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# Euclidean distance

## What is Euclidean distance?

In about 300 B.C.E., the Greek mathematician Euclid was examining the relationships between angles and distances. Euclidean geometry is still widely used and taught even today and applies to spaces of two or three dimensions, but it can be generalized to higher-order dimensions.

He came up with the concept of Euclidean space. Euclidean space is a two- or three-dimensional space to which the axioms and postulates of Euclidean geometry apply.

Euclidean distance refers to the distance between two points in Euclidean space. It essentially represents the shortest distance between two points.

## The three Euclidean tools

Euclidean distance is the distance from each cell in a raster to the closest source.

Euclidean allocation helps identify the cells which should be allocated to a source depending on proximity.

And lastly, Euclidean direction shows us the direction from each cell to the nearest source.

## How is Euclidean distance applied in machine learning?

In machine learning, the Euclidean distance is most commonly used to understand and measure how similar observations are to each other.

## How does Euclidean distance work?

Here's how the Euclidean algorithm works: the distance from each cell to each source cell is found by calculating the Hypotenuse with x_max and y_max as the other two sides of the triangle.

This method helps us find the actual Euclidean distance, rather than the cell distance. If the shortest distance to a source is less than the specified maximum distance, we assign the value to the cell location on the output raster.

## How do you calculate the Euclidean distance?

Let’s say that (x1, x2) and (y1, y2) exist in a two-dimensional space. If a line segment formed between these two points, it could be the hypotenuse of a right-angled triangle.

Then the other two sides of the right-angled triangle would be |x1 — y1| and |x2 — y2|.

The Euclidean distance between (x1, x2) and (y1, y2) could be considered to be the length of the hypotenuse of the right-angled triangle. Now, we can make use of the Pythagorean Theorem to find the Euclidean distance.

The Euclidean distance between (x1, x2) and (y1, y2) would be (x1 — y1)2+(x2 — y2)2.

Now, if points (x1, x2, x3) and (y1, y2, y3) were in a three-dimensional space, the Euclidean distance between them would be (x1 — y1)2+(x2 — y2)2+(x3 — y3)2.

Taking this further, to calculate the Euclidean distance between, (x1, x2,..., xn) and (y1, y2,..., yn) in an n-dimensional space, the formula would be (x1 — y1)2+(x2 — y2)2+...+ (xn — yn)2

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# Euclidean distance

October 14, 2020

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## What is Euclidean distance?

In about 300 B.C.E., the Greek mathematician Euclid was examining the relationships between angles and distances. Euclidean geometry is still widely used and taught even today and applies to spaces of two or three dimensions, but it can be generalized to higher-order dimensions.

He came up with the concept of Euclidean space. Euclidean space is a two- or three-dimensional space to which the axioms and postulates of Euclidean geometry apply.

Euclidean distance refers to the distance between two points in Euclidean space. It essentially represents the shortest distance between two points.

## The three Euclidean tools

Euclidean distance is the distance from each cell in a raster to the closest source.

Euclidean allocation helps identify the cells which should be allocated to a source depending on proximity.

And lastly, Euclidean direction shows us the direction from each cell to the nearest source.

## How is Euclidean distance applied in machine learning?

In machine learning, the Euclidean distance is most commonly used to understand and measure how similar observations are to each other.

## How does Euclidean distance work?

Here's how the Euclidean algorithm works: the distance from each cell to each source cell is found by calculating the Hypotenuse with x_max and y_max as the other two sides of the triangle.

This method helps us find the actual Euclidean distance, rather than the cell distance. If the shortest distance to a source is less than the specified maximum distance, we assign the value to the cell location on the output raster.

## How do you calculate the Euclidean distance?

Let’s say that (x1, x2) and (y1, y2) exist in a two-dimensional space. If a line segment formed between these two points, it could be the hypotenuse of a right-angled triangle.

Then the other two sides of the right-angled triangle would be |x1 — y1| and |x2 — y2|.

The Euclidean distance between (x1, x2) and (y1, y2) could be considered to be the length of the hypotenuse of the right-angled triangle. Now, we can make use of the Pythagorean Theorem to find the Euclidean distance.

The Euclidean distance between (x1, x2) and (y1, y2) would be (x1 — y1)2+(x2 — y2)2.

Now, if points (x1, x2, x3) and (y1, y2, y3) were in a three-dimensional space, the Euclidean distance between them would be (x1 — y1)2+(x2 — y2)2+(x3 — y3)2.

Taking this further, to calculate the Euclidean distance between, (x1, x2,..., xn) and (y1, y2,..., yn) in an n-dimensional space, the formula would be (x1 — y1)2+(x2 — y2)2+...+ (xn — yn)2

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