The simplest measure would be a statistic that uses the residuals. Either sum-of-squared errors (SSQ) or mean squared error (MSE) could be used. The difficulty, I think, is that the residuals ought to be computed in the direction orthogonal to the curve. For fitting a circle, this means that the residuals are in the radial direction. For ellipses and spirals, the definitions for the residuals are in the comments.
A more complex statistic would be an adjusted R-squared statistics, which includes a "penalty term" for fitting more complex models. A circle has two parameters, an ellipse has four parameters, and the spiral has four.
