Generalization of the Gluckstern formulas I: Higher orders, alternatives and exact results

The Gluckstern formulas for the covariance matrix of track slope and curvature are generalized to circular tracks, with both plane and cylindrical detector surfaces. The formulas are extended to higher-order expansions of the track model and to the exact circular track model. An improved second-order formula is derived. Optimal detector spacings are obtained for minimal determinant of the covariance matrix and minimal variances of slope and curvature. The user interface to the MATLAB implementation of all formulas is described.