# Euclidean Distance Calculator

- What is the Euclidean distance?
- How do I calculate the Euclidean distance?
- Euclidean distance between two points
- Euclidean distance of three points
- Euclidean distance between a point and a line
- Euclidean distance between two parallel lines
- A different point of view: Euclidean distance as a distance between points in a set
- How do I use the Euclidean distance calculator?
- Euclidean distance and beyond
- FAQ

It may sound fancy, but the Euclidean distance is nothing more than the distance the way we think of it every day: discover more on how to calculate it with our Euclidean distance calculator!

Here you will learn:

**What is the Euclidean distance**?- Formulas:
- The
**distance between two points**and the distance between**three points**; - The
**distance between a point and a line**; - The
**distance between two lines**.

- The
- Other uses of the Euclidean distance.

## What is the Euclidean distance?

The Euclidean distance is a **metric** defined over the **Euclidean space** (the physical space that surrounds us, plus or minus some dimensions). In a few words, the Euclidean distance measures the **shortest path** between two points in a smooth n-dimensional space.

We can define the Euclidean distance only in flat spaces: on curved surfaces, strange things happen, and straight lines are not necessarily the shortest!

## How do I calculate the Euclidean distance?

The Euclidean distance is defined through the **Cartesian coordinates** of the points under analysis. We can think of it as the translation vector between two points. In our Euclidean distance calculator, we teach you how to calculate:

- The Euclidean distance between two or three points in spaces form
**one to four dimensions**; - The Euclidean distance between a point and a line in a 2D space; and
- The Euclidean distance between two parallel lines in a 2D space.

## Euclidean distance between two points

To find the Euclidean distance between two points. you need to know the coordinates of these points.

Take a generic point $p$. We can write its coordinates as:

The number of components depends on the dimensionality of the space.

To calculate the distance between the point $p$ and the point $q$, we apply a generalized form of the Pythagorean theorem:

Where $n$ is the dimensionality of the space.

As you can see, it's pretty easy!

## Euclidean distance of three points

If we add a point, we will find **three possible distances**; three points, after all, always define a triangle (possibly degenerated).

Taking the triple of points $p$, $q$, and $r$, we can find the following **two points distances**:

This concept can obviously expand to increasingly larger **n-tuples**.

## Euclidean distance between a point and a line

To find the distance from a point to a line (in **two dimensions**), we need to consider the fact that a line is nothing but a collection of points satisfying an equation. The distance between a point and a line is then the **minimum distance between the point and the line**.

The segment connecting the point to the point on the line satisfying the condition above is perpendicular to the line itself. To find the distance, we use this formula:

Where $a$, $b$, and $c$ are the **coefficient** of the equation of the line in the form: $a\cdot x + b\cdot y + c = 0$. If we are considering the more commonly found expression $y = m\cdot x + c$, we would use this formula (equivalent in every way):

## Euclidean distance between two parallel lines

To calculate the distance between two parallel lines we use the following equation:

The lines have equations:

- $a_1\cdot x+b_1\cdot y_1 + c_1$; and
- $a_2\cdot x+b_2\cdot y_1 + c_2$.

However, there is a strong constraint over the choices of coefficients. Since two **non-parallel lines** would cross at a certain point in the plane, giving a trivial distance $d=0$, we define the Euclidean distance only in the case of parallel lines. This gives us quite some helpful rules:

- $a_1=a_2=a$;
- $b_1=b_2=b$;
- However, $c_1\neq c_2$.

Using the slope-intercept form of the equation of a line, the formula for the distance becomes:

## A different point of view: Euclidean distance as a distance between points in a set

The Euclidean distance is becoming an important concept in **machine learning** (the less sci-fi version of AI), where the distance between points in **arbitrary spaces of features** is measured with **metrics**. The most commonly used ones are:

- The
**Minkowski distance**; - The
**Manhattan distance**; and - The
**Euclidean distance**.

Each of them is appropriate for certain data: the Manhattan distance for integer values and the Euclidean distance for real-valued data. The Minkowski distance is a generalization of both.

It is interesting to notice how a physical, and concrete, concept (the distance between two points) is translated into a distance in a space of features: the difference is from calculating the distance between your house and the one of a friend to a generalized distance between two colors, two car models...

## How do I use the Euclidean distance calculator?

First, choose the objects you are calculating the distance: for points, you can also choose the **dimensionality of the space**.

Then, insert the coordinates of the points or the parameters of the lines. We will show both the distance and the **steps** to calculate it. There is nothing more to it!

## Euclidean distance and beyond

The Euclidean distance is a particular — yet common — way to measure the distance. We made more calculator specific to certain topics and problems:

## FAQ

### What is the distance between two parallel lines with equations y = 2x + 3 and y = 2x - 4?

The distance is `3.13`

. To calculate it,

- We identify the parameters of the lines:
`m=2`

(the value is the same for both lines since they are parallel);`c₁ = 3`

, and`c₂ = -4`

.

- We apply the equation for the distance between two parallel lines:

`d = | c₁ - c₂ |/(√[m² + 1])`

`d = | 3 + 4 |/(√[2² + 1] = 3.13`

### How do I calculate the distance between a point and a line?

To calculate the distance between a point and a line, follow these steps:

1 Define the coordinates and parameters of the objects;

2 Calculate the distance using the formula:

`d = | m × p₁ + q₁ + c |/(√[m² + 1])`

3. There's nothing more to it.

To do this, we had to calculate the area of a triangle in coordinates space and then calculate its height.

### Is the distance formula still valid in a 4-dimensional space?

Yes: the concept of distance exists in any Euclidean space with an arbitrary number of dimensions. It may be hard to visualize a 4-dimensional space, though. Imagine a car that moved from a point `A`

to a point `B`

in a certain time `t`

. If you take time as your fourth dimension, you can see that, if you freeze the movement, the car is not moving: we need the fourth dimension to define the distance!

### What is the distance between the point p = (1,2) and the point q = (2,3)?

`d = 1.414`

. To calculate the distance between these two points on a plane, we simply apply the formula:

`√[(p₂ - p₁)² + (q₂ - q₁)²]`

In this case:

`√[(2 - 1²) + (3 - 2²] = √2 = 1.414`

Notice how the distance between these two particular points corresponds both to the hypotenuse of a right triangle and the diagonal of a square.