Simulating soft bodies like fabric requires managing the intricate connections between thousands of points. If you try simulating a piece of cloth using traditional numerical methods like Euler integration, errors accumulate incredibly fast, causing interconnected springs to jitter, build up energy, and eventually explode.

To solve this, we can rely on a different approach to calculate motion and maintain connections: Verlet integration paired with distance constraints.

Verlet Integration

Unlike standard Euler integration which stores position and velocity, Verlet integration infers velocity from the difference between the current and previous positions:

$$ x_{t+\Delta t} = 2x_t - x_{t-\Delta t} + a_t \Delta t^2 $$

This approach offers phenomenal numerical stability, essential for constraint-based physics systems like cloth where many interconnected springs might otherwise accumulate catastrophic energy errors.

Distance Constraints

The cloth is constructed as a grid of discrete point masses. Each adjacent pair is constrained to maintain a resting distance $d$. If the current distance $l$ diverges from $d$, the points are pushed along their axis by a corrective factor proportional to the difference:

$$ \Delta x = \frac{l - d}{l} \times 0.5 \times (x_2 - x_1) $$

By iterating this constraint solver multiple times per frame (relaxation), the grid globally stiffens into a cohesive fabric.

Try it Yourself!

Check out Experiment 032: Cloth Simulation to see this in action. Try dragging the fabric to tear constraints, and see how the remaining sections dynamically ripple as they fall.