engleski

Harmonic Evolutes of Timelike Ruled Surfaces in Minkowski Space

Let S be a ruled surface in 3-dimensional Minkowski space parameterized by x(u, v) = c(u) + ve(u), where c(u) is a base curve and e(u) a non- vanishing vector field along c which generates the rulings. Ruled surfaces in Minkowski space are classified with respect to the casual character of their rulings which can be either space-like, timelike or null (light- like). A time-like ruled surface inherits the pseudo-Riemannian metric of index 1 from the ambient space. It is generated in the following cases: when c is a space-like curve and e(u) a time-like field (then e′(u) is space- like) or vice-versa, when c is a time-like curve and e(u) a space-like field (with e ′ (u) either null or non-null). It is also generated when c′ (u), e(u) are both null. The last ruled surfaces are called the null-scrolls, or in the special case, the B-scrolls. In this presentation we investigate properties of harmonic evolutes of time-like ruled surfaces in Minkowski space. The harmonic evolute of a surface is the locus of points which are harmonic conjugates of a point of a surface with respect to its centers of curvature p1 and p2.

harmonic evolute ; b-scrolls

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