🧶 Rope & Chain Physics

It uses Verlet integration combined with Jakobsen constraint relaxation, a technique known as position-based dynamics (PBD). Each particle only stores its position and previous position; gravity and damping displace particles directly each frame, then a separate relaxation pass repeatedly nudges connected particles back toward their rest distance. This avoids solving stiff differential equations directly and is the same method used for cloth, hair, and rope in many real-time games.

This is the Störmer–Verlet form of integration: velocity is implicit, recovered as v ≈ x − xPrev rather than tracked as a separate quantity. It is numerically stable and, crucially, makes constraint solving simple, since the Jakobsen relaxation pass only has to move particle positions directly; the implied velocity for the next frame updates automatically from the new position and the position it just came from.

Each iteration is one pass over every stick constraint in the rope, and a single pass only partially corrects each link's length error. With few iterations the rope behaves like a soft elastic band that visibly stretches under load; with many iterations the repeated corrections converge closer to an inextensible rope, where every segment sits almost exactly at its rest length.

Jakobsen relaxation is an approximate iterative solver, not an exact one. With a finite number of iterations K and simultaneous pulls from gravity, neighbouring constraints, pinned points, and the peg, the rope rarely reaches a perfectly zero-error state within one frame. Raising K reduces stretch at the cost of more computation per frame, while lowering gravity, damping, or the peg's push has a similar effect.

The peg does not use full rigid-body collision. After each constraint iteration the simulation checks whether an unpinned particle now lies inside the peg's circle and, if so, projects it directly back onto the circle's edge. Repeating this alongside the stick constraints, K times per frame, makes the rope appear to wrap and slide around the peg convincingly, even though it is just a positional clamp rather than a physical response with impulses or friction.