A GEOMETRIC STUDY OF BERTRAND CURVES IN EUCLIDEAN 3-SPACE

It is a differential geometrical work on Bertrand curves in the Euclidean space, in three dimensions. The so-called Bertrand curves are regarded as special curves of space; the two curves have a mate curve of which the two curves are believed to coincide at the same principal normal vectors. It is devoted to the investigation of the most significant correspondence between curvature and torsion which defines the appearance and the labours of Bertrand curves. The article applies the Freenet-Serret model to describe the geometrical properties of curves, and to study what conditions to apply to obtain Bertrand curves. The results show that the dependence between curvature and torsion must be linear to have Bertrand property. Such curves are analysed under various analysis situations with the objective of establishing their stability, predictability and consistency of structure. The results reveal that the shapes of the Bertrand curves are much more regular and stable geometries than the underlying space curves, and may be used in geometric modelling, in computer graphics and in mechanical design. Moreover, the study concludes that parameters are to be controlled because despite even the smallest variations, the presence of Bertrand mate curves can be affected. Such is a piece, which has led to the study of the subject of differential geometry in giving a clear and detailed insight to the Bertrand curves in the classical view and in the modern-day analysis view of the object. New research in non-Euclidean geometries as well as applied mathematics is also a result of the research.