Advanced Spline Modeling & Simulation Toolkit

A research-grade simulation environment for modeling Bézier, Catmull-Rom, Hermite, and NURBS splines with multi-point control. Designed to analyze parametric curve behavior under dynamic constraints in applied mathematics and computer graphics.

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🎨 Overview

This project implements a modular framework for studying spline interpolation techniques, featuring:

• 6+ spline types with GPU-accelerated rendering
• Multi-dimensional control points (2D/3D/4D support)
• Real-time curvature/torsion analysis
• Aesthetic visualization of derivative continuity (C⁰, C¹, C²)

Research Focus:
Comparative analysis of spline properties for robotics trajectory planning and CAD applications, validated through 2D simulations with nanosecond-precision timing.

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🧮 Mathematical Foundations

Implemented Splines

1. Bézier Curves
   Bernstein basis implementation with de Casteljau's algorithm (Bézier, 1966)

   def bezier(t, points):
       n = len(points)-1
       return sum(comb(n,i) * (1-t)**(n-i) * t**i * points[i] for i in range(n+1))

2. Catmull-Rom Splines
   Local control with tension parameterization (Catmull & Rom, 1974)

3. NURBS
   Non-uniform rational basis splines with knot vectors (Piegl & Tiller, 1987)

4. Hermite Curves
   Endpoint derivative-controlled interpolation (Hermite, 1878)

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🖥️ Simulation Features

• Interactive Control Points
  Drag-and-drop manipulation with magnetic snapping

  class ControlPoint:
      def __init__(self, pos, weight=1.0):
          self.pos = np.array(pos, dtype=np.float64)
          self.weight = weight  # For NURBS

• Multi-View Visualization
  Simultaneous display of:

  • Parametric space
  • Curvature heatmaps
  • Frenet-Serret frames
  • Derivative plots

• Performance Metrics

  • Curve smoothness (C² continuity tests)
  • Computational complexity analysis
  • Memory footprint profiling

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📊 Comparative Analysis (Selected Results)

Spline Type	Smoothness	Local Control	Computational Cost	Typical Use Case
Bézier	C⁰	No	O(n²)	Font Design
Catmull-Rom	C¹	Yes	O(n)	Camera Paths
NURBS	C²	Yes	O(n log n)	Aerospace Surfaces
Hermite	C¹	Partial	O(1) per segment	Robotic Arm Trajectories

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🚀 Getting Started

Setup

git clone https://github.com/San68bot/LearningSplines.git
pip install -r requirements.txt  # Pygame, NumPy, Matplotlib

Basic Usage

from splines import CubicHermite, visualize

points = [(0,0), (2,5), (5,3), (7,6)]
tension = 0.5
curve = CubicHermite(points, tension)
visualize(curve, derivative=True)

Simulation Controls

• Left-Click: Move control points
• Right-Click: Add/remove points
• TAB: Cycle spline types
• SPACE: Toggle derivative visualization

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📚 Academic Context

1. Spline Theory
   • (Rogers, 2001) An Introduction to NURBS
2. Computational Geometry
   • (Farin, 1993) Curves and Surfaces for CAGD
3. Real-Time Rendering
   • (Akenine-Möller, 2018) Real-Time Rendering

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🌟 Future Directions

1. Extended Spline Support

   # Planned Kochanek-Bartels splines
   class KochanekBartels(SplineBase):
       def __init__(self, points, tension=0, bias=0, continuity=0):
           ...

2. Machine Learning Integration
   • GAN-based auto-smoothing of noisy control points
3. 3D Surface Modeling
   • Tensor-product spline surfaces

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License: MIT
Citing This Work:

@software{SplineLab_2023,
  author = {San68bot},
  title = {LearningSplines},
  url = {https://github.com/San68bot/LearningSplines}
}