The growth curve fitting method has been initiated and refined in the successive versions of Reference Catalogues (de Vaucouleurs & de Vaucouleurs, 1964 = RC1, de Vaucouleurs et al. 1976 = RC2; de Vaucouleurs et al. 1991 RC3). In this line of work, the growth curves (i.e. fractional flux vs normalized aperture) are chosen dependent on the morphological type of the galaxy, i.e. a different growth curve is adopted for each morphological type. The net of growth curves was adopted by averaging the growth curves obtained for small sets of template galaxies of each morphological type.
In the RC3, when the photometric data were numerous and good enough the fit was performed with the different growth curves and the best fit adopted for the determination of the effective radius and total magnitude. Hence, this determined a "photometric'' type (see Buta et al. 1995), by definition correlated with the morphological type (because the growth curves were constructed by averaging the observed growth curves of template galaxies sampling the range of morphological types). However, the correlation was not good enough for the photometric type being considered as a measurement of the morphological type.
Because the present database includes a large amount of aperture photometry derived from CCD observations (characterized by a very high internal consistency and well sampled growth curves), it will be possible to generalize the determination of the photometric type.
The growth curves are
expressed as functions of the photometric type 
,
and of the normalised radius of the circular aperture
where A is the aperture diameter
and, 
the diameter of the effective aperture, in tenths of an arcmin.
They are defined as:
where m(x) is the integrated magnitude within the aperture x, and,
is the asymptotic (or total) magnitude.
The adopted nets of growth curves are described in Sect. 4. The photometric type is defined to match the morphological type coded as in RC2 by setting two conditions: (1) g(-5,x) corresponds to the de Vaucouleurs law and (2) g(10,x) to the exponential law.
Considering a set of measured magnitudes, mi,
the fit, by varying and , consists in minimizing the quantity:
where 
, the residual (in mag) for the measurement i, is:
, being determined from:
The correction term ci is:
The different components of ci are defined as follows:
relations and is detailed in Sect. 3.2
and, in Eq. (3 (click here)), wi, the weight attached to measurement i, is:
where,
is a weight globally affected to all the measurements (in all the colours)
from reference r.
accounts for the density of the sampling of reference r.
is a clipping function aimed at rejecting the discrepant measurements.
reduces the weight of the small apertures.
For each galaxy,
the fit was performed using a downhill method,
starting from guesses of 
and . The guess for
was an average of the measured apertures,
and for it was the morphological type (hereafter 
) taken
from LEDA.
When was not available in LEDA, we assumed
for galaxies noted as "Compact" and 
when classified
as "Diffuse". For
300 remaining galaxies without any structure indication reported in LEDA, we
arbitrarily assumed initial 
, but determined the photometric
parameters only if the aperture data were sufficient to fit .
The morphological type of a galaxy in LEDA is the weighted average of the
different estimations available in the database. According to
Naim et al. (1995), the dispersion around such determination is
typically 1.8.
Because the growth curves are monotonic,
the choice of the starting values only changes the rapidity of the convergence
of the fit.
We fitted the photometric type () if at least 10 apertures were
available and if the range in exceeded 0.7. Otherwise,
we adopted the morphological type for the determination of
the other parameters, but these galaxies were not included in the figures
or in the analysis of the residuals.
The fitting procedure is automatic. When the routine does not
converge to stable values we only used the upper part of the growth curve to
get the total magnitude. At each step
cc(i), cr0(i), wc(i), and
were recomputed, and wr0(i) was progressively rised to 1
as cr0 was refined.
We initially started from equal weights 
and wr0,
and no crc corrections
for all the references, and we computed these weights and
corrections by analysing the residuals from the fit for all the
galaxies of our sample.
The determinations of the different parameters, correction
terms and their weights converged to stable values after about 10 iterations.
As new references were incorporated in the database, they were
given initial low weights afterwards set to their stable values.
The growth curve fitting is done in the B-band, and the measurements
done in other bandpasses are converted to the B-band by fitting a
linear relation between the observed colour indices, coc, and 
:
where 
and 
are the two parameters fitted for the colour c
by minimizing:
The colour index entering Eq. (5 (click here)) is thus:
This fit provides with the colours reduced at the effective aperture and the colour gradients.
When the direct calibration in the B-band was not possible, for example, if
a galaxy was only observed in the R-band, the conversion toward the
B-band was done using the mean colour index corresponding to the
adopted value of . These were determined
a posteriori, see Sect. 5 (click here).
The reduction to the B-band was recomputed at each iteration of the
growth curve fitting to account for the detection of discrepant
measurements (through the clipping function 
).
An important characteristic of our database is the mixing of photoelectric
and CCD photometry. While the latter has a very high internal
consistency (
against 
for photoelectric
photometry) its external precision (error on the zero-point) is not better,
and often worse than that of the former.
To account for this characteristic, we computed a correction, cr0 of the zero-point of each reference during the fit of the growth curve.
cr0 is evaluated as the difference between
the mean residual for reference r and for all the others.
This comparison is restricted to the range in 
where measurements from
more than one reference are available.
The correction is adopted if it is computed from at least 3 measurements
and if it is decided statistically significant.
The weight wr0 is determined from the distribution of the cr0 for all the sample.
The adopted wr0 are listed in Table 3.
) associated
with each reference
We analysed the for all the sample,
searching for systematic effects associated
with the reference.
We determined colour-dependent corrections to individual references, crc, whenever a significant systematic residual was found.
The weights, , are computed from the rms dispersion of the
residuals for reference r.
These corrections and weights were computed after
several iterations over the whole sample, until stability was reached.
The corresponding and crc are listed in Table 3.
)
A major source of uncertainty in the photometric measurements results
from the error in determining the sky background to be subtracted. In
general, each photoelectric measurement of an object is accompanied with
a sky measurement, hence, even in a series of observations (multi-aperture),
all the observed magnitudes are independent. At the contrary, the
surface photometry proceeds to a single (and likely more precise)
sky determination. Consequently, the magnitudes measured through the
different apertures are not completely independent. Hence, a finer sampling
of the aperture range would unduely increase the effective weight
of the corresponding reference. To tackle this effect, the weight
is decreased by the factor 
,
where 
is the average number of aperture measured for each
galaxy in reference r, if 
.
)
After each downhill step of the growth curve fit, the residuals are
searched for the detection of discrepant measurements. Data farther
than 
are clipped, i.e. 
, and to avoid instability,
is reduced for data in the range 
. For other
measurements, 
.
)
On the one hand, the small apertures are affected by seeing effect and, for
the photoelectric photometry, by centering errors.
On the other hand, the large apertures are affected by sky subtraction errors.
To take into account these two effects, the analysis of the residuals for the
whole sample lead us to compute a weight 
from the variation of
the rms residuals with 
.
A different system of weighting was adopted for the preparation of the RC3 (Buta et al. 1995). It is function of the reference but also of the telescope aperture and measured magnitude. We did not perform a detailed comparison with our weighting function, but qualitatively, the zero-point corrections agree and both approaches show the highest quality of the most recent observations.
The residuals for each references have been analysed separately. In a couple of cases, this lead to corrections which are not included in Eq. (5 (click here)).
For example, the reference 
(see Table 2) is better corrected
by subtracting 0.03 to than changing the magnitude zero-point.
It may be due to an error on the telescope scale, or to the use of an
octogonal diaphram. Apertures were also modified in references
and 
. For the reference
(CCD in R), we individually
recalibrated the zero-point when it was possible.
The determination of the errors is derived from the distribution of the residuals for a considered galaxy:
where N is the number of observations and f the number degrees
of freedom: f = 3 +h , (if was fitted) where h is the number of
references corrected for zero-point.
Hence, the errors on the three parameters were determined by estimating
the variations in , and associated with e1,
respectively e1, e2 and e3.
Because of the non-linearity of the fit, the errors on each parameter
are not independent, i.e. the error box is not an ellipsoid with
axes parallels to the axes of the parameter space. In particular,
the correlation of the errors on and
is well documented,
see e.g. Hamabe & Kormendy (1987). Thus, we computed the errors,
e4 and e5, on
resulting from e2 and e3.
The total error on is:
The correlation between the internal errors on and on
is shown in Fig. 1 (click here).
Figure 1: Correlation of the internal errors on and 
.
In abscissae: The error on (expressed in tenth of arcmin).
In ordinates: The error on (in mag).
The straight line is the regression 
