Resonance as a nearest-point problem: the response parabola of the driven damped oscillator

The steady-state displacement amplitude of a sinusoidally driven damped oscillator is reciprocal to the magnitude of its dynamic stiffness. The reactive and dissipative stiffness components trace a parabola when sweeping the drive frequency. Displacement resonance is therefore a nearest-point problem: maximum displacement corresponds to the smallest origin-centered circle touching this parabola. The construction explains the shift of the displacement peak below the natural frequency and its disappearance at a damping ratio ζ = 1/ 2, before critical damping is reached. Velocity resonance gives a contrasting nearest-point problem on a straight line and remains at the natural frequency for every positive damping. Circle intersections in the companion plane give the exact absorbed-power bandwidth, while reciprocal frequency coordinates relate the displacement and acceleration peaks. The core treatment uses quadratic equations and elementary plane geometry and is accompanied by a suggested undergraduate teaching sequence. Optional extensions use curvature and inversion to obtain the amplitude as a function of phase lag and show that at critical damping the inverse parabola is a cardioid.