The helix r(t)=< 2 cos(2t), 2 sin(2t), t >, 0 ≤ t ≤ 2π . Red: unit tangent vector, T(t) Green: unit normal vector, N(t) Blue: unit binormal vector, B(t) (Repeating in an endless loop.) The helix r(t)=< 2 cos(2t), 2 sin(2t), t >, 0 ≤ t ≤ 2π . Red: unit tangent vector, T(t) Green: unit normal vector, N(t) Blue: unit binormal vector, B(t) Yellow: Osculating Circle (Repeating in an endless loop.) The elliptical helix r(t)=< 4 cos(2t), 2 sin(2t), t >, 0 ≤ t ≤ 2π . Red: unit tangent vector, T(t) Green: unit normal vector, N(t) Blue: unit binormal vector, B(t) Yellow: Osculating Circle (Repeating in an endless loop.) The "tornado" r(t)=< 2t cos(10πt), 2t sin(10πt), t >, 0 ≤ t ≤ 1 . Yellow: Osculating Circle (Repeating in an endless loop.) The "spiral" r(t)=< e-t cos(t), e-t sin(t), e-t >, 0 ≤ t ≤ 2π . Yellow: Osculating Circle Question: why does the motion slow down so much as the particle approaches the bottom? Will it ever reach the bottom? (Repeating in an endless loop.) The "twisted cubic" r(t)=< t, t2, t3 >, -1 ≤ t ≤ 1; . Red: unit tangent vector, T(t) Green: unit normal vector, N(t) Blue: unit binormal vector, B(t) Yellow: Osculating Circle (Repeating in an endless loop.)