clothoid

Clothoid, also known as the Euler spiral or Cornu spiral, is a plane curve whose curvature varies linearly with arc length. If the curve is parameterized by arc length s, its curvature is κ(s) = s / a^2, where a is a constant with dimensions of length. The tangent angle θ(s) satisfies dθ/ds = κ(s), hence θ(s) = s^2/(2 a^2). The curve can be described by Cartesian coordinates x(s) and y(s) obtained by integrating the unit tangent: x(s) = ∫0^s cos(θ(u)) du and y(s) = ∫0^s sin(θ(u)) du. Equivalently, in closed form via Fresnel integrals: x(s) = a√(π/2) C(s/(√π a)), y(s) = a√(π/2) S(s/(√π a)), where C and S are the Fresnel integrals.

Properties include a starting straight segment (zero curvature at s = 0), a smoothly increasing curvature with arc length, and a self-similar scaling: scaling the curve by a factor λ is equivalent to changing a to λ a. The clothoid approaches a straight line as s→0 and gradually bends into tighter curves as s grows, making it ideal as a transition curve between straight paths and circular arcs.

Applications are common in civil engineering for road and rail transitions, where gradual curvature minimizes jerk, as well as in roller coasters, track design, computer graphics, and robotics path planning.