The distance formula is a mathematical equation that is used to determine the distance between two points in a plane. In Python, this formula can be implemented using the builtin math library, as well as through various thirdparty libraries such as NumPy and SciPy.
The basic distance formula is:
d = √((x2 – x1)² + (y2 – y1)²)
Where (x1, y1) and (x2, y2) are the coordinates of the two points, and d is the distance between them.
To calculate the distance between two points using Python, you can use the math library's sqrt() function to calculate the square root of the sum of the squares of the differences of the x and y coordinates. Here is an example of a Python function that implements the distance formula:
import math
def distance(x1, y1, x2, y2):
return math.sqrt((x2  x1)**2 + (y2  y1)**2)
print(distance(1, 2, 4, 6)) # Output: 5.0
Alternatively, you can use the NumPy library's linalg.norm() function to calculate the distance between two points. Here is an example:
import numpy as np
def distance(x1, y1, x2, y2):
point1 = np.array([x1, y1])
point2 = np.array([x2, y2])
return np.linalg.norm(point1  point2)
print(distance(1, 2, 4, 6)) # Output: 5.0
The above examples are for 2D coordinates, but you can also calculate the distance between points in higher dimensions. Here is an example of calculating the distance between two points in 3D space using the NumPy library:
import numpy as np
def distance(x1, y1, z1, x2, y2, z2):
point1 = np.array([x1, y1, z1])
point2 = np.array([x2, y2, z2])
return np.linalg.norm(point1  point2)
print(distance(1, 2, 3, 4, 5, 6)) # Output: 8.666003
It's worth noting that in case of large dataset of coordinates calculating distance using above method can be computationally expensive. Therefore, to improve performance, you can use the KDTree class from the SciPy library's spatial module. It uses a data structure called a kd tree, which allows for efficient querying of nearest neighbors.
from scipy.spatial import KDTree
data = np.array([[1, 2], [3, 4], [5, 6], [7, 8]])
tree = KDTree(data)
# Find the nearest point to [0, 0]
dist, idx = tree.query([0, 0])
print(f"Nearest point: {data[idx]} Distance: {dist}")
In conclusion, there are multiple ways to calculate the distance between two points in Python, using the builtin math library, NumPy, and SciPy. Depending on the size of your dataset and the specific requirements of your
In addition to calculating the distance between two points, there are several other related concepts and techniques that are commonly used in computational geometry and computer science.
One such concept is the Manhattan distance, also known as the "taxi cab" distance, which is the distance between two points measured along the axes at right angles. This is calculated by taking the absolute difference of the xcoordinates plus the absolute difference of the ycoordinates. Here is an example of a Python function that calculates the Manhattan distance:
def manhattan_distance(x1, y1, x2, y2):
return abs(x1  x2) + abs(y1  y2)
print(manhattan_distance(1, 2, 4, 6)) # Output: 6
Another related concept is the Euclidean distance, which is the square root of the sum of the squares of the differences of the coordinates. This is the most common distance measure used in machine learning, and it is the default distance measure used in the KNN algorithm.
Another important topic is Clustering. Clustering is the process of grouping similar data points together. The KMeans algorithm is one of the most widely used clustering algorithms. It is a centroidbased algorithm, or a distancebased algorithm, where we calculate the distances to assign a point to a cluster. The KMeans algorithm is sensitive to the initial placement of the centroids. To overcome this problem, we have another variation of KMeans algorithm called KMeans++ which tries to overcome this issue by placing the centroids far away from each other.
Finally, the Convex Hull is an important concept in computational geometry. It is the smallest convex polygon that contains all the points of a given set of points in the plane. The Jarvis march algorithm, also known as the "gift wrapping" algorithm, is a common method for computing the convex hull. It starts with an extreme point (a point with the smallest or largest xcoordinate) and then "wraps" a line around the other points, always taking the point with the smallest or largest angle with respect to the previous line segment.
Python has several libraries for working with computational geometry, including Shapely, Geopy, Pyclipper and more. These libraries provide additional functionality for working with geometric shapes, such as calculating the area, centroid, and intersection of shapes, as well as performing spatial analysis and manipulation.
In conclusion, the distance formula is an important concept in computational geometry, and there are several related concepts and techniques that are commonly used in computer science and machine learning, such as the Manhattan distance, Euclidean distance, Clustering and Convex Hull. Python has several libraries that make working with these concepts easy, efficient and accurate.
Popular questions

How do I calculate the distance between two points in Python?
Answer: You can use the builtin math library's sqrt() function to calculate the square root of the sum of the squares of the differences of the x and y coordinates, or use the NumPy library's linalg.norm() function. 
Can the distance formula be used for points in more than 2 dimensions?
Answer: Yes, the distance formula can be used for points in higher dimensions by simply adding additional coordinates to the formula. 
What is the difference between the Euclidean distance and Manhattan distance?
Answer: Euclidean distance is the square root of the sum of the squares of the differences of the coordinates, while Manhattan distance is the distance between two points measured along the axes at right angles, calculated by taking the absolute difference of the xcoordinates plus the absolute difference of the ycoordinates. 
How can we improve performance when working with large dataset of coordinates?
Answer: To improve performance when working with large dataset of coordinates, you can use the KDTree class from the SciPy library's spatial module which uses a data structure called a kd tree, which allows for efficient querying of nearest neighbors. 
Are there any libraries in Python that can help with working with computational geometry?
Answer: Yes, there are several libraries in Python that can help with working with computational geometry such as Shapely, Geopy, Pyclipper and more. These libraries provide additional functionality for working with geometric shapes, such as calculating the area, centroid, and intersection of shapes, as well as performing spatial analysis and manipulation.
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ComputationalGeometry.