Arc length of curve from data points in python

When working with data points in Python, it is often necessary to calculate the arc length of a curve. The arc length represents the distance along the curve between two points. In this article, we will explore three different ways to solve this problem using Python.

Option 1: Using Numerical Integration

One way to calculate the arc length of a curve is by using numerical integration. This method involves dividing the curve into small segments and approximating the length of each segment. By summing up the lengths of all segments, we can obtain an estimate of the total arc length.


import numpy as np

def arc_length_numerical_integration(x, y):
    n = len(x)
    length = 0.0
    
    for i in range(1, n):
        dx = x[i] - x[i-1]
        dy = y[i] - y[i-1]
        length += np.sqrt(dx**2 + dy**2)
    
    return length

In this code snippet, we define a function arc_length_numerical_integration that takes two arrays x and y representing the x and y coordinates of the data points. The function iterates over the points and calculates the length of each segment using the Euclidean distance formula. The lengths are then summed up to obtain the total arc length.

Option 2: Using Simpson’s Rule

Another approach to calculating the arc length is by using Simpson’s rule, which is a numerical integration method that provides a more accurate approximation compared to simple numerical integration. This method involves dividing the curve into smaller segments and using a quadratic polynomial to approximate the curve within each segment.


from scipy.integrate import simps

def arc_length_simpsons_rule(x, y):
    n = len(x)
    length = 0.0
    
    for i in range(1, n):
        dx = x[i] - x[i-1]
        dy = y[i] - y[i-1]
        length += simps([dy, dy + (dy - y[i-1]), y[i]], [0, dx/2, dx])
    
    return length

In this code snippet, we use the simps function from the scipy.integrate module to perform Simpson’s rule integration. The function takes two arrays x and y representing the x and y coordinates of the data points. Similar to the previous method, the function iterates over the points and calculates the length of each segment using Simpson’s rule. The lengths are then summed up to obtain the total arc length.

Option 3: Using the Chord Length Formula

The third option for calculating the arc length is by using the chord length formula. This formula provides an approximation of the arc length by assuming that the curve between two points is a straight line. While this method may not be as accurate as the previous two, it is computationally simpler and can be sufficient for certain applications.


def arc_length_chord_length(x, y):
    n = len(x)
    length = 0.0
    
    for i in range(1, n):
        dx = x[i] - x[i-1]
        dy = y[i] - y[i-1]
        length += np.sqrt(dx**2 + dy**2)
    
    return length

In this code snippet, we define a function arc_length_chord_length that takes two arrays x and y representing the x and y coordinates of the data points. The function iterates over the points and calculates the length of each segment using the Euclidean distance formula. The lengths are then summed up to obtain the total arc length, similar to the first option.

After exploring these three different methods for calculating the arc length of a curve from data points in Python, it is important to consider the specific requirements of your application. If accuracy is crucial, using Simpson’s rule (Option 2) would be the best choice. However, if simplicity and computational efficiency are more important, the chord length formula (Option 3) can provide a reasonable approximation. Ultimately, the choice depends on the specific needs of your project.

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