  This work presents an analytical solution for a general three-dimensional
track fit based on hit triplets. Input to the fit are triplet parameters, which
contain information about the triplet geometry (hit positions), the radiation
length of the material and the magnetic field. The general fit considers
spatial hit and multiple Coulomb scattering uncertainties, and can also be
extended to include energy losses.
  The output of the fit, which is given by an analytical closed-form solution,
contains the total momentum and the hit residuals, including the full
covariance matrix, thus allowing for easy software alignment of the detector.
The fit qualities are calculated for the global track fit as well as for the
local hit triplets. This feature allows filtering out triplets with poor fit
quality at an early stage of track reconstruction. The fit of local triplets is
fully parallelizable, enabling accelerated computation with parallel hardware
architectures.
  The triplet track fit is detector-independent, making it possible to use the
same fitting code for all tracking detectors. Only the detector-specific
triplet parameters (fit input) depend on the triplet geometries and the
magnetic field. Formulas for the calculation of the triplet parameters are
given for the most common tracking detector setups, namely a homogeneous
magnetic field and a spectrometer using a dipole magnet. Furthermore, an
algorithm is presented to calculate tracking parameters for an arbitrary
magnetic field configuration.
  Moreover, this work includes a discussion of track fit biases and presents an
extension of the fit to include energy losses. Last but not least, it is
proposed to use triplet-based scale parameters that characterize different
tracking regimes to accelerate track fits and to optimize the design of future
tracking detectors.
