The Behavior of Midsets When Repeatedly Taking the Midset of Two Lines in Geometry

We study the outcome of taking midsets of two lines in geometry. We establish the algorithm for repeatedly finding these midsets and characterize the limiting midsets. We discuss the issue of angle measurement in Minkowski geometries, especially with respect to the limiting midsets. 1. Introduction An important class of planar Minkowski geometries are those that are determined by the unit circles for [1, 2]. These are designated or simply geometries. The most important example is Euclidean geometry with . Our concern is with geometry, which is sometimes called taxicab, Manhattan, or city block geometry [3]. Our main interest is in the midsets of two lines in this geometry. The midsets of two points or of a point and a line are less nuanced than the midsets of two lines. The midset of two points, which is all points equidistant from the two points, can have different shapes, but generally it is a piecewise linear set of three segments [3, pages 8-9]. The midset of a point and a line is an parabola, which has the appearance of an approximation of an Euclidean parabola by line segments [3, page 39]. The midset of two lines is all points equidistant from the two lines. In Euclidean geometry, repeatedly taking midsets settles at the first or second step into a simple oscillation between pairs of orthogonal lines. The midset of the two intersecting and distinct lines and is the perpendicular lines and . Repeatedly finding midsets oscillates between these two orthogonal lines and a pair of lines rotated 45° from them. If the original lines are orthogonal, that is, , the oscillatory behavior begins at the first step. As seen in Section 2, in geometry, repeatedly finding midsets of two lines generally does not produce this oscillatory behavior, unless the original lines are orthogonal. Subsequently, “midset” indicates “ midset.” In Minkowski geometries, normality to a line is not reflexive; that is, the line perpendicular to the line that is perpendicular to the original line is not parallel to the original line, except in the case that the space’s unit circle is a Radon curve [2, pages 233-234], [4, pages 137-138], and [5, page 145]. Analogously, we show that in general taking midsets is not reflexive in geometry; that is, the midset of the midset of two lines is not the original lines. Without loss of generality, place the origin at the intersection of the original lines. In geometry the distance from a point to a line is measured either parallel to the -axis or parallel to the -axis. For , if , then the distance is measured parallel to the -axis; if , then