HARMONI is the first light visible and near-IR integral field spectrograph for the ELT. It covers a large spectral range from 450nm to 2450nm with resolving powers from R (≡λ/Δλ) 3500 to 18000 and spatial sampling from 60mas to 4mas. It can operate in two Adaptive Optics modes - SCAO (including a High Contrast capability) and LTAO - or with NOAO. The project is preparing for Final Design Reviews. HARMONI slices the input light beam in subfields and then into slitlets and rearranges them to obtain spectra on its detectors. The Data Reduction software (DRS) handles calibration and scientific raw data from HARMONI and computes a fully reduced and calibrated science data cube. The challenge is to develop robust methods suitable for each of the 44 scale/band combinations of HARMONI. The geometrical calibration, one of the steps of the DRS, determines the coordinate transformation from detector pixels to wavelength and relative spatial position in the input focal plane. This paper provides a mathematical description of the algorithms involved in the geometrical calibration and presents validations on mock data simulated with the HARMONI Instrument Numerical Model (HINM). Briefly, to cope with a possible overlap of slitlets, we locate the slitlets using a global fitting method on flat-field exposures. The wavelength solution is computed using arc exposures. To compute the geometrical transformation we choose to use specific masks illuminated with a white continuum lamp. A trace mask exposure provides the transformation along the slitlets. A pinhole mask exposure determines the transformation in the perpendicular direction by fitting the flux within each slitlet.
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