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Check if two given circles touch or intersect each other

In the field of computational geometry, determining the relative position of geometric shapes is a fundamental problem. One of the common tasks is to figure out whether two given circles touch or intersect each other. This problem has numerous practical applications, such as collision detection in video games, traffic analysis for autonomous vehicles, and many more. In this blog post, we will explore the mathematical concepts behind determining the relationship between two circles and provide implementations in Python to help you solve the problem.

2026-07

Table of Contents#

  1. Mathematical Background
  2. Conditions for Circle Intersection and Touching
  3. Python Implementation
  4. Example Usage
  5. Best Practices and Common Pitfalls
  6. Conclusion
  7. References

Mathematical Background#

A circle in a two - dimensional plane is defined by two main attributes: its center point $(x, y)$ and its radius $r$. Given two circles $C_1$ and $C_2$, let the center of $C_1$ be $(x_1,y_1)$ with radius $r_1$, and the center of $C_2$ be $(x_2,y_2)$ with radius $r_2$.

The distance $d$ between the centers of two circles $C_1$ and $C_2$ can be calculated using the Euclidean distance formula: [d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}]

Conditions for Circle Intersection and Touching#

Based on the distance $d$ between the centers of the two circles and their radii $r_1$ and $r_2$, we can define the following conditions:

1. Circles are disjoint (do not touch or intersect)#

  • Externally disjoint: If $d>r_1 + r_2$, the two circles are completely separated from each other.
  • Internally disjoint: If $d<|r_1 - r_2|$, one circle is completely inside the other without touching.

2. Circles touch each other#

  • Externally touching: When $d = r_1 + r_2$, the two circles touch each other at exactly one point from the outside.
  • Internally touching: When $d=|r_1 - r_2|$, one circle touches the other from the inside at exactly one point.

3. Circles intersect each other#

If $|r_1 - r_2|<d<r_1 + r_2$, the two circles intersect at two distinct points.

Python Implementation#

import math
 
def check_circle_relationship(x1, y1, r1, x2, y2, r2):
    # Calculate the distance between the centers of the circles
    distance = math.sqrt((x2 - x1)**2+(y2 - y1)**2)
    
    if distance > r1 + r2:
        return "The circles are externally disjoint."
    elif distance < abs(r1 - r2):
        return "One circle is internally disjoint from the other."
    elif distance == r1 + r2:
        return "The circles are externally touching."
    elif distance == abs(r1 - r2):
        return "The circles are internally touching."
    else:
        return "The circles intersect each other."

Example Usage#

# Example 1: Externally disjoint circles
x1,y1,r1 = 0,0,1
x2,y2,r2 = 5,5,1
print(check_circle_relationship(x1,y1,r1,x2,y2,r2))
 
# Example 2: Intersecting circles
x1,y1,r1 = 0,0,3
x2,y2,r2 = 2,0,2
print(check_circle_relationship(x1,y1,r1,x2,y2,r2))
 
# Example 3: Internally touching circles
x1,y1,r1 = 0,0,3
x2,y2,r2 = 1,0,2
print(check_circle_relationship(x1,y1,r1,x2,y2,r2))

In the first example, the distance between the centers is much larger than the sum of the radii, so the circles are externally disjoint. In the second example, the distance between the centers falls between the difference and sum of the radii, indicating that the circles intersect. In the third example, the distance between the centers is equal to the difference of the radii, so the circles are internally touching.

Best Practices and Common Pitfalls#

Best Practices#

  • Use appropriate data types: When dealing with coordinates and radii, make sure to use appropriate data types. Floating - point numbers are usually a good choice since coordinates can take non - integer values.
  • Error handling: In a real - world application, you may need to handle cases where the input values are negative or invalid. For example, a negative radius is not a valid input for a circle.

Common Pitfalls#

  • Rounding errors: When using floating - point numbers to calculate the distance between the centers, rounding errors may occur. It's a good practice to define a small tolerance value $\epsilon$ and check if $|d-(r_1 + r_2)|<\epsilon$ or $|d - |r_1 - r_2||<\epsilon$ instead of strict equality for determining the touching cases.

Conclusion#

Determining whether two circles touch or intersect each other is a relatively straightforward problem that can be solved using basic Euclidean geometry concepts. By calculating the distance between the centers of the circles and comparing it with the sum and difference of their radii, we can easily classify the relationship between two circles. We have also provided a Python implementation along with example usage, best practices, and common pitfalls to help you solve this problem in real - world scenarios.

References#

  • Computational Geometry: Algorithms and Applications by Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars.
  • Wikipedia: Circle