Pi Estimator: Monte Carlo Method

Overview

This Python program estimates the value of $\pi$ using the Monte Carlo method. By randomly generating points within a square and checking whether they fall inside an inscribed circle, the program calculates an approximation of $\pi$. The estimate improves as the number of points increases.

Concept and Formula

1. Setup:

   • The square has side length $2r$, and the circle has radius $r$, with the circle perfectly inscribed in the square.
   • The area of the square is $4r^2$, and the area of the circle is $\pi r^2$.

2. Monte Carlo Method:

   • Random points $(x, y)$ are generated within the square, where $x, y \in [-r, r]$.
   • A point is considered inside the circle if it satisfies $x^2 + y^2 \leq r^2$.

3. Estimation of $\pi$:

   • The ratio of points inside the circle ($N_\text{circle}$) to the total number of points ($N_\text{total}$) is approximately equal to the ratio of the circle's area to the square's area: $\frac{N_\text{circle}}{N_\text{total}} \approx \frac{\pi}{4}$
   • Rearranging, we estimate $\pi$ as: $\pi \approx 4 \cdot \frac{N_\text{circle}}{N_\text{total}}$

Control Flow and Description

1. Random Point Generation:

   • Points are generated randomly within the square's bounds.

2. Classification:

   • Each point is classified as either inside the circle or outside the circle based on the condition $x^2 + y^2 \leq r^2$.

3. Estimation:

   • $\pi$ is calculated using the formula: $\pi \approx 4 \cdot \frac{N_\text{circle}}{N_\text{total}}.$
   • As the number of points ($N_\text{total}$) increases, the estimate converges toward the true value of $\pi$, though fluctuations may occur.

4. Visualization:

   • A graph displays the randomly generated points, showing their distribution within the square and the circle.
   • Another graph tracks:
     • The total number of trials ($N_\text{total}$).
     • The number of points inside the circle ($N_\text{circle}$).
     • The number of points outside the circle ($N_\text{total} - N_\text{circle}$).
     • The running estimate of $\pi$.

GUI

The figure below shows a screenshot of the GUI at runtime. It includes:

• A graphical representation of the points distributed in the square and circle.
• Real-time statistics, including:
  • Number of points generated.
  • Points inside the circle.
  • Points outside the circle.
  • The current estimate of $\pi$.

Notes

• The accuracy of the estimate improves as more points are generated.
• Results may vary slightly due to the stochastic nature of the method.