## 1 Introduction
Amendment of this circumstance has encouraged routing further away from [PERSON]'s perception on covariation among different traits, in order to conceive allometry as centered on covariation between size and shape [73; 74]. Alongside this it is necessary to consider of multiple-parameter complex allometry (MCA). Related formulations can admit all sorts of nonlinear or discontinuous relationships intended to be fitted in arithmetical scales by means of DNLR protocols [75; 76; 77; 78]. However, opposing MCA-DNLR slants, defenders of a TAMA approach state that as conceived in the original theoretical standpoint of allometry, a logarithmic transformation is deemed necessary in the analysis [79; 80; 81; 82; 83; 84; 85; 86; 87]. Thus, embracing MCA-DNLR protocols feeds one of the most central discrepancies among schools of allometric examination. Furthermore, from a traditional stance MCA-DNLR schemes sacrifice appreciation of biological theory in order to privilege statistical correctness [32; 59; 87]. Besides, a DNLR approach could stand unreliable results, for instance, an inadequate consideration of intrinsic error structure can lead to substantial bias [88]. In addition, largest values of covariate can be influential of parameter estimates [18]. From a practical point of view depending on the complexity of MCA to be fitted in direct scales, there could be issues related to initial parameter estimates as well as convergence of non-linear regression algorithms. Therefore, there are also caveats in efficiency of allometric projection of plant biomass units derived from MCA-DNLR arrangements. Then, defenders of traditional allometry assert that overcoming MCA-DNLR inconveniences could be achieved by embracing suitable complexity in geometrical space. [PERSON] marked a breakpoint in the log-log plot of chela mass vs. body mass of fiddler crabs (_Uca pugnax_). This was endorsed by an abrupt change in relative growth of the chela approximately when crabs reach sexual maturity [34; 89; 90]. Admittance of a break point for transition to succeeding growth phases readily adapts a notion of curvature in geometrical space. This paradigm is also referred as non-loglinear allometry in [PERSON]'s original interpretation [70; 91; 92]. Extension of [PERSON]'s break point idea allows consideration of polyphasic loglinear allometry (PLA) [91; 93; 94; 95]. This approach characterizes heterogeneity of the response in geometrical space by composing the range of covariate into sectors separated by break points. Each tract associates to a linear sub model. Therefore, as opposed to examination through MCA-DNLR schemes, endorsing PLA break-point borne curvature offers a way to add complexity while maintaining the essence of traditional allometry. However, curvature as comprehended in the traditional allometric perspective is also controversial. For instance, its manifestation has been associated to distortion produced by the use of a logarithmic transformation itself [64; 96]. However, [PERSON] et al. [88] emphasizes that the occurrence of curvature has nothing to do with the use of logarithmic transformations since deviations from perfect log-linear allometry can be explained on grounds of methodological factors related to data. In conceiving the aims of the present research we abided by this perspective and offer methods that in our view allow efficient identification of MCA patterns through logtransformation methods. Nevertheless, as we shall explain ahead, present methods could also embrace efficient identification of MCA forms in direct scales. Following [PERSON] et al. [77] and [PERSON] et al. [92], we can conceive MCA as a generalization of [PERSON]'s formula of simple allometry, namely,  
\[y=\beta(x)x^{\alpha(x)} \tag{2}\]  
with \(y\) and \(x\) belonging to domains \(Y\) and \(X\) one to one, both contained in \(R^{+}\), and where \(\beta(x)\) and \(\alpha(x)\) are real functions, with \(\beta(x)\) having a range in \(R^{+}\).