The present paper deals with some characterizations of rectifying and osculating curves on a smooth surface respect to the reference frame {T?,N?,T?×N?}. We have computed components position vectors along T?,N?,T?×N? then investigated their invariancy under isometry surfaces, it is shown that they are invariant iff either normal curvature curve or vector in direction tangent curve.

This paper aims to design a generalized cylinder with geodesic base curve according the Darboux frame in Euclidean 3-space. A is special ruled surface that constructed by continuous fixed motion of generator line called ruling along given curve. The necessary and sufficient conditions for be are studied. main results show an osculating whose helical geodesic, rulings directed unit vector.

In this work, we give parallel transport frame of a curve and we introduce the relations between the frame and Frenet frame of the curve in 4-dimensional Euclidean space. The relation which is well known in Euclidean 3-space is generalized for the …rst time in 4-dimensional Euclidean space. Then we obtain the condition for spherical curves using the parallel transport frame of them. The conditi...

1. Linear series on curves 1.1. Terminology and notation 1.2. Morphisms from linear series; Castelnuovo’s genus bound 1.3. Linear series from morphisms 1.4. Relation between linear series and morphisms 1.5. Hermitian invariants; Weierstrass semigroups I 2. Weierstrass point theory 2.1. Hasse derivatives 2.2. Order sequence; Ramification divisor 2.3. D-Weierstrass points 2.4. D-osculating spaces...

A k–nucleus of a normal rational curve in PG(n, F ) is the intersection over all k–dimensional osculating subspaces of the curve (k ∈ {−1, 0, . . . , n− 1}). It is well known that for characteristic zero all nuclei are empty. In case of characteristic p > 0 and #F ≥ n the number of non–zero digits in the representation of n+ 1 in base p equals the number of distinct nuclei. An explicit formula ...