Using the tangent vector, the angle of the tangent to the curve, say θ, is given by: sin ⁡ ⁡ θ θ = y ′ ( s ) x ′ ( s ) 2 + y ′ ( s ) 2 = y ′ ( s ) ; and cos ⁡ ⁡ θ θ = x ′ ( s ) x ′ ( s ) 2 + y ′ ( s ) 2 = x ′ ( s ) . The radius of curvature is introduced completely formally (without need for geometric interpretation) as: 1 ρ ρ = d θ θ d s . The derivative of θ can be found from that for sin θ: d sin ⁡ ⁡ θ θ d s = cos ⁡ ⁡ θ θ d θ θ d s = 1 ρ ρ cos ⁡ ⁡ θ θ = 1 ρ ρ x ′ ( s ) . Now: d sin ⁡ ⁡ θ θ d s = d d s y ′ ( s ) x ′ ( s ) 2 + y ′ ( s ) 2 = y ″ ( s ) x ′ ( s ) 2 − − y ′ ( s ) x ′ ( s ) x ″ ( s ) ( x ′ ( s ) 2 + y ′ ( s ) 2 ) 3 / 2 , in which the denominator is unity according to Eq. 1. With this formula for the derivative of the sine, the radius of curvature becomes: d θ θ d s = 1 ρ ρ = y ″ ( s ) x ′ ( s ) − − y ′ ( s ) x ″ ( s ) = y ″ ( s ) x ′ ( s ) = − − x ″ ( s ) y ′ ( s ) , where the equivalence of the forms stems from differentiation of Eq. 1 : x ′ ( s ) x ″ ( s ) + y ′ ( s ) y ″ ( s ) = 0 . Having set up the description of any position on the path in terms of its associated value for s, and having found the properties of the path in terms of this description, motion of the particle is introduced by stating the particle position at any time t as the corresponding value s (t).