Example of force calculation for the barred galaxy NGC 4548. Panel A): the de-projected image of NGC 4548 in magnitude scale with a range of 17−25 μ_{3.6 μm}(AB). The blue dotted line indicates the visually estimated length and position angle of the bar. The light green ellipse corresponds to the bar radius from the isophotal fit (maximum ellipticity). The purple and light blue circles have radii equal to r_A2 and r_Qb, respectively. Panel B): image of the galaxy after subtracting the axisymmetric m = 0 component, together with contours tracing the force ratio F_T/ ⟨F_R⟩ (separated by 0.1 intervals). The dashed lines indicate the regions where the tangential forces change sign. The outer circle delimits a region of radius twice the size of the bar (the same as in Panel A)). Panel C): the F_T/ ⟨F_R⟩ force map (butterfly pattern) of the galaxy. The length and ellipticity of the bar are traced with black lines. The inner dotted circle corresponds to the bulge effective radii from P4 decompositions. Panel D): the thick solid curve indicates the normalized tangential force amplitude (Q_T in Eq. (6)) calculated from the force maps, using the nominal vertical scale height estimated from the radial scale length. The dashed lines correspond to the assumed upper (red) and lower bounds (blue) for the scale height. The thin green line corresponds to the bar-only force profile (see the text). Panel E): the normalized m = 2 Fourier density amplitude vs. radius. The dashed lines show higher order even Fourier amplitudes. Panel F): the phase of m = 2 Fourier amplitude vs. radius. Panel G): the solid line indicates the circular velocity curve calculated from the image (dashed lines correspond to the same upper and lower bounds as in panel D)). The horizontal dotted lines indicate the maxima of the observed H i velocity amplitude (green) and that of our calculated stellar component of the velocity curve. The solid orange line indicates the slope inferred from the linear term of the third degree polynomial fit of the inner part of the rotation curve (dashed orange line). The dotted vertical orange line delimits the region where this fit is made, which is taken between the galactic centre and the radius of the maximum rotation within one fourth of R_25.5. The dark green vertical dotted line corresponds to a radius of 2.2 times the disk scale length (radius of the maximum velocity of an exponential disk; Freeman 1970), while the vertical dashed line traces the bar length.