5 Best Ways to Check Whether a Given Circle Resides in the Boundary Maintained by Two Other Circles in Python

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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

center1 = (1, 1)
center2 = (5, 5)
center0 = (3, 2)

# Check if circle0 is within the boundary
```

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
```

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
```

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
```

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

Pythonβs 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
Output: `True`