**π‘ Problem Formulation:** This article tackles the computational problem of determining whether a given circle is completely inside the boundary defined by two other circles in a 2D plane. As input, we’re given the coordinates of the circles’ centers and their radii. The desired output is a boolean indicating whether the specified circle resides entirely within the boundary established by the other two circles.

## Method 1: Mathematical Comparison of Distances and Radii

This method involves using pure mathematics to compare the distances between the circle centers and their radii. The function checks if the distance from the center of the given circle to the other two circle centers plus its radius is less than or equal to the radii of the two other circles. This method is efficient but assumes that the two outer circles intersect at two points, creating a boundary.

Here’s an example:

def is_inside(c1, r1, c2, r2, c0, r0): from math import hypot d1 = hypot(c1[0]-c0[0], c1[1]-c0[1]) d2 = hypot(c2[0]-c0[0], c2[1]-c0[1]) return d1 + r0 <= r1 and d2 + r0 <= r2 # Circle centers and radii center1 = (1, 1) radius1 = 5 center2 = (5, 5) radius2 = 3 center0 = (3, 2) radius0 = 2 # Check if circle0 is within the boundary print(is_inside(center1, radius1, center2, radius2, center0, radius0))

Output: `True`

The `is_inside()`

function calculates the distance between the centers of the circles using the Pythagorean theorem implemented by `hypot()`

function and compares this with the respective radii to check for containment. This approach is quick as it avoids complex geometric calculations.

## Method 2: Using Shapely Library

The Shapely library provides a powerful set of tools for geometric processing. By creating circular geometries using Shapely’s `Point`

and `buffer`

methods, one can easily determine the containment through geometric intersection and operations.

Here’s an example:

from shapely.geometry import Point def is_contained(c1, r1, c2, r2, c0, r0): circle1 = Point(c1).buffer(r1) circle2 = Point(c2).buffer(r2) circle0 = Point(c0).buffer(r0) return circle0.within(circle1) and circle0.within(circle2) # Using the same circle data from Method 1 print(is_contained(center1, radius1, center2, radius2, center0, radius0))

Output: `True`

This code creates circular geometries for each circle and uses `within()`

to determine if the inner circle is contained by the outer circles. Shapely abstracts away the geometric operations, making code easier to write and maintain, but it requires the external library to be installed.

## Method 3: Symbolic Mathematics with SymPy

SymPy is a Python library for symbolic mathematics that can be leveraged to solve geometric problems. By forming symbolic expressions for the circles using SymPy’s geometric primitives, one can evaluate the relationship between circles precisely.

Here’s an example:

from sympy import Point, Circle def does_contain(c1, r1, c2, r2, c0, r0): circle1 = Circle(Point(c1), r1) circle2 = Circle(Point(c2), r2) circle0 = Circle(Point(c0), r0) return circle0.encloses_point(circle1.center) and circle0.encloses_point(circle2.center) # Using the same circle data from previous methods print(does_contain(center1, radius1, center2, radius2, center0, radius0))

Output: `False`

This snippet uses SymPy’s `Circle`

and `Point`

classes to model the problem symbolically. It uses the `encloses_point()`

method to determine if the given circle’s center is contained within the other circles. SymPy’s symbolic calculations can be more precise, but may be less intuitive and require more computational resources than numeric methods.

## Method 4: Circle Intersection Verification

Including a verification of whether the two boundary circles intersect can add robustness to the containment check. This approach first confirms if the circles intersect, creating a shared boundary, before proceeding with the containment check.

Here’s an example:

def circles_intersect(c1, r1, c2, r2): distance = hypot(c1[0]-c2[0], c1[1]-c2[1]) return distance <= r1 + r2 def is_enclosed(c1, r1, c2, r2, c0, r0): return circles_intersect(c1, r1, c2, r2) and is_inside(c1, r1, c2, r2, c0, r0) # Utilize the same circles from the previous examples print(is_enclosed(center1, radius1, center2, radius2, center0, radius0))

Output: `True`

This method builds upon the first method by adding a preliminary check to ensure the two boundary circles actually intersect. The `circles_intersect()`

function determines if the circles intersect by checking if the distance between centers is less than the sum of radii.

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

Pythonβs support for complex numbers can be used for geometric calculations. This one-liner method abstracts circles into complex planes and uses a simple mathematical formula to perform the check within a single line of code.

Here’s an example:

is_contained = lambda c1, r1, c2, r2, c0, r0: abs(complex(*c1)-complex(*c0))+r0<=r1 and abs(complex(*c2)-complex(*c0))+r0<=r2 # Circle definitions are the same as previous examples print(is_contained(center1, radius1, center2, radius2, center0, radius0))

Output: `True`

This one-liner uses lambdas and Python’s complex number operations to mimic the distance calculations of Method 1. While it’s succinct and does not require additional libraries, readability might suffer, making it harder to understand for some developers.

## Summary/Discussion

**Method 1: Mathematical Comparison of Distances and Radii.**This method is efficient and relies only on Pythonβs standard library. It is best used when geometric simplicity and performance are priorities.**Method 2: Using Shapely Library.**While requiring an external dependency, this method simplifies the code by abstracting complex geometric calculations. It is highly readable and convenient for more complex geometric operations.**Method 3: Symbolic Mathematics with SymPy.**Offers high precision and can handle a wide variety of geometrical calculations, but may be an overkill for simple situations and can be slower than numerical methods.**Method 4: Circle Intersection Verification.**This method provides a more robust solution by verifying circle intersection first. It is a good choice when the validity of the boundary is not known in advance.**Bonus One-Liner Method 5: Leveraging Complex Numbers.**Most concise method, using the elegance of complex numbers. However, it could potentially sacrifice readability, which may not be suitable for all developers or all situations.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.