that is not straight. But in maths, the word curve has a slightly different meaning. Any line where we start at a point, we move our pencil, and then we stop at a point is called a curve.
So that includes things like this. Interestingly, it also includes things like this. We started at a point, we moved the pencil, and then we stopped at another point. We can even draw curves where we start and stop at exactly the same place. A circle is a type of curve. Although it may not seem like it, a rectangle is a type of curve. We start at a point, we move our pencil without taking it off the page, and then we arrived at a point. In this case, it’s the same point we started at.
Our question gives us an example of a curve and asks us: is it open or closed? Let’s have a think about what those two words mean. An open curve is one where the start and the end point are different. These are points where it’s impossible to get back to the starting point without taking our pencil off the paper. A closed curve is where the start and the end points are the same. In other words, we are right back where we started.
One way of checking whether a curve is closed is to ask ourself: can I colour the space inside it? And if we can colour the space inside it, it usually means we have a closed curve. And so, is the curve in the question open or closed? A good way to find out is to put our finger on one end of the curve and then to trace it. And as we move our finger along the curve, we arrive at a different point than where we started. And so we know, this is not a closed curve. We have different start and end points. The curve is open.