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, like using a ruler. This idea started with old Greek mathematicians like Euclid and Pythagoras.

We can find the Euclidean distance using the Pythagorean theorem. This rule helps us with triangles. It lets us calculate the distance if we know the Cartesian coordinates of the points. These coordinates are like addresses for where each point is.

This way of measuring distance is useful in many areas, such as statistics and optimization. Even in more advanced math, the idea of distance is used to study many things, but Euclidean distance is one of the most basic 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 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 used in three or more dimensions 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. It is symmetric, meaning the distance from point A to point B is the same as from B to A. It is always positive, so the distance between two different points is a number greater than zero. 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 simpler and helps avoid small mistakes with numbers.

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. This norm tells us how far a point is from the starting point, called the origin. It can 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. His book Elements was a very important geometry book for many years. People have thought about length and distance for thousands of years, but Euclid did not define distance as a number between two points.

The Pythagorean theorem became important for measuring distances after René Descartes introduced Cartesian coordinates in 1637. The actual distance formula was first shared in 1731 by Alexis Clairaut. Later, in the 1800s, mathematicians found other ways to measure distances, called non-Euclidean geometry.