Euclidean distance

In mathematics, the Euclidean distance is a way to measure how far apart two points are in space. It tells us the straight-line distance between them, just like measuring with a ruler. This idea comes from old Greek mathematicians like Euclid and Pythagoras, who studied shapes and distances long ago.

We can find the Euclidean distance using something called the Pythagorean theorem, a famous rule that helps us work with triangles. It lets us calculate the distance if we know the Cartesian coordinates of the points, which are like addresses that tell us where each point is located.

This way of measuring distance is very useful in many areas, such as statistics and optimization. Even in more advanced math, the idea of distance has been expanded to study all sorts of spaces and objects, but the Euclidean distance remains one of the most basic and important ways we measure how far things are from each other.

Distance formulas

The distance between two points tells us how far apart they are. In one dimension, like on a straight number line, the distance is simply the absolute difference between their positions. For example, if one point is at position 3 and another is at position 7, the distance between them is |3 − 7| = 4.

In two dimensions, such as on a flat piece of paper, we can find the distance using a rule similar to the Pythagorean theorem. If one point has coordinates (x1, y1) and another has (x2, y2), the distance between them is the square root of [(x1 − x2)² + (y1 − y2)²]. This works because we can imagine forming a right triangle where the line between the points is the hypotenuse, and the differences in x and y coordinates form the other two sides. The same idea can be extended to three or more dimensions, just by adding more squared differences together before taking the square root.

Main article: Euclidean distance

Properties

The Euclidean distance has special properties that make it useful in math. First, it is symmetric, meaning the distance from point A to point B is the same as from B to A. It is also positive, so the distance between two different points is always a number greater than zero, and the distance from a point to itself is zero.

Another important property is the triangle inequality. This means that if you travel from point A to point C through point B, the total distance will always be at least as long as going directly from A to C. This helps us understand how distances work in space.

Squared Euclidean distance

The squared Euclidean distance is a way to measure how far apart two points are, but without taking the square root at the end. This makes calculations easier and helps avoid small errors that can happen with numbers. For example, when we want to compare distances, we can just look at the squared values because they tell us the same order as the real distances.

Squared Euclidean distance is important in statistics, where it helps find the best fits for data. It is also used in cluster analysis to group points together. Even though it doesn’t follow all the rules of a true distance measure, it is very useful in optimization theory because it makes problems easier to solve.

Generalizations

In higher mathematics, Euclidean distance is linked to a special measurement called the Euclidean norm, which tells us how far a point is from the starting point, called the origin. This norm stays the same even if we turn the space around. It can also be used in spaces with many dimensions and even infinite dimensions, known as the L² norm.

There are other ways to measure distance besides Euclidean distance. For example, the Chebyshev distance looks at the biggest difference between points, the Taxicab distance adds up the differences, and the Minkowski distance is a general rule that includes these and Euclidean distance. When measuring distances on curved surfaces like Earth, we use different methods such as the haversine distance or Vincenty's formulae.

History

Euclidean distance is named after the ancient Greek mathematician Euclid, whose book Elements was a key geometry textbook for many years. Though ideas about length and distance go back thousands of years, Euclid did not directly define distance as a number between two points.

The important role of the Pythagorean theorem in measuring distances only became clear after René Descartes introduced Cartesian coordinates in 1637. The actual distance formula was first shared in 1731 by Alexis Clairaut, leading some to call it Pythagorean distance. Later, in the 1800s, mathematicians discovered other ways to measure distances, known as non-Euclidean geometry.