Parallel Line Segments Research Articles

On Outwardly Simple Line Families

In (5) Hammer and Sobczyk defined an outwardly simple line family in the plane as a family of lines in the plane having the property that each point outside some given circle lies on exactly one line of the family, and they characterized planar outwardly simple line families as follows (5): the extended diameters of a convex body, whose boundary has no pair of parallel line segments in it, form an outwardly simple line family; moreover, each outwardly simple line family is the family of extended diameters of a convex body having constant width.

Diameters of convex bodies

In [2], we have shown that the extended diameter family of a convex body C in n-space has the property of outward simplicity if and only if there is no pair of parallel line segments in the boundary of C respectively in an opposite pair of parallel contact planes of C. In [3] Professor Sobczyk and I show that every outwardly simple family of lines in the plane is the extended diameter family of a constant breadth convex body and incidentally give a simple construction for all constant breadth planar convex bodies. The purpose of this note is to show that not every continuous outwardly simple line family in Euclidean 3-space is the extended diameter family of a convex body. A diameter of a convex body in En (n > 2) is a longest chord in its direction. In [2] it is proved that the diameters of a convex body cover the body.

Formulas for the Geometric Mean Distances of Rectangular Areas and of Line Segments

Formulas are derived, by direct integration of the defining definite integrals, for the general geometric mean distance between (i) a point and a line segment; (ii) a point and a rectangular area; (iii) two parallel line segments; (iv) two orthogonal line segments; (v) a line segment and a parallel-sided rectangular area; (vi) two parallel-sided rectangular areas. Certain useful identities of the formula for the last-named case are presented.