Interactive application to calculate and visualize the optimal dimensions of Norman windows
- Developer: Carlos Gabriel Magallanes López
- Email: cgmagallanes23@gmail.com
- GitHub: @TheNarratorVIMMXX
- Date: December 2025
- 📖 Project Description
- 🎯 Mathematical Problem
- 📐 Mathematical Foundation
- ⚡ Features
- 🔧 Technical Implementation
- 💾 Installation
- 🖥️ Display Configuration
- 🚀 Usage
- 📜 License
GeoFrame is a scientific application developed in Python that solves a classic constrained optimization problem: finding the optimal dimensions of a Norman window that maximize its area given a fixed perimeter.
This project was conceived as an educational tool to visualize and understand mathematical concepts such as optimization, derivatives, and practical applications of calculus in architectural design.
A Norman window (also known as a semicircular or Romanesque window) is an architectural structure composed of:
- A rectangle with base
xand heighty - A semicircle with radius
r = x/2placed on top of the rectangle
This classic architectural design combines structural stability with aesthetics, and presents an interesting mathematical challenge: How do you distribute a limited perimeter to obtain the maximum possible area?
Imagine you have a window frame with a fixed perimeter P (for example, 12 meters of material). How should you design the window to allow the greatest amount of light (area) possible?
This is not an intuitive problem because:
- If the window is too wide, the height decreases
- If it is too tall, the width and semicircle shrink
- The optimal balance requires calculus and optimization techniques
The perimeter of a Norman window is composed of:
- Rectangle base:
x - Two vertical sides:
2y - Upper semicircle:
πr = π(x/2)
Perimeter constraint:
P = x + 2y + πx/2
P = x(1 + π/2) + 2y
Solving for y:
y = (P - x(1 + π/2)) / 2
The total area is the sum of:
- Rectangle area:
A_rect = xy - Semicircle area:
A_semi = πr²/2 = π(x/2)²/2 = πx²/8
Total area function:
A(x) = xy + πx²/8
Substituting y from the constraint:
A(x) = x · [(P - x(1 + π/2))/2] + πx²/8
A(x) = (Px)/2 - x²(1 + π/2)/2 + πx²/8
A(x) = (Px)/2 - x²/2 - πx²/4 + πx²/8
A(x) = (Px)/2 - x²/2 - πx²/8
To find the maximum, we take the derivative and set it to zero:
dA/dx = P/2 - x - πx/4 = 0
P/2 = x(1 + π/4)
x_optimal = P / (2 + π/2)
Once x_optimal is obtained, we calculate:
y_optimal = (P - x_optimal(1 + π/2)) / 2
A_max = x_optimal · y_optimal + π(x_optimal)²/8
To confirm it is a maximum (and not a minimum):
d²A/dx² = -1 - π/4 < 0 ✓ (Confirms it is a maximum)
- Automatic optimization: Calculates optimal dimensions (x, y, r) using advanced numerical methods (SciPy)
- Real-time updates: All visualizations update dynamically when the perimeter changes
- Interactive slider: Adjust the perimeter from 1 to 100 meters with 0.1 precision
- Quick default values: Buttons for common values (5, 12, 25, 50, 100 meters)
1. 🖼️ Norman Window Panel
- Interactive geometric representation
- Annotated dimensions with arrows
- Three visualization modes:
- Normal: Clean basic view
- Detailed: Grid lines and multiple layers
- Technical: Full specifications with partial areas
2. 📈 Area vs Width Graph
- Curve of the area function A(x)
- Visual identification of the maximum point
- Reference lines toward optimal dimensions
- Dynamic annotation with coordinates
3. 📋 Numerical Results Panel
- Input data display
- Optimal dimensions (x, y, r)
- Partial areas (rectangle and semicircle)
- All values with 4-decimal precision
4. 🔍 Sensitivity Analysis Graph
- Shows how maximum area varies with perimeter
- Range from 1 to 100 meters
- Highlight of the current point
- Useful for hypothetical scenario analysis
- Smart cache: Stores previously calculated results to avoid redundant computations
- Precalculation: Sensitivity data is calculated once and reused
- History management: Keeps a record of the last 20 calculations
| Library | Version | Purpose |
|---|---|---|
| Python | 3.13.0 | Main programming language |
| Matplotlib | 3.10.7 | Advanced graphing and visualization |
| Seaborn | 0.13.2 | Professional graph styling |
| PyQt5 | Latest | GUI and window management |
| NumPy | Latest | Numerical computing and array operations |
| SciPy | Latest | Advanced optimization algorithms (minimize_scalar) |
The application follows Object-Oriented Programming principles:
GeoFrameClass: Main application controller- Separation of concerns: Each panel has its own update method
- Event-driven design: Responds to user interactions (slider, buttons, radio buttons)
- Modular structure: Easy to maintain and extend
No installation or dependencies required! The game is available as a ready-to-use executable.
- Go to the Releases section
- Download the
GeoFrame.exefile - Double-click to run
- Done!
⚠️ Note: Python does not need to be installed or any environment configured.
For the best visual experience with GeoFrame, we recommend the following display configuration:
| Setting | Value |
|---|---|
| Recommended | 1920 × 1080 (Full HD) |
| Minimum | 1280 × 720 |
This resolution ensures all panels, graphs, and controls display correctly without overlapping.
The application interface is optimized for two specific zoom levels:
| Option | Scale | Best For |
|---|---|---|
| Option 1 (Recommended) | 150% | Best balance between visibility and screen space |
| Option 2 | 100% | Maximum screen usage, all panels visible simultaneously |
- Right-click on the desktop and select Display settings
- Under Scale & layout, find the Display resolution dropdown
- Set to 1920 × 1080 (Recommended)
- In the same section, find the Scale dropdown
- Choose 100% or 150% based on your preference
- Click Apply and restart the GeoFrame application
- Other resolutions: The app will work but layout may not be optimal
- Other scales: Using 125% or 175% may cause minor alignment issues
- Multiple monitors: Ensure GeoFrame runs on the monitor with the recommended configuration
- Open the app: Double-click to run the application
- Adjust the perimeter: Use the interactive slider or preset buttons
- Read the results: All panels update automatically
- Change view mode: Use the radio buttons (Normal / Detailed / Technical)
- Analyze sensitivity: Observe how the area changes with different perimeters
Example with P = 12 meters:
Optimal Width (x): 4.2667 m
Optimal Height (y): 1.6234 m
Semicircle Radius: 2.1333 m
Maximum Area: 13.5752 m²
This means that with 12 meters of perimeter, the design that allows the most light is a window 4.27 meters wide and 1.62 meters tall, with a total area of 13.58 square meters.
Copyright © 2025 Carlos Gabriel Magallanes López
All Rights Reserved
This license allows you to:
✅ Download and use the software for personal and educational purposes
✅ Install and run the application on your personal devices
✅ Create derivative works based on the software
You may NOT:
❌ Modify, reverse engineer, decompile, or disassemble the software
❌ Redistribute, share, or make copies available to others
❌ Use the software for commercial purposes without written permission
❌ Remove or modify copyright notices or proprietary marks
This software is provided "as is", without warranties of any kind. The author is not responsible for damages or issues arising from the use of the software.
For licensing inquiries, commercial use, or permissions:
📧 Email: cgmagallanes23@gmail.com
Last Updated: December 16, 2025
© 2025 Carlos Gabriel Magallanes López. All rights reserved.