The Behavior of Normality when Iteratively Finding the Normal to a Line in an lp Geometry

The normal direction to the normal direction to a line in Minkowski geometries generally does not give the original line. We show that in lp geometries with repeatedly finding the normal line through the origin gives sequences of lines that monotonically approach specific lines of symmetry of the unit circle. Which lines of symmetry that are approached depends upon the value of p and the slope of the initial line. 1 p 


Introduction
Minkowski geometries are completely characterized by their unit circle, which is centrally symmetric about the origin and convex [1: p. 17].The spaces are homogeneous (all points are the same) and generally anisotropic (the yard stick for distance is not the same in all directions).Our principal interest is the planar Minkowski l p geometries with .Their unit circles are 1 p  1 The exponent p must be at least 1 in order for the unit circle to be convex.Convexity is required for the triangle inequality [1: pp. 22,23].If , this is Euclidean geometry.If , the circle is not strictly convex.As discussed in Section 2, since in Minkowski geometries a necessary and sufficient condition for uniqueness of normal directions to lines is that the unit circle be strictly convex, we do not consider the case.Convex unit circles are strictly convex if they contain no line segments.The l 1 geometry is well studied and is sometimes called taxicab, Manhattan, or city-block geometry [2].Since the unit circle for the limiting case is the square with vertices , we do not consider that geometry, as well.Figure 1 shows some l p unit circles and the circle with , which does not produce a Minkowski geometry since the circle is not convex.the Euclidean distance between P 1 and P 2 and the unit of measurement or scale in the direction of L. There are many applications of l p geometries.The shape is called a Lamé curve after some work by Gabriel Lamé.Ruane and Swartzlander [5] considered apertures for light with shape (2) with , which give a larger area than for their constraints.Piet Hein designed a large traffic island for Stockholm, Sweden using (2) with and , saying that it gives a smooth traffic flow.He called the curves (2) with superellipses.The shape (2) has been extensively used for furniture design and elsewhere [6:  In the next section, we define normality in Minkowski geometries.Since the normal line to the normal line of a line is usually not the original line, in Section 3 we determine the behavior of the lines obtained by successively finding normal lines of normal lines.The limiting behavior is in Theorems 3.4 and 3.5.In Section 4, we create a circle, called a Radon curve, using portions of two l p geometries' unit circles, for which the normal to the normal of any line is the original line, which is called reflexivity of normality.

Definition of Normality
There are two equivalent, intuitive ways to define normality in Minkowski geometries with smooth unit circles.One is that line L 2 is normal to the given line L 1 with L 2 meeting L 1 at point Q if for every point P on line L 2 , the distance from P to Q is the minimum of all distances from P to any point on L 1 [1: p. 78, 3: p. 228].
For easier expression, we give the other definition in terms of unit vectors.It says that a unit vector is normal to a second unit vector if the first vector contains the origin and a point where the slope of the unit circle is the same as the slope of the second vector [1: p. 125, 7: p. 145].An application of this definition is illustrated in Figure 2, where 6 p  .We use the second definition, since in practice finding the tangent lines to (1) is easier than minimizing a distance.
In l p geometries, the axes and are mutually normal lines, as are and .However, in general, the normal line to the normal line of a line is not the original line.
In any Minkowski geometry, the unit circle is strictly convex if and only if normality is unique [1: p. 257, 3: p. 232].If the unit circle contains a line segment S, then normality is not unique for any line parallel to that segment.Take such a line L through the origin.Any line through the origin and intersecting S is normal to L, since the distance from the origin to the segment is one for all the normal lines.Hence, we do not consider l 1 or l  geometries.

Repeatedly Finding Normal Lines
The purpose of this section is to explore the behavior of the lines found by repeatedly finding normal lines in l p geometries.The origin O is placed at the point on the initial line where the normal is found.
Lemma 3.1 Consider l p geometry with .For , the slope of the normal line to is For 0 m  , the slope of the normal line to y mx  is Proof.For , we find the point of tangency to the unit circle, where the tangent is parallel to and for odd n, For even n, and for odd n, These formulas can be appropriately altered for 0 0 m  .Proof.To obtain (5), for even n, using (3), and using ( 4) and ( 9), and also the main induction step to give (7).Similarly, obtain ( 6) and (8).□ Lemma 3.3 Consider l p geometry with .If where the second subscript indicates the identity of the line.
Proof.Take m 0,1 and m 0,2 to be positive.The proof for negative initial slopes is similar.For even n, using (7) for line 1, (10), and then (7) for line 2 give The proof of (11) for odd n uses ( 8) and ( 10) in a similar manner.□ Because of the symmetries of the l p unit circle about the axes, only 0 need be considered.The condition between the slopes of two initial lines means that the lines have the same angle with the respective axes.Lemma 3.3 shows the symmetries about y x   in the behavior of the iterated normal lines, so only initial slopes between 0 and 1 need to be considered.for even n has values and monotonically approaches 1, and the subsequence for odd n has values and monotonically approaches -1.
give the behavior of the iterated normal lines for .□ As an example of Theorem 3.4, Table 1 contains the slopes of the first eight iterated normal lines for

A Geometry with Reflexive Normality
Although our focus is on l p geometries with , portions of the unit circles (1) for different values of p can be joined to obtain interesting geometries.Theorem 4.1 shows how to make normality reflexive for all lines, that is, the normal to the normal of a line is the initial line.Reflexivity is sometimes called symmetry. 1 p  Theorem 4.1 Given the portion of the l p unit circle that is in the first and third quadrants, the only way to complete a unit circle in the second and fourth quadrants for a Minkowski geometry with reflexive normality is with the portions of the l q unit circle in the second and fourth quadrants for 1  1 p q 1   .Proof.Since Minkowski unit circles are symmetric about their centers, we can reference only the first and second quadrants.Take the center to be the origin, and construct all normal lines at the origin.In the first quadrant, the unit circle is .In the second quadrant, the unit circle is . The original line L 1 is , , which intersects at the point 1 1 1 .The construction is illustrated in Figure 7 for p = 4.For reflexivity, demand that the slope of line L 1 equals the slope of the tangent line L 3 at the point Then, Equating the slopes of the lines L 1 and L 3 gives  Schäffer's theorem says that dual unit circles have the same circumferences, when the circumferences are measured with their own distance functions [1: pp. 111-118, 7: p. 153, 8,9].Because of the symmetry of the unit circle in Theorem 4.1, it has the same circumference as the dual l p and l q unit circles whose arcs compose it.
Radon curves are equivalently defined as either unit circles for which normality is reflexive for all lines or unit circles that have arcs of dual circles in alternating quadrants as in Theorem 4.

3 . 5
that the entries in the table's two columns are inverses, since the values of the m 0 s are inverses.The normal lines monotonically approach the lines 5 m  y x   , as shown by the arrows in their graphs in Figures4 and 5.Theorem Consider l p geometry with 1 even n has values and monotonically approaches 0, and the subsequence for odd n has values and monotonically approaches -∞.For the initial line 0 with , the subsequence of for even n has values and monotonically approaches , and the subsequence for odd n has values and monotonically approaches 0.

Table 2 .
Figure 6.The lines L 4 , which is to be orthogonal to line L 1 .The goal is to find the function   g x .The slope of L 2 is found by taking the derivative of

Figure 7 . 4 .
Figure 7.The unit circle in this Minkowski geometry is 4 4 + = x y 1 in quadrants 1 and 3 and the expressions for t in (12) and (14) and dropping the subscript 2 give the differential equation unit circles and dual spaces are central to Minkowski geometry[1,3,7].
The unit of measurement is the Euclidean distance from the O to point Q where L intersects the unit circle[3: p. 225, 4].Equivalently, translate the axes so that the origin is at P 1