Euclidean distance

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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. At this point, the Euclidean distance is pretty much the most common use of distance. In most situations in which people are talking about distance, they are referring to the Euclidean distance. It examines the root of square distances between the co-ordinates of a pair of objects. To derive the Euclidean distance, you would have to compute the square root of the sum of the squares of the differences between corresponding values.

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.

By making use of the Pythagorean formula for distance, Euclidean space (or even any inner product space) would become a metric space. The associated norm is referred to as the Euclidean norm, which is defined as the distance of each vector from the origin. One of the important properties of the Euclidean norm, relative to other norms, is that it remains unchanged under arbitrary rotations of space around the origin. According to Dvoretzky's theorem, every finite-dimensional normed vector space has a high-dimensional subspace on which the norm is more or less Euclidean. The Euclidean norm is the only norm that posesses this property.

Earlier, this metric was also known as the Pythagorean metric. Due to the fact that it is possible for you to find the Euclidean distance by making use of the coordinate points and the Pythagoras theorem, it is also sometimes referred to as the Pythagorean distance.

Euclidean distance
Source: Wikipedia

Which are the three prime Euclidean terms?

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

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

3. Euclidean direction shows us the direction from each cell to the nearest source.

How is Euclidean distance applied in machine learning?

In machine learning, it is most commonly used to understand and measure how similar observations are to each other. With vision AI, euclidean distance can be used for the camera to infer specific movements based on distance changes. This is applicable in many fields including AI exercise motion capture and analysis.

How does Euclidean distance work?

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 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.

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How do you calculate Euclidean distance (formula)?

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. Since it is nothing but the straight line distance between two given points, we can make use of the Pythagorean Theorem.

The 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

What is the Euclidean Squared Distance Metric?

The Euclidean squared distance metric makes use of the same equation as the Euclidean distance metric, but it does not take the square root. Because of this, clustering can be performed at a faster pace with the Euclidean Squared Distance Metric than it can be carried out with the regular Euclidean distance. 

Even if you replace the Euclidean distance with the Euclidean squared distance metric, the output of Jarvis-Patrick clustering and of K-Means clustering will not be affected. But if you do this, the output of hierarchical clustering will be very likely to change.

Since squared Euclidean distance does not satisfy the triangle inequality, it does not form a metric space. Instead, it is a smooth, strictly convex function of the two points, unlike the distance, which is non-smooth (near pairs of equal points) and convex but not strictly convex. 

The Euclidean squared distance is preferred in optimization theory because it enables convex analysis to be used. Due to the fact that squaring happens to be a monotonic function of non-negative values, minimizing squared distance is equivalent to minimizing the Euclidean distance. Therefore the optimization problem is equivalent in terms of either, but it is easier to solve by making use of squared distance.

The collection of all squared distances between pairs of points from a finite set could be stored in a Euclidean distance matrix and can be used in that form in distance geometry.

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