Constraints (Joints)

Joints connect bodies with fixed relationships:

• Revolute (hinge): two bodies share an anchor; can rotate freely.
• Prismatic (slider): bodies translate along an axis.
• Distance (rope, spring): two anchors maintain distance (or spring force).
• Weld: rigid connection.
• Pulley, gear: complex coupled motion.

Solving constraints: keep bodies satisfying the constraint each step.

Iterative solver (sequential impulses, Gauss-Seidel style):

1. For each constraint, compute impulse to satisfy.
2. Apply to bodies.
3. Repeat for several iterations (8-15 typical).
4. Constraints settle.

Spring constraint (Hooke's law):

Damping: f = -c * relative_velocity. Prevents oscillation.

Rope:

• Distance constraint with max_length.
• Constraint only when stretched (slack rope = inactive).

Chains:

• Sequence of bodies + joints.
• Solver iterates from one end to other.
• Many iterations needed for long chains.

Cloth simulation:

• Grid of bodies + structural + shear + bending springs.
• Verlet integration + position-based dynamics.

Game examples:

• Vehicle: 4 wheels + body + 4 distance constraints (springs for suspension).
• Ragdoll: bones + revolute joints (limbs, spine).
• Door: revolute joint with motor + limit.

For our toy: distance + spring constraints. Skip rotation for simplicity.