With this notation, the velocity becomes: v = z ˙ ˙ = d d t ( R e i θ θ [ t ] ) = R d d t ( e i θ θ [ t ] ) = R e i θ θ ( t ) d d t ( i θ θ [ t ] ) = i R θ θ ˙ ˙ ( t ) e i θ θ ( t ) = i ω ω R e i θ θ ( t ) = i ω ω z and the acceleration becomes: a = v ˙ ˙ = i ω ω ˙ ˙ z + i ω ω z ˙ ˙ = ( i ω ω ˙ ˙ − − ω ω 2 ) z = ( i ω ω ˙ ˙ − − ω ω 2 ) R e i θ θ ( t ) = − − ω ω 2 R e i θ θ ( t ) + ω ω ˙ ˙ e i π π 2 R e i θ θ ( t ) .