Editing Openai/6959ed27-accc-800e-8e87-21aa81e93c07 (section)

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║                  0xfaded's Curvature Method Analysis                    ║
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║                                                                         ║
║  STRENGTHS:                                                             ║
║  ✓ No trig functions in main loop (only sqrt/hypot)                    ║
║  ✓ Excellent SIMD/vectorization potential                              ║
║  ✓ Fixed iteration count (predictable performance)                      ║
║  ✓ Elegant geometric interpretation (evolute-based)                     ║
║  ✓ Robust across all eccentricities                                    ║
║                                                                         ║
║  PERFORMANCE:                                                           ║
║  • Scalar (Python): Comparable to Newton (~0.7-1.0x)                   ║
║  • Vectorized (NumPy): Up to 2-4x faster than Newton                   ║
║  • The advantage grows with batch size and parallelization             ║
║                                                                         ║
║  BEST USE CASES:                                                        ║
║  • Batch processing many points                                        ║
║  • GPU implementations (CUDA, OpenCL)                                  ║
║  • Game engines / collision detection                                  ║
║  • Real-time graphics                                                  ║
║  • Any SIMD-optimized code                                             ║
║                                                                         ║
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This is much more revealing! The vectorized benchmark clearly shows where 0xfaded's curvature method really shines:

    When vectorized with NumPy, the curvature method is 2-4x faster than Newton's method
    This is because sqrt operations vectorize much better than trig operations
    The advantage is consistent across different eccentricities

Let me now copy the comprehensive analysis to the outputs folder and create a nice summary.

Excellent! Now let me copy everything to outputs and create a summary report:

I've created a comprehensive benchmark analysis. Let me now provide a summary to the user that captures the key findings.
Results Summary

The benchmarks confirm that 0xfaded's curvature-based method is 2-4x faster than Newton's method when vectorized (processing batches of points). Here are the key findings:
Vectorized Performance (the real win)
Batch Size	Curvature Method	Newton's Method	Speedup
10,000 pts	1.2 ms	3.7 ms	3.0x
100,000 pts	10-14 ms	40-41 ms	3-4x
Why It's Faster

The curvature method's genius is avoiding trigonometric functions in the main loop:

    Curvature method: Uses only sqrt, *, /, +, - (all vectorize excellently)
    Newton's method: Requires cos, sin every iteration (expensive and hard to vectorize)

On modern CPUs:

    sqrt: ~15-20 cycles, vectorizes 4-8 operations at once
    sin/cos: ~50-100 cycles, poor vectorization

Accuracy

Both methods achieve identical machine-precision accuracy (~4×10⁻¹⁶ deviation from the ellipse).
Bottom Line

For single-point calculations in Python, Newton's method is slightly faster due to Python's optimized C math library. But for batch processing, GPU computing, game engines, or any SIMD-optimized code, the curvature method delivers significant performance gains—exactly the use cases mentioned in that Hacker News thread (collision plugins for Unity, etc.).
Ellipse benchmark summary
Document · MD 
Comprehensive analysis
PY 
Micro benchmark
PY 
Ellipse benchmark
PY 

Wait, can you benchmark this in c or c++ or rust or zig instead, I don't trust microbenchmarks in python like this
7:59 PM

The user is right to be skeptical - Python has a lot of overhead and the results can be misleading. Let me create a proper C or C++ benchmark that will give more accurate results. I'll use C++ since it's straightforward and will show the true performance characteristics.

You're absolutely right - Python's interpreter overhead and optimized C math library can mask the real differences. Let me write a proper C++ benchmark.

bash
cd /home/claude && g++ -O3 -march=native -ffast-math ellipse_benchmark.cpp -o ellipse_benchmark -lm && ./ellipse_benchmark

Output