Physics Engine (3D Rigid Body)

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Introduction:

A lightweight 3D rigid body physics engine written in Python, focusing on collision detection, contact generation, and impulse-based resolution.

This project is designed as a learning-oriented and research-oriented engine with clear algorithmic structure and explicit physical modeling.

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Usage:

Precompiled executables for this engine can be found at the releases page for Windows and Linux. Or you can just:

git clone https://github.com/EMEEEEMMMM/AtlasPhys.git
uv sync
uv run python main.py

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Features:

• 3D rigid body dynamics (linear + angular)
• Semi-implicit Euler integrator
• Gravity and basic for accumulation
• Collision detection:
  • Plane-Cube
  • Plane-Sphere
  • Cube-Sphere
  • Cube-cube
  • Sphere-Sphere
• Contact generation
• Acceleration & Impulse visualization (Arrows in red and green)
• Based on OpenGL and Pyqt6
• Interactive demo (camera control, object generation)

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Simulation pipeline

Each physics step/frame follows this order:

1. Integration

   • Semi-implicit Euler
   • Updates velocity -> position
   • Updates angular velocity -> orientation
   • Updates for the Arrows

2. Broad Phase

   • Simple AABB overlap test

3. Narrow Phase

   • GJK for collision detection
   • EPA for penetration depth & contact normal

4. Contact Generation

   • Analytical contacts for simple shapes
   • Face selection + polygon clipping for Cube-Cube

5. Constraint Solving

   • Impulse-based normal resolution
   • Iterative solver
   • Split impulse formulation no Baumgarte bias in velocity solve

6. Positional Correction

   • Pure position-level correction

flowchart TD
    A[Start] --> B[Integrate Motion]
    B --> C[Broad Phase AABB Test]
    C -->|Overlap| D[GJK Collision Detection]
    D -->|Colliding| E[EPA Penetration Solve]
    E --> F[Contact Generation]
    F --> G[Impulse Solver]
    G --> H[Positional Correction]
    H --> I[End Step]

Loading

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Physics Model

Integrator:

$$ a = g $$ $$ {v_{t + \Delta t}} = {v_t} + a\Delta t $$ $$ {x_{t + \Delta t}} = {x_t} + {v_{t + \Delta t}}\Delta t $$ $$ \tau = I\alpha $$ $$ {\omega _{t + \Delta t}} = {\omega _t} + I_{world}^{ - 1}\tau \Delta t $$ $$ {\theta _{t + \Delta t}} = {\theta _t} + {\omega _{t + \Delta t}}\Delta t $$ $$ I_{world}^{ - 1} = RI_{body}^{ - 1}{R^T} $$

Impulse:

$$ \begin{align} {r_A} = P - {x_A},{r_B} = P - {x_B}\\ {v_{P,A}} = {v_A} + {\omega _A} \times {r_A},{v_{P,B}} = {v_B} + {\omega _B} \times {r_B}\\ {v_{rel}} = {v_{P,B}} - {v_{P,A}}\\ {v_n} = {v_{rel}} \cdot n\\ J = jn\\ v_A' = {v_A} - \frac{j}{{{m_A}}}n,v_B' = {v_B} + \frac{j}{{{m_B}}}n\\ \omega _A' = {\omega _A} - I_A^{ - 1}\left( {{r_A} \times jn} \right),\omega _B' = {\omega _B} + I_B^{ - 1}\left( {{r_B} \times jn} \right)\\ v_{P,A}' = v_A' + \omega _A' \times {r_A},v_{P,B}' = v_B' + \omega _B' \times {r_B}\\ v_{rel}' = v_{P,B}' - v_{P,A}'\\ v_{re{l^\prime }} \cdot n = - e\left( {{v_{rel}} \cdot n} \right)\\ j = - \frac{{\left( {1 + e} \right){v_{rel}} \cdot n}}{{\frac{1}{{{m_A}}} + \frac{1}{{{m_B}}} + n \cdot \left[ {\left( {I_A^{ - 1}\left( {{r_A} \times n} \right)} \right) \times {r_A} + \left( {I_B^{ - 1}\left( {{r_B} \times n} \right)} \right) \times {r_B}} \right]}} \end{align} $$

Positional Correction:

$$ \ Correction = percent \times \max \left( {penetration - slop,0} \right)/InvMassSum $$