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During 20 years the analytical solution for determining feasible regions of self-modeling curve resolution (SMCR) method was only slightly researched. After publishing of the famous papers of Borgen et al. in 1985 and1986, pure generalizations have not been given and even investigated. The reason might have been that the published algorithms and descriptions have been hard to interpret (to code) and to understand respectively. In this paper, several theoretical reasoning is revised giving clearer and more general proofs for principles of SMCR, that is the normalized responses are embraced in the (N À 1)-dimensional simplex with the vertices being the N-normalized pure profiles. For the first time in the chemometric literature, computational geometry tools are introduced instead of, for example, linear programming tools, for developing our algorithm to draw Borgen plots of any three-component systems. Numbers of illustrations are given for the sake of clarity. As it has turned out, the highly cited and used data matrix published by Lawton and Sylwestre has not two but more components as it is presented in detail. For a simulated three-component system, Borgen plots are drawn and some explanations are given.

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Self modeling curve resolution (SMCR) was introduced by Lawton and Sylvestre (LS) [Technometrics 1971; 13: 617-633] to decompose raw spectroscopic data of two component systems into product of two physically interpretable profile matrices provided that both concentrations and absorbances are non-negative, accepting both as minimal constraints. Later Borgen et al. in 1985-86 [Anal. Chim. Acta 1985 174: 1-26; Microchim. Acta 1986; 11: 63-73] generalized LS method for three-component systems with the same minimal constraints. The concepts of Borgen were rather difficult to understand and to implement, that is why several chemometricians turned to developing approximation methods. Very recently, Rajkó and István [J. Chemom. 2005; 19: 448-463] have revisited Borgen's method and they have given clearer interpretation and used computational geometry tools to find inner and outer polygons. In the meantime Henry [Chemom. Intell. Lab. Syst. 2005; 77: 59-63] has introduced the duality relationship, but he has described it only for multivariate receptor modeling of compositional data of airborne pollution. Generalization of his duality principle will be given in this paper for universally using it in SMCR which is based on singular value decomposition (SVD) or principal component analysis (PCA). Duality in SMCR 165 ≥ = J N×1 J N×1 N×N J

Lawton and Sylvestre, and later Borgen et al. provided first the analytical solution for determining feasible regions of self-modeling curve resolution (SMCR) method for two-and three-component systems, respectively. After 20 years, Rajkó and Istvá n recently revitalized Borgen's method given a clear interpretation and algorithm how to draw Borgen plots using computer geometry tools; later Rajkó proved the existence of the natural duality in minimal constrained SMCR. In both latter cases, 1-norm was used to normalize raw data; however Borgen et al. introduced a more general class of normalization. In this paper, the definition and detailed descriptions of Borgen norms are given firstly appearing in the chemical literature. Some theoretical and practical studies on the adaptability of some Borgen norms used for SMCR method are also provided.

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