fv_base_point_tangent

t3toolbox.backend.geometry.fv_base_point_tangent(frame)

def fv_base_point_tangent(
        frame:  typ.Tuple,    # (U, O, P, Q) -- an orthogonal frame
) -> typ.Tuple:               # v_X: the attachment point X as a gauged tangent (tucker_var, tt_var)

The attachment point X = (U, P) as a gauged tangent v_X – the DIRECT construction: all variations zero except the last TT variation, set to the frame’s last left core P_last. This is already gauged (the last TT variation is the one slot with no gauge condition), so it needs neither an ambient projection nor a gauge projection: dense(v_X) = X and ‖v_X‖_coord = ‖X‖_HS exactly. (It equals tv_project_t3_onto_tangent_space(frame, (U, P)) – verified – but avoids that roundabout computation, whose environment contractions all collapse since the projected T3 IS the frame’s own cores. See docs/contributor/fitting_internals.md §Regularization.)

Parameters:

frame (t3toolbox.backend.common.typ.Tuple)

Return type:

t3toolbox.backend.common.typ.Tuple