The curvature of the cardioid r = a(1 − cos(θ)) at the point (a, π/2) is:

The equation of the tangent line at the point parameter λ = 1 to the curve r = (1 + λ, -λ2, 1 + λ3) is:

If the tangential and normal accelerations of a particle describing a plane curve are constant throughout, then the radius of curvature ρ at any point is given by (where a and b are arbitrary constants, and t is time):