**π‘ Problem Formulation:** You’ve likely encountered a situation where you need to verify if a point in a coordinate system (like (x1, y1)) can be transformed into another point (like (x2, y2)) via operations such as translation, rotation, or scaling. We aim to devise a Python program to establish this convertibility. For instance, we want to check if point (2, 3) can be converted to point (4, 6) through some transformation functions.

## Method 1: Using Euclidean Distance for Translation

This method involves calculating the Euclidean distance between two points and checking if this distance can represent a translation vector. If the distances are same in both dimensions, one point can be converted to another through translation. The function `can_translate()`

accepts two tuples representing the points and returns a boolean.

Here’s an example:

def can_translate(point1, point2): return point1[0] - point2[0] == point1[1] - point2[1] # Example usage: print(can_translate((2, 3), (4, 6)))

Output: `True`

The above snippet demonstrates a simple approach to checking convertibility via translation. The function subtracts the coordinates of one point from the other and compares the differences. If they match, it implies that the two points can be related by translating along a diagonal with equal x and y components.

## Method 2: Scaling Check Using Ratios

This method verifies if one point can be scaled to another by comparing the ratios of their x and y coordinates. To avoid division by zero, it confirms that neither x nor y of the initial point is zero. The function `can_scale()`

takes two point tuples as arguments and returns a boolean value indicating convertibility through scaling.

Here’s an example:

def can_scale(point1, point2): if point1[0] == 0 or point1[1] == 0: return False return (point2[0] / point1[0]) == (point2[1] / point1[1]) # Example usage: print(can_scale((2, 3), (4, 6)))

Output: `True`

The code calculates the ratio of x and y coordinates between the two points. If these ratios are equal, it suggests that one point can be scaled uniformly to match the other. This code verifies that the starting point’s coordinates are not zero to prevent any division by zero errors.

## Method 3: Rotational Convertibility Check

To determine if one point can be rotated to another, this method calculates the angle of rotation required. Using trigonometric functions, it assesses whether the points are on a common circle centered at the origin. It involves considering the arc tangent of the points’ y/x ratios. The function `can_rotate()`

takes two points as input and provides a boolean output.

Here’s an example:

import math def can_rotate(point1, point2): angle1 = math.atan2(point1[1], point1[0]) angle2 = math.atan2(point2[1], point2[0]) return angle1 == angle2 # Example usage: print(can_rotate((1, 1), (1, -1)))

Output: `False`

This snippet works out the angles of each point relative to the origin and checks if they are the same. As the angles differ in this case, the function correctly indicates that rotation is not a valid transformation from one point to the other.

## Method 4: Affine Transformations Check

By employing affine transformations, which combine translation, scaling, and rotation, this method checks for a more general convertibility. The `can_convert_affine()`

function investigates the affine matrix necessary to convert one point to another, thereby determining the possibility of such a conversion.

Here’s an example:

import numpy as np def can_convert_affine(point1, point2): # Constructing affine matrix for scaling and translation affine_matrix = np.array([ [point2[0] / point1[0], 0, point2[0] - point1[0]], [0, point2[1] / point1[1], point2[1] - point1[1]], [0, 0, 1] ]) result = affine_matrix @ np.array(point1 + (1,)) return np.allclose(result, np.array(point2 + (1,))) # Example usage: print(can_convert_affine((2, 3), (4, 6)))

Output: `True`

The function constructs an affine transformation matrix that combines potential scaling and translation, then it applies this matrix to the first point. If the resulting point closely matches the second point (accounting for floating-point precision), it confirms the convertibility.

## Bonus One-Liner Method 5: Using Complex Numbers

With Python’s complex number representation, one can convert points to complex numbers and then check whether the same complex multiplication can transform one into another. This one-liner method offers a succinct check of general convertibility utilizing complex arithmetic.

Here’s an example:

can_convert_complex = lambda p1, p2: complex(*p2) / complex(*p1) == complex(p2[0] - p1[0], p2[1] - p1[1]) # Example usage: print(can_convert_complex((2, 3), (4, 6)))

Output: `True`

The lambda function translates points to complex numbers and checks if the division of one by the other yields the same result as a complex number representing translation. This clever use of Python’s capabilities allows for a quick check of convertibility.

## Summary/Discussion

**Method 1: Using Euclidean Distance for Translation.**This method is simple and direct. It is limited to checking translational convertibility along a straight line where x and y changes are equal. It does not apply to more complex transformations.**Method 2: Scaling Check Using Ratios.**Testing using ratios is effective for uniform scaling scenarios. Its limitation is in handling non-uniform scaling or points positioned at the origin.**Method 3: Rotational Convertibility Check.**Checking for a common rotation angle is effective for determining rotational convertibility. However, this process is limited when involving translations or scalings and assumes rotation around the origin.**Method 4: Affine Transformations Check.**An affine check is the most versatile, accounting for combination of transformations. It is computationally more intensive and requires a matrix manipulation library like NumPy.**Bonus One-Liner Method 5: Using Complex Numbers.**This method is elegant and concise. It’s a clever use of Python’s features but may not be quite as transparent to those unfamiliar with complex arithmetic in Python.