On the definition of tangents to sets of infinite linear measure

Given a plane set E of positive and finite linear measure, the tangent at a point x at which the upper density of E is positive (which is so at almost all points of E and is not so at almost all points outside E) is defined in the following way. The line l through x is said to be the tangent to the set at the point x if in any angle vertex x that leaves the line outside the density of E is zero. This definition when applied to sets of infinite linear measure leads often to the conclusion that no tangent exists in cases when the structure of the set singles out some lines that have strong claim to be tangents to the set.