In this article, we study the so-called rectifying curves in an arbitrary dimensional Euclidean space. A curve is said to be a rectifying curve if, in all points of the curve, the orthogonal complement of its normal vector contains a fixed point. If this fixed point is chosen to be the origin, then this condition is equivalent to saying that the position vector of the curve in every point lies in the orthogonal complement of its normal vector. Here we characterize rectifying curves in the n -dimensional Euclidean space in different ways: using conditions on their curvatures, with an expression for the tangential component, the normal component, or the binormal components of their position vector, and by constructing them starting from an arclength parameterized curve on the unit hypersphere.

In the paper, three types of surfaces of revolution in the Galilean 3space are defined and studied. The construction of the well-known surface of revolution, defined as the trace of a planar curve rotated about an axis in the supporting plane of the curve, is given for the Galilean 3-space. Then we classify the surfaces of revolution with vanishing Gaussian curvature or vanishing mean curvature in the Galilean 3-space.

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In this work, we define twisted surfaces in Galilean 3-space. In order to construct these surfaces, a planar curve is subjected to two simultaneous rotations, possibly with different rotation speeds. The existence of Euclidean rotations and isotropic rotations leads to three distinct types of twisted surfaces in Galilean 3-space. Then we classify twisted surfaces in Galilean 3-space with zero Gaussian curvature or zero mean curvature.

We examine curvature properties of twisted surfaces with null rotation axis in Minkowski 3-space. That is, we study surfaces that arise when a planar curve is subject to two synchronized rotations, possibly at dierent speeds, one in its supporting plane and one of this supporting plane about an axis in the plane. Moreover, at least one of the two rotation axes is a null axis. As is clear from its construction, a twisted surface generalizes the concept of a surface of revolution.We classify at, constant Gaussian curvature, minimal and constant mean curvature twisted surfaces with a null rotation axis. Aside from pseudospheres, pseudohyperbolic spaces and cones, we encounter B-scrolls in these classications. The appearance of B-scrolls in these classications is of course the result of the rotation about a null axis. As for the cones in the classication of at twisted surfaces, introducing proper coordinates, we prove that they are determined by so-called Clelia curves. With a Clelia curve we mean a curve that has linear dependent spherical coordinates.

A new type of surfaces in 4-dimensional Euclidean and Lorentz-Minkowski space is constructed by performing two simultaneous rotations on a planar curve. In analogy with rotational surfaces, the resulting surfaces are called double rotational surfaces. Classification theorems of flat double rotational surfaces are proved. These classifications contain amongst other cones over 4-dimensional Clelia curves. As a side product these new kinds of curves in 4-space are defined.

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