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Swim along the tangent

A swimming school and a breaching dolphin

3D particle system

Each fish rides a phase-shifted ellipse, so the school does not collide. Heading is the path derivative, not a rigid spin of the whole group.

The body wave A u² sin(ks − ωt) grows toward the tail, which is how fish actually swim. The dolphin takes the inner lap and rises on a sin² breach.

Related surfaces