Tangent-Normal Coordinates

🧭 Motion in a Plane Using the Normal–Tangential Coordinate System

1. Introduction

The normal–tangential coordinate system describes particle motion using directions defined by the trajectory itself rather than a fixed Cartesian coordinate system. The tangential direction is always aligned with the particle's velocity, while the normal direction is perpendicular to the trajectory and is associated with the instantaneous centre of curvature. 

This coordinate system provides a natural decomposition of acceleration into two independent effects: the tangential component changes the magnitude of velocity, while the normal component changes its direction.

2. Animation Details

Coordinate System: Normal–Tangential (eₜ, eₙ)

Animation: Planar curvilinear motion

Trajectory: r(u)=3+1.2cos(2u), x(u) = r(u) cos(u), y(u) = r(u) sin(u)

Parameter–Time Relationship: u(t) = 0.1t + 0.01t²

Vectors Displayed: Velocity, Acceleration, eₜ, eₙ

Learning Objective: Visualize the decomposition of velocity and acceleration into normal–tangential components and understand how the local curvature of the trajectory governs normal acceleration.

3. Kinematic Equations

4. Key Observations

• The coordinate system moves with the particle and is defined by the local geometry of the trajectory.

• The tangential unit vector eₜ is always aligned with the instantaneous velocity.

• The normal unit vector eₙ is perpendicular to eₜ and is directed according to the local curvature of the trajectory.

• The tangential component of the acceleration at controls the rate of change of the magnitude of velocity.

• The normal component of the acceleration an controls the rate of change of the direction of velocity.

• The centre of curvature and osculating circle continuously change as the particle moves along the trajectory.

• At an inflection point, the curvature becomes zero κ = 0, and consequently an = 0.

• As the curvature changes sign through an inflection point, the normal direction changes accordingly.

5. Watch For

• Observe how eₜ continuously rotates as the particle follows the closed trajectory.

• Notice that eₙ remains perpendicular to eₜ, while its direction changes according to the local curvature.

• Observe the changing osculating circle and the corresponding movement of the centre of curvature C.

• Watch how the radius of curvature ρ changes along the trajectory.

• At the inflection points, observe the osculating circle moving away from the particle as ρ → ∞, while the normal acceleration approaches zero.

• Notice that after crossing an inflection point, the direction associated with the normal component reverses.

• Observe how the continuously increasing speed makes the normal acceleration aₙ = v²κ increasingly sensitive to the local curvature.

• Compare the moving normal–tangential frame with the fixed Cartesian coordinate system.