Rope/Chain Physics

The core of the rope simulation uses Verlet integration to update each segment's position. Unlike traditional Euler integration which tracks velocity explicitly, Verlet derives motion from the difference between the current and previous positions. This makes the simulation inherently more stable and resistant to energy accumulation errors.

Each frame, every non-fixed segment computes its new position using the formula:

newPos = 2 * currentPos - previousPos + acceleration * dt^2

The previous position is then stored for the next frame. This approach naturally handles damping and produces smooth, physically plausible motion without needing to explicitly manage velocity vectors.

Inputs:

Delta Time: The time step for the current frame.
Position: The segment's current world position.
Previous Position: Where the segment was last frame.
Acceleration: External forces like gravity acting on the segment.

Output:

Updated position and previous position for each rope segment, producing continuous, stable motion.

After Verlet integration moves each segment freely, the rope's structural integrity is restored through the Jakobsen method. This technique iterates over all adjacent segment pairs multiple times, pushing or pulling them to maintain a fixed rest length between each connection.

The number of iterations directly controls accuracy versus performance. More iterations produce a stiffer, more accurate rope, while fewer iterations allow more elasticity and stretch. Even a small number of passes (3-5) produces convincing results for most gameplay scenarios.

Purpose:

• Enforce fixed distances between connected segments.
• Prevent the rope from stretching or compressing unnaturally.
• Converge toward physically correct positions through repeated relaxation passes.

The RelaxConstraint function handles the actual correction between two connected segments. It calculates the current distance between them, compares it to the desired rest length, and shifts both segments equally along the connecting axis to satisfy the constraint.

Fixed endpoints (like attachment points or anchors) are skipped during correction, ensuring the rope stays connected to its anchor while the rest of the chain adjusts naturally. This is what allows one end to be pinned to a wall or a player's hand while the rest swings freely.

Inputs:

Segment A & B: Two adjacent rope segments to constrain.
Rest Length: The target distance between the segments.
Rope Part Type: Whether a segment is fixed (anchor/end) or free.

Output:

Corrected positions for both segments, maintaining the rope's structural length without breaking connectivity.