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A method for diagnosing the physical properties of a time-varying ellipse is presented. This essentially involves extending the notion of instantaneous frequency to the bivariate case. New complications, and possibilities, arise from the fact that there are several meaningful forms in which a time-varying ellipse may be represented. A perturbation analysis valid for the near-circular case clarifies these issues. Diagnosis of the ellipse properties may then be performed using wavelet ridge analysis, and slowly-varying changes in the ellipse structure may be decoupled from the fast orbital motion through the use of elliptic integrals, without the need for additional explicit filtering. The theory is presented in parallel with an application to a position time series of a drifting subsurface float trapped in an oceanic eddy.