Here is when the concept of perpendicular line becomes crucial. In these cases, we first need to define what point on this line or circumference we will use for the distance calculation, and then use the distance formula that we have seen just above. Sometimes we want to calculate the distance from a point to a line or to a circle. We often don't want to find just the distance between two points. The distance formula we have just seen is the standard Euclidean distance formula, but if you think about it, it can seem a bit limited. You can try to understand it by thinking of the so-called lines of longitude that divide the Earth into many time zones and cross each other at the poles. Another very strange feature of this space is that some parallel lines do actually meet at some point. The shortest distance from one point to another is not a straight line, because any line in this space is curved due to the intrinsic curvature of the space. In this case, very strange things happen. This curved space is hard to imagine in 3D, but for 2D we can imagine that instead of having a flat plane area, we have a 2D space, for example, curved in the shape of the surface of a sphere. There are, however, other types of mathematical spaces called curved spaces in which space is intrinsically curved and the shortest distance between two points is no a straight line. This means that space itself has flat properties for example, the shortest distance between any two points is always a straight line between them (check the linear interpolation calculator). This space is very similar to Euclidean space, but differs from it in a very crucial feature: the addition of the dot product, also called the inner product (not to be confused with the cross product).īoth the Euclidean and Minkowski space are what mathematicians call flat space. The reason we've selected this is because it's very common in physics, in particular it is used in relativity theory, general relativity and even in relativistic quantum field theory. The first example we present to you is a bit obscure, but we hope you can excuse us, as we're physicists, for starting with this very important type of space: Minkowski space. Also, you will hopefully understand why we are not going to bother calculating distances in other spaces. However, we can try to give you some examples of other spaces that are commonly used and that might help you understand why Euclidean space is not the only space. We do not want to bore you with mathematical definitions of what is a space and what makes the Euclidean space unique, since that would be too complicated to explain in a simple distance calculator. Euclidean space can have as many dimensions as you want, as long as there is a finite number of them, and they still obey Euclidean rules. Let's also not confuse Euclidean space with multidimensional spaces. This is something we all take for granted, but this is not true in all spaces. In Euclidean space, the sum of the angles of a triangle equals 180º and squares have all their angles equal to 90º always. The Euclidean space or Euclidean geometry is what we all usually think of 2D space is before we receive any deep mathematical training in any of these aspects. Since this is a very special case, from now on we will talk only about distance in two dimensions. If you wish to find the distance between two points in 1D space you can still use this calculator by simply setting one of the coordinates to be the same for both points. For each point in 2D space, we need two coordinates that are unique to that point. These points are described by their coordinates in space. To find the distance between two points, the first thing you need is two points, obviously. If you are looking for the 3D distance between 2 points we encourage you to use our 3D distance calculator made specifically for that purpose. For this calculator, we focus only on the 2D distance (with the 1D included as a special case). In most cases, you're probably talking about three dimensions or less, since that's all we can imagine without our brains exploding. If we stick with the geometrical definition of distance we still have to define what kind of space we are working in. You will see in the following sections how the concept of distance can be extended beyond length, in more than one sense that is the breakthrough behind Einstein's theory of relativity. This definition is one way to say what almost all of us think of distance intuitively, but it is not the only way we could talk about distance. The most common meaning is the 1D space between two points. Before we get into how to calculate distances, we should probably clarify what a distance is.
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