The determination of the detailed 3D structure of PNe encounters certain limitations in the methods described previously. On the one hand, hydrodynamical models, while producing self-consistent results, require numerous assumptions regarding the physical processes involved for which there are no direct observational constraints other than the observed morphologies. The assumptions can lead to degeneracy and uncertainties into the reconstructed 3D structure. On the other hand, kinematical modeling approaches, which make use of a limited number of high-resolution slit observations, often necessitate the use of idealized assumptions about the 3D structures which will be combined, such as cylinders, toruses, cones, and other geometric shapes. These idealizations may not fully capture the complex and diverse details of the morphologies of real PNe. Consequently, these inherent limitations should be carefully considered when interpreting the results of 3D structural studies of PNe.

For the case of PNe, there is ample evidence that, in general, the velocity vector at any given point is proportional to the position vector of that point in the nebula \citep[among many others]{Wilson1950,Weedman1968, 2004ASPC..313..148C, 2008MNRAS.385..269M, 2014MNRAS.442.3162U}. According to \cite{Zijlstra01}, this type of velocity distribution can be expected if the nebula has evolved from a relatively short mass-loss event and is now moving ballistically. Also, \cite{Steffen04} argue that a continuous interaction of a wind with small-scale structures can also develop a positive velocity gradient with distance. The result of this type of expansion is that the nebula conserves its shape over time. 

The assumption described above has been used in numerous studies aimed at reconstructing the spatial structure of planetary nebulae. In particular, \cite{Sabbadin06} focused extensively on this technique, successfully obtaining three-dimensional structures for a sample of objects. Another important effort is the work of \cite{Shape06}, in which they developed a computational tool to analyze and untangle the three-dimensional geometry and kinematic structure of gaseous nebulae. Subsequent studies, widely cited and still in use today, have applied this methodology to a diverse range of complex objects, recovering detailed information about their kinematics and structure.

Here we present preliminary results obtained by applying the same strategy to analyze data from high resolution scanning Fabry-Perot interferometer, SAM-FP, mounted on the SOAR telescope adaptive module for the PN NGC~3132. Observations were obtained in the [NII] filter (SAMI 6584-20) and H$\alpha$ filter (BTFI 6569-20) with a spectral resolution of about 11200 at H$\alpha$. The observations employed an e2v CCD detector with dimensions of 4096 × 4112 pixels, covering a total field of view of 180 arcseconds × 180 arcseconds and a pixel size of 0.18 arcseconds.

To obtain the final 3D density structure from this data we started from an initial guess of the proportionality constant used to translate the velocity field information to distance. With the inferred structure we start the iterative procedure to fit the photoionization model to the observational data constraints available. In this sense, the proportionality constant is taken as a free parameter of the modeling process. A visualization of the 3D density structure obtained is shown in Fig. \ref{fig:3D-dens}.

We start by adopting that, to first order, the $H\alpha$ emission intensity can be approximated by $I_{H\alpha} \propto n^2$, where n is the gas density. Using the velocity and position proportionality, we convert $H\alpha$ velocity datacubes into a three-dimensional density structure. With this density structure we start the iterative process of fitting the photoionization model to the observational constraints. Once a preliminary agreement is achieved, we further extend our analysis by establishing a relation between the [NII] 6584 line intensity and density finding that, for this particular case,  $I_{[NII]} \propto n^{2.5}$ which enabled us to convert [NII] 6584 velocity datacubes into a corresponding 3D density structure. These two density structures are then combined as the iterative process is continues until a reasonable fit to the observational constraints is found. It is important to note here that in the fitting process we found that adopting density fluctuations in the outer regions of the structure was necessary to better reproduce the fainter emission in those regions.

The ionizing source with its $T_{eff}$ and luminosity is one of the critical free parameters that is fit using the model. We started with the usual blackbody sources and quickly found that they were not able to reproduce the observations completely, specially when considering the gas temperature as measure through the diagnostic diagrams, always giving significantly hotter plasmas. Reproducing the gas temperature is critical to guarantee that the abundances found with the model fitting are indeed accurate. We then explored more detailed NLTE stellar model atmospheres provided by T. Rauch  \footnote{\url{http://astro.uni-tuebingen.de/~rauch/}} which gave better results but still lead to gas temperatures that were few hundred degrees too hot when compared to observational diagnostics.

At this point, the observations obtained by the JWST telescope became available and were analyzed by a group of researchers in \cite{2022NatAs...6.1421D}. One of the striking results presented was the detection of a dusty envelope on the central star of the nebula. The JWST data provided critical observational constraints including precise photometry for the central source. In \cite{2022NatAs...6.1421D} they found that the observations were consistent with a dust disk with an inner radius of 55 au, an outer radius of 140 au, and a dust mass of $2 \times 10^{-7}M_{\odot}$. In \cite{2023ApJ...943..110S}, using the same observations, the authors find that a $3.9 \times 10^{-8}M_{\odot}$, a radius of 1785 au, with a composition of 70\% silicate and 30\% amorphous carbon could explain the available photometry. However, it was already clear from the work in \cite{2022NatAs...6.1421D} that a dust only shell or disk was not capable of improving the photoionization models results, reducing the gas temperature obtained. These results motivated us to explore the influence of a dust and gas shell in our models.

We included a dust and gas spherical shell with uniform density surrounding our model ionizing source. To accomplish this with the available resolution of the model density grid, we first calculated a photoionization model for the central region, and then used the calculated spectral energy distribution (SED) as input ionization source for the PN photoionization model. 

After iterating with the free parameters of the introduced shell, we found that it allowed the model to reproduce the line diagnostic ratios, in particular for the temperature. A shell with a radius of 334 AU, $4.3 \times 10^{-5}M_{\odot}$ gas and $1.5 \times 10^{-10}M_{\odot}$ dust was found to produce the best fit to the observational constraints. We note that the fit was also sensitive to the mass fractions used in the shell abundances. The best fit corresponds to a He-poor, C and O-rich shell with mass fractions of $X_{He} = 5.0\times10^{-7}$, $X_{C} = 0.5$, $X_{N} = 1.0\times10^{-7}$, $X_{O} = 0.49$. Interestingly, these mass fractions are consistent with what is expected for the inner layers of a $3M_{\odot}$ progenitor star remnant.

The constraints used in the model fitting were spatially resolved VLT-MUSE observations obtained by \cite{Monreal-Ibero2020}, long-slit scans from \cite{Tsamis04}, and long-slit observations from \citep{Krabbe2005,Krabbe2006} and remeasured International Ultraviolet Explorer (IUE) large aperture (10.3 by 23 arcseconds) observations. Model integrated line intensities are presented and compared to observations in Table \ref{tab:mod-res}. To scale the UV to optical line data, we used the theoretical intensity ratio for HeII~$\lambda$1640/$\lambda$5411, measuring the HeII~$\lambda$5411 flux using the MUSE data on the same aperture configuration to ensure accurate scaling factors without introducing errors due to the ionization structure or aperture position. The IUE observations were crucial in constraining the C abundance. In Fig. \ref{fig:hb-modcomp} we show the comparison of the emission line map of $H_{\beta}$ obtained from the model and from MUSE data. A detailed description of the constraints and model results will be presented in a forthcoming paper.

The final abundances obtained by the model fit for the most important elements are presented in Table \ref{tab:abun-comp}. It is important to clarify here that model abundances are free parameters in the fitting process and are not adopted from the empirical method determinations. The results show that the He abundances in general agree, but significant differences in the C,N and O abundances are present, just as in the PNe described in previous sections. The largest discrepancy is found for the C abundance; however, inspecting the ratios of C/O and N/O obtained from the model, we see that again, the values are consistent with what is expected for a $3M_{\odot}$ star.