Multivariable Calculus: Arc Length and Curvature

Arc Length

L = ∫_a^b√((dx/dt)² + (dy/dt)² + (dz/dt)²) dt = ∫_a^b||r’(t)||dt

Arc Length Function

s(t) = ∫_a^t||r’(u)||du. Solving for t and reinserting it back into the original equation is known as reparameterizing

Curvature

Sharpness of a curve. κ = ||T’(t)|| / ||r’(t)|| = ||r’(t) x r’’(t)|| / ||r’(t)||³ = |f’’(x)| / (1 + (f’(x)²)^3/2.

Unit Normal Vector

N(t) = T’(t) / ||T’(t)||

Binormal Vector

B(t) = T(t) x N(t)

Normal Plane

Plane containing the normal and binormal vectors, with the unit tangent vector being normal to the plane.

Osculating Plane

Plane containing the normal and unit tangent vectors, with the binormal vector being normal to the plane.

Rectifying plane

Plane containing the unit tangent and binormal vectors, with the normal vector being normal to the plane.

Torsion

Twist of a curve. Given by 𝜏 = - dB/ds • N = - (B’(t) • N(t)) / ||r’(t)|| = ((r’(t) × r’’(t)) • r’’’(t)) / ||r’(t) × r’’(t)||²