Android Particle Simulation

• 2D Newtonian Particle Simulation
• Particle simulation that uses phone gyroscope and accelerometer to influence the particle.
• Written using OpenGL Graphics, OpenGL Mathematics and c++, using java as a wrapper.

Particle Integration and Collision Resolution

For each particle $i \in [0, N)$ with position $\mathbf{p}_i = (x_i, y_i)$ and velocity$\mathbf{v}_i = (x_i, y_i)$:

1. Spatial Cell Mapping

Each particle is mapped to a uniform grid cell:

$$ c_x = \left\lfloor \frac{x_i}{s} \right\rfloor, \quad c_y = \left\lfloor \frac{y_i}{s} \right\rfloor $$

where $( s )$ is the cell size.

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2. Velocity and Position Integration

Each frame (or substep), the velocity and position are updated as:

$$ \mathbf{v}_i \leftarrow (\mathbf{v}_i + \mathbf{g} + \mathbf{a}) , d $$

$$ \mathbf{p}_i \leftarrow \mathbf{p}_i + \mathbf{v}_i , \Delta t_{\text{sub}} $$

where:

• $( \mathbf{g} )$ = gravity vector
• $( \mathbf{a} )$ = external acceleration
• $( d )$ = damping factor
• $( \Delta t_{\text{sub}} )$ = substep time interval

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3. Neighbor Cell Search

For collision handling, each particle checks neighboring cells within offsets:

$$ (\Delta x, \Delta y) \in {-1, 0, 1}^2 $$

and for each valid neighbor cell $((n_x, n_y))$, all particles $( j )$ in that cell are considered.

We only process pairs with $( j > i )$ to avoid duplicates.

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4. Particle–Particle Collision

For a neighbor particle ( j ):

Compute relative displacement:

$$ \boldsymbol{\delta} = \mathbf{p}_i - \mathbf{p}_j $$

$$ d^2 = \boldsymbol{\delta} \cdot \boldsymbol{\delta} $$

If $( d^2 < (2r)^2 )$, a collision occurs (where $( r )$ is particle radius).

Then:

$$ d = \sqrt{d^2}, \quad \mathbf{n} = \frac{\boldsymbol{\delta}}{d} $$

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5. Positional Correction (Overlap Resolution)

Penetration depth:

$$ p = 2r - d $$

Apply equal and opposite corrections:

$$ \mathbf{p}_i \leftarrow \mathbf{p}_i + \frac{1}{2} p,\mathbf{n}, \quad \mathbf{p}_j \leftarrow \mathbf{p}_j - \frac{1}{2} p,\mathbf{n} $$

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6. Velocity Impulse (Elastic Collision)

Relative velocity along the collision normal:

$$ \mathbf{v}_{\text{rel}} = \mathbf{v}_i - \mathbf{v}_j $$

$$ v_{\text{sep}} = \mathbf{v}_{\text{rel}} \cdot \mathbf{n} $$

If $( v_{\text{sep}} < 0 )$ (particles moving toward each other):

Apply impulse with restitution coefficient $( e )$:

$$ \mathbf{J} = -e,v_{\text{sep}},\mathbf{n},\frac{1}{2} $$

Update velocities:

$$ \mathbf{v}_i \leftarrow \mathbf{v}_i + \mathbf{J} $$

$$ \mathbf{v}_j \leftarrow \mathbf{v}_j - \mathbf{J} $$

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Summary

This procedure models:

• Continuous gravity and damping integration
• Spatial partitioning for efficient neighbor lookup
• Overlap correction (position-based collision)
• Elastic response via velocity impulse

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Symbols:

Symbol	Meaning
$( \mathbf{p}_i )$	Position of particle ( i )
$( \mathbf{v}_i )$	Velocity of particle ( i )
$( \mathbf{g} )$	Gravity vector
$( \mathbf{a} )$	External acceleration
$( d )$	Damping factor
$( e )$	Coefficient of restitution
$( r )$	Particle radius
$( c )$	Particle assigned cell
$( \Delta t_{\text{sub}} )$	Substep timestep