Euclidean distance

Euclidean geometry is a mathematical system credited to ancient Greek mathematician Euclid and documented in his classic ‘The Elements’. Euclid’s method entails starting with a limited collection of intuitively acceptable axioms and deducing a large number of other propositions (theorems) from them. Although many of Euclid’s findings were already known, he was the first to organise them into a logical framework in which each result is supported by axioms and previously verified theorems. 

Because no other type of geometry had been invented in over two thousand years, the word “Euclidean” was unneeded. Any theorem proved from Euclid’s axioms was judged true in an absolute, frequently metaphysical sense (with the probable exception of the parallel postulate). However, numerous other self-consistent non-Euclidean geometries exist today, with the first being discovered in the early nineteenth century.

The distance between two points in coordinate geometry is known as Euclidean distance. The length of the line segment connecting the two sites should be measured in order to determine the distance between them.

Definition:-

The distance between two points is known as the Euclidean distance in mathematics. In other terms, the length of the line segment between two points is defined as the Euclidean distance between two points in Euclidean space. The Pythagorean distance is sometimes termed the Euclidean distance since it can be calculated using coordinate points and the Pythagoras theorem.

Euclidean distance formula:-

As previously stated, the Euclidean distance formula aids in determining the distance between two line segments. Assume two locations in the two-dimensional coordinate plane, such as (p, q) and (r, s).

As a result, the Euclidean distance formula is given by :

         d=√(r- p)2+(s- q)2

Where, d is the Euclidean distance.

(p,q) is the coordinate of first point and (r,s) is the coordinate of the second point.

Euclidean distance formula derivation:-

Consider two locations, P(x1, y2) and Q(x2, y2), and d is the distance between them to get the formula for Euclidean distance. Using a line, connect the points A and B. Construct a right triangle with a hypotenuse of AB to calculate the Euclidean distance formula. Now, as indicated in the diagram below, draw a horizontal and vertical line from A and B that intersect at C.

To find the distance between two points, we must apply the Pythagoras theorem to the triangle PQR.

Using Pythagoras’ theorem as a guide,

Base² + Perpendicular² = Hypotenuse²

PR² + QR² = PQ²

As a result, d² = (x2 – x1)² + (y2 – y1)²

Taking the square root on both sides of the equation gives us

d = √{(x2 – x1)² + (y2 – y1)²}

As a result, the Euclidean distance formula is derived.

Euclidean Distance Matrix:-

An n × n matrix expressing the separation of a set of n points in Euclidean space is called a Euclidean distance matrix in mathematics. The elements of their Euclidean distance matrix A are squares of distances between points display style x1, x2, …….., xn in k-dimensional space Rk.

Application of Euclidean Distance formula:-

In geometry, the Euclidean distance formula is commonly employed in the following situations:

To prove that the four sides are equal, as well as the diagonals, to prove that a given figure is a square.

To prove that the four sides of a given figure are equal to prove that it is a rhombus.

To prove that opposite sides are equal and diagonals are likewise equal to prove that a given figure is a rectangle.

To prove that the opposite sides of a given figure are equal to prove that it is a parallelogram.

To prove that the opposite sides of a given figure are equal but the diagonals are not equal to prove that it is a parallelogram but not a rectangle.

On a 20-D space of genes, the Euclidean distance measurement method is used to measure the distance between genes in order to distinguish between healthy and malignant breast genes, and a PCA model based on 20-D amino acid composition is used to predict genes.

Conclusion:-

The length of a line segment between two locations in Euclidean space is known as the Euclidean distance in mathematics. It is sometimes referred to as Pythagorean distance since it could be determined from cartesian coordinates of the points using the Pythagorean Theorem. Although Euclid did not describe distances as numbers, and the relationship between the Pythagorean theorem and distance calculation was not realised until the 18th century, these names are derived from ancient Greek mathematicians Euclid and Pythagoras.

The smallest distance between pairs of points from two things is commonly described as the distance between two objects that aren’t points. Distances between different sorts of objects, such as the distance from a point to a line, can be calculated using formulas. Distance has been generalised to abstract metric spaces in advanced mathematics, and other distances besides Euclidean distances have been researched. The square of the Euclidean distance is used instead of the distance itself in various applications in statistics and optimization.