# 5 Best Ways to Compute Element Wise Arc Tangent of x1 and x2 Choosing the Quadrant Correctly in Python

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๐ก Problem Formulation: Computing the element-wise arc tangent of two arrays, X1 and X2, involves determining the angle theta such that tan(theta) = X2/X1, while taking into account the correct quadrant of the angle based on the signs of X1 and X2. The desired output is an array of angles in radians, ranging from -ฯ to ฯ. For example, given X1 = [1, -1] and X2 = [1, 1], the correct quadrant-aware result should be [ฯ/4, 3ฯ/4].

## Method 1: Using NumPyโs `arctan2()` Function

NumPy library has a function `arctan2()` that calculates the arc tangent of the quotient of two arrays element-wise. It returns the angles in radians and considers the quadrant of the inputs, making it a robust choice for this problem. `numpy.arctan2(x1, x2)` needs two arrays of the same shape as input and returns the angles, considering the signs of both inputs to determine the correct quadrant.

Here’s an example:

```import numpy as np

x1 = np.array([1, -1])
x2 = np.array([1, 1])
angles = np.arctan2(x2, x1)

print(angles)```

Output of this code snippet:

`[ 0.78539816  2.35619449]`

This code snippet demonstrates the use of the `np.arctan2()` function from the NumPy library to compute the correct quadrant-wise arc tangent values for the given arrays X1 and X2. The printout shows the angles in radians, resulting in ฯ/4 and 3ฯ/4 respectively, corresponding to the correct quadrants for the input values.

## Method 2: Using Math Libraryโs `atan2()` Function

The math library in Python also provides an `atan2()` function that computes the arc tangent of two numbers. Unlike NumPy, it handles numerical inputs rather than arrays, but it can be used in a loop or with list comprehension to achieve the same element-wise behavior. It correctly handles the signs of the arguments to compute the arc tangent in the correct quadrant.

Here’s an example:

```import math

x1 = [1, -1]
x2 = [1, 1]
angles = [math.atan2(y, x) for x, y in zip(x1, x2)]

print(angles)```

Output of this code snippet:

`[0.7853981633974483, 2.356194490192345]`

In this code example, we use list comprehension to apply the `math.atan2()` function to each pair of elements drawn from X1 and X2. The output is the same as the previous method: an array of angles in radians that correctly reflects the quadrants of the input coordinates.

## Method 3: Using a Custom Function with Element-wise Operations

For a more hands-on approach, one can define a custom function that manually checks the signs of X1 and X2 elements and uses the `math.atan()` function to compute the arc tangent, adjusted for the correct quadrant. This method requires additional logic but is good for deep understanding.

Here’s an example:

```import math

def custom_arctan2(y, x):
angle = math.atan(abs(y)/abs(x))
if x >= 0 and y >= 0:
return angle
elif x = 0:
return math.pi - angle
elif x < 0 and y < 0:
return -math.pi + angle
else:
return -angle

x1 = [1, -1]
x2 = [1, 1]
angles = [custom_arctan2(y, x) for x, y in zip(x1, x2)]

print(angles)```

Output of this code snippet:

`[0.7853981633974483, 2.356194490192345]`

This code snippet introduces a custom function, `custom_arctan2()`, that calculates the arc tangent while considering the quadrant manually. The function adjusts the angle depending on the signs of X and Y. When used in combination with list comprehension, it generates the correct quadrant-aware angles for the pairs of X1 and X2.

## Method 4: Using SymPyโs `atan2()` Function

SymPy is a Python library for symbolic mathematics. It includes an `atan2()` function that can operate element-wise on symbolic objects or numerical values. SymPy’s `atan2()` carefully respects mathematical rigor, which can be beneficial for applications needing symbolic computation capabilities.

Here’s an example:

```from sympy import atan2

x1 = [1, -1]
x2 = [1, 1]
angles = [atan2(y, x) for x, y in zip(x1, x2)]

print(angles)```

Output of this code snippet:

`[pi/4, 3*pi/4]`

This example utilizes SymPy’s `atan2()` function within a list comprehension to compute the quadrant-aware arc tangent of each pair of elements from the lists X1 and X2. SymPy returns the result as symbolic expressions which are useful for further symbolic manipulations.

## Bonus One-Liner Method 5: Using NumPy with Lambda and Map

For a quick one-liner, NumPy’s `arctan2()` function can be coupled with the `lambda` function and `map()` to apply the operation in an element-wise manner. This approach is concise but might lack clarity for those unfamiliar with lambda functions.

Here’s an example:

```import numpy as np

x1 = [1, -1]
x2 = [1, 1]
angles = list(map(lambda pair: np.arctan2(pair[1], pair[0]), zip(x1, x2)))

print(angles)```

Output of this code snippet:

`[0.7853981633974483, 2.356194490192345]`

This snippet succinctly demonstrates combining the use of a lambda function with `map()` to calculate the angles using NumPy’s `arctan2()`. The zip function pairs up elements from X1 and X2, which are then processed by lambda to give us the desired result.

## Summary/Discussion

• Method 1: NumPy’s `arctan2()` function. Strengths: Vectorized operations, performance, and simplicity. Weaknesses: Dependency on an external library.
• Method 2: Math libraryโs `atan2()`. Strengths: Part of the standard library, no additional installation required. Weaknesses: Not designed for element-wise array operations, less performance-efficient.
• Method 3: Custom Function. Strengths: Deep understanding of the quadrant adjustment process. Weaknesses: Verbose and prone to human error.
• Method 4: SymPyโs `atan2()`. Strengths: Provides symbolic results, which is useful for certain types of mathematical problems. Weaknesses: Overhead of symbolic computation, not as performance-efficient as numerical methods.
• Method 5: NumPy with Lambda and Map. Strengths: Concise and functional-programming style. Weaknesses: May be less readable for those not familiar with lambda and map functions.