On the Plane Geometry with Generalized Absolute Value Metric

Abstract

Metric spaces are among the most important widely studied topics in mathematics. In recent years, Mathematicians began to investigate using other metrics different from Euclidean metric. These metrics also find their place computer age in addition to their importance in geometry. In this paper, we consider the plane geometry with the generalized absolute value metric and define trigonometric functions and norm and then give a plane tiling example for engineers underlying Schwarz's inequality in this plane.

1. Introduction

In the late 1800's, Minkowski [1] published a whole family of metrics providing new insight into the study of plane geometry. In recent years, mathematicians and engineers begin to investigate non-Euclidean metrics used in real-life applications. For example, Eisenberg and Khabbaz [2] discussed the applications of taxicab metric in geometry and network theory. Similarly, Burman et al. [3] utilized the concept of 𝜆-geometry and discussed its applications in the problem of global routing of multiterminal nets.

Our goal is to further develop the plane geometry with the generalized absolute value metric𝑑_𝑔 suggested by Kaya et al. [4], and to see how it can lead to future researches in related areas. In ℝ², the 𝑑_𝑔-distances between 𝑋=(𝑥₁,𝑦₁) and 𝑌=(𝑥₂,𝑦₂) are defined by𝑑_𝑔(𝑋,𝑌)=𝑑_𝑔((𝑥₁,𝑦₁),(𝑥₂,𝑦₂))=𝑎max{|𝑥₁−𝑥₂|,|𝑦₁−𝑦₂|}+𝑏min{|𝑥₁−𝑥₂|,|𝑦₁−𝑦₂|} (1.1)for all 𝑎,𝑏∈ℝ,𝑎≥𝑏≥0,𝑎≠0. According to the definition of 𝑑_𝑔-distances, the shortest way between two points 𝑋 and 𝑌 is the union of a vertical or a horizontal line segment and a line segment with the slope ±2𝑎𝑏/(𝑎²−𝑏²).

We will show that the 𝑑_𝑔-distance defined above is a metric. First, it is positive definite and symmetric, and it can be shown that 𝑑_𝑔 satisfies the triangle inequality as follows. For 𝑋=(𝑥₁,𝑦₁),𝑌=(𝑥₂,𝑦₂), and 𝑍=(𝑥₃,𝑦₃), since|𝑥₁−𝑥₂|=|𝑥₁−𝑥₃+𝑥₃−𝑥₂|≤|𝑥₁−𝑥₃|+|𝑥₃−𝑥₂|,|𝑦₁−𝑦₂|=|𝑦₁−𝑦₃+𝑦₃−𝑦₂|≤|𝑦₁−𝑦₃|+|𝑦₃−𝑦₂|, (1.2)one obtains𝑑_𝑔(𝑋,𝑌)=𝑎max{|𝑥₁−𝑥₂|,|𝑦₁−𝑦₂|}+𝑏min{|𝑥₁−𝑥₂|,|𝑦₁−𝑦₂|}=𝑎max{|𝑥₁−𝑥₃+𝑥₃−𝑥₂|,|𝑦₁−𝑦₃+𝑦₃−𝑦₂|}+𝑏min{|𝑥₁−𝑥₃+𝑥₃−𝑥₂|,|𝑦₁−𝑦₃+𝑦₃−𝑦₂|}≤𝑎max{|𝑥₁−𝑥₃|+|𝑥₃−𝑥₂|,|𝑦₁−𝑦₃|+|𝑦₃−𝑦₂|}+𝑏min{|𝑥₁−𝑥₃|+|𝑥₃−𝑥₂|,|𝑦₁−𝑦₃|+|𝑦₃−𝑦₂|}=𝑘. (1.3)

Let us examine the following four cases:

(i)|𝑥₁−𝑥₃|≥|𝑦₁−𝑦₃| and |𝑥₃−𝑥₂|≥|𝑦₃−𝑦₂|;(ii)|𝑥₁−𝑥₃|≥|𝑦₁−𝑦₃| and |𝑥₃−𝑥₂|≤|𝑦₃−𝑦₂|;(iii)|𝑥₁−𝑥₃|≤|𝑦₁−𝑦₃| and |𝑥₃−𝑥₂|≤|𝑦₃−𝑦₂|;(iv)|𝑥₁−𝑥₃|≤|𝑦₁−𝑦₃| and |𝑥₃−𝑥₂|≥|𝑦₃−𝑦₂|.

We will only prove the first two cases since the remaining cases can be proved similarly.

If |𝑥₁−𝑥₃|≥|𝑦₁−𝑦₃| and |𝑥₃−𝑥₂|≥|𝑦₃−𝑦₂|, then𝑑_𝑔(𝑋,𝑌)≤𝑘=𝑎(|𝑥₁−𝑥₃|+|𝑥₃−𝑥₂|)+𝑏(|𝑦₁−𝑦₃|+|𝑦₃−𝑦₂|)=𝑎|𝑥₁−𝑥₃|+𝑏|𝑦₁−𝑦₃|+𝑎|𝑥₃−𝑥₂|+𝑏|𝑦₃−𝑦₂|=𝑑_𝑔(𝑋,𝑍)+𝑑_𝑔(𝑍,𝑌). (1.4)If |𝑥₁−𝑥₃|≥|𝑦₁−𝑦₃| and |𝑥₃−𝑥₂|≤|𝑦₃−𝑦₂|, then there are two possible situations.

(i) Let |𝑥₁−𝑥₃|+|𝑥₃−𝑥₂|≥|𝑦₁−𝑦₃|+|𝑦₃−𝑦₂|. Then𝑑_𝑔(𝑋,𝑌)≤𝑘=𝑎(|𝑥₁−𝑥₃|+|𝑥₃−𝑥₂|)+𝑏(|𝑦₁−𝑦₃|+|𝑦₃−𝑦₂|). (1.5)Using |𝑥₁−𝑥₃|≥|𝑦₁−𝑦₃| and |𝑥₃−𝑥₂|≤|𝑦₃−𝑦₂|,𝑑_𝑔(𝑋,𝑍)=𝑎|𝑥₁−𝑥₃|+𝑏|𝑦₁−𝑦₃𝑑|,_𝑔(𝑍,𝑌)=𝑎|𝑦₃−𝑦₂|+𝑏|𝑥₃−𝑥₂|. (1.6)Also, 𝑑_𝑔(𝑋,𝑌)≤𝑑_𝑔(𝑋,𝑍)+𝑑_𝑔(𝑌,𝑍) if and only if𝑑_𝑔(𝑋,𝑌)≤𝑘=𝑎(|𝑥₁−𝑥₃|+|𝑥₃−𝑥₂|)+𝑏(|𝑦₁−𝑦₃|+|𝑦₃−𝑦₂|)𝑘≤𝑎(|𝑥₁−𝑥₃|+|𝑦₃−𝑦₂|)+𝑏(|𝑦₁−𝑦₃|+|𝑥₃−𝑥₂|)⟺𝑎(|𝑥₃−𝑥₂|−|𝑦₃−𝑦₂|)+𝑏(|𝑦₃−𝑦₂|−|𝑥₃−𝑥₂|)≤0⟺(𝑎−𝑏)(|𝑥₃−𝑥₂|−|𝑦₃−𝑦₂|)≤0⟺|𝑥₃−𝑥₂|≤|𝑦₃−𝑦₂|. (1.7)

(ii) Let |𝑥₁−𝑥₃|+|𝑥₃−𝑥₂|≤|𝑦₁−𝑦₃|+|𝑦₃−𝑦₂|. Then𝑑_𝑔(𝑋,𝑌)≤𝑘=𝑎(|𝑦₁−𝑦₃|+|𝑦₃−𝑦₂|)+𝑏(|𝑥₁−𝑥₃|+|𝑥₃−𝑥₂|). (1.8)Using |𝑥₁−𝑥₃|≥|𝑦₁−𝑦₃| and |𝑥₃−𝑥₂|≤|𝑦₃−𝑦₂|,𝑑_𝑔(𝑋,𝑍)=𝑎|𝑥₁−𝑥₃|+𝑏|𝑦₁−𝑦₃𝑑|,_𝑔(𝑍,𝑌)=𝑎|𝑦₃−𝑦₂|+𝑏|𝑥₃−𝑥₂𝑑|,_𝑔(𝑋,𝑌)≤𝑑_𝑔(𝑋,𝑍)+𝑑_𝑔𝑑(𝑌,𝑍)⟺_𝑔(𝑋,𝑌)≤𝑘=𝑎(|𝑦₁−𝑦₃|+|𝑦₃−𝑦₂|)+𝑏(|𝑥₁−𝑥₃|+|𝑥₃−𝑥₂|)𝑘≤𝑎(|𝑥₁−𝑥₃|+|𝑦₃−𝑦₂|)+𝑏(|𝑦₁−𝑦₃|+|𝑥₃−𝑥₂|)⟺𝑎(|𝑦₁−𝑦₃|−|𝑥₁−𝑥₃|)+𝑏(|𝑥₁−𝑥₃|−|𝑦₁−𝑦₃|)≤0⟺(𝑎−𝑏)(|𝑦₁−𝑦₃|−|𝑥₁−𝑥₃|)≤0⟺|𝑦₁−𝑦₃|≤|𝑥₁−𝑥₃|. (1.9)

Note that the generalized absolute value metric 𝑑_𝑔 is a generalization of the taxicab metric 𝑑_𝑇, maximum metric 𝑑_max, and Chinese checker metric 𝑑_𝐶 in the following manner: 𝑑_𝑔=𝑑_𝑇 if 𝑎=𝑏=1;𝑑_𝑔=𝑑_max if 𝑎=1 and 𝑏=0;𝑑_𝑔=𝑑_𝐶 if 𝑎=1 and 𝑏=√2−1, respectively. Furthermore, the unit circle in (ℝ²,𝑑_𝑔) is the set of points (𝑥,𝑦) in the plane which satisfies𝑎max{|𝑥|,|𝑦|}+𝑏min{|𝑥|,|𝑦|}=1,𝑎≥𝑏≥0,𝑎≠0. (1.10)

It is well known that Schwarz's inequality is a fundamental inequality of the Euclidean plane, which states that the magnitude of the inner product of two vectors cannot exceed the product of the lengths of these two vectors in the plane. Our main goal is to show the validity of Schwarz's inequality in (ℝ²,𝑑_𝑔). To achieve this goal, our presentation is organized as follows. After the definitions of trigonometric functions in (ℝ²,𝑑_𝑔) are given, we define the norm, prove Schwarz's inequality, and give an area formula of a triangle in this plane (Section 3). Finally, we give a plane tiling example underlying Schwarz's inequality in Section 4.

2. The Trigonometric Functions

We know that if 𝑋=(𝑥,𝑦) is a point on the Euclidean unit circle, then 𝑥=cos𝜃 and 𝑦=sin𝜃, where 𝜃 is the angle with the positive 𝑥-axis as the initial side and the radial line passing through point (𝑥,𝑦) as the terminal side. Now, we can define the trigonometric functions cosine and sine of the angle 𝜃 in (ℝ²,𝑑_𝑔) in the same way as the Euclidean cosine and sine functions. Let (𝑥,𝑦) be a point on the unit circle in (ℝ²,𝑑_𝑔). Since the unit circle is an octagon in (ℝ²,𝑑_𝑔) (see [4]), trigonometric functions can be defined according to eight quadrants. If we show the cosine and sine functions by cos_𝑔𝜃 and sin_𝑔𝜃, they can be given as follows with the method in [5]:cos_𝑔𝜃=cos𝜃𝑎|cos𝜃|+𝑏|sin𝜃|if|tan𝜃|≤1,cos𝜃𝑏|cos𝜃|+𝑎|sin𝜃|if|tan𝜃|≥1,sin_𝑔𝜃=sin𝜃𝑎|cos𝜃|+𝑏|sin𝜃|if|tan𝜃|≤1,sin𝜃𝑏|cos𝜃|+𝑎|sin𝜃|if|tan𝜃|≥1. (2.1)One can easily see that if 𝜃^′ is the related angle in first or second quadrant of given angle 𝜃 in quadrant 𝑖,(𝜋/4)(𝑖−1)≤𝜃≤(𝜋/4)𝑖,𝑖=1,…,8. The relationship between cos_𝑔𝜃 and cos_𝑔𝜃^′ or sin_𝑔𝜃 and sin_𝑔𝜃^′ is the same as the Euclidean one.

We note that there is the nonuniform increment in arc length when the angle is incremented by a fixed amount whereas on the Euclidean unit circle. So, we want to define trigonometric functions any angle 𝜃 between any two lines by using reference angle 𝛼 of 𝜃 [6]. Definition 2.1. Let 𝜃 be an angle with the reference angle 𝛼 which is the angle between 𝜃 and the positive direction of the 𝑥-axis in 𝑑_𝑔-unit circle. Then the cosine and sine functions of angle 𝜃 with the reference angles 𝛼, 𝑔cos𝜃, and 𝑔sin𝜃 are defined by𝑔cos𝜃∶=cos_𝑔(𝛼+𝜃)⋅cos_𝑔𝛼+sin_𝑔(𝛼+𝜃)⋅sin_𝑔𝛼,𝑔sin𝜃∶=sin_𝑔(𝛼+𝜃)⋅cos_𝑔𝛼−cos_𝑔(𝛼+𝜃)⋅sin_𝑔𝛼. (2.2)One can easily see that if 𝛼=0, then 𝑔cos𝜃=cos_𝑔𝜃 and 𝑔sin𝜃=sin_𝑔𝜃.

2.1. Variation of 𝑑_𝑔-Lengths Under Rotations

It is well known that all translations and rotations of the plane preserve the Euclidean distance. It is obvious that 𝑑_𝑔-distances are invariant under all translations . Now, we want to investigate the change of 𝑑_𝑔-lengths of a line segment after rotations as in [7]. Theorem 2.2. Let any two points be 𝐴 and 𝐵 in (ℝ²,𝑑_𝑔), and let the line segment 𝐴𝐵 be not parallel to the 𝑥-axis and the angle 𝛼 between the line segment 𝐴𝐵 and the positive direction of 𝑥-axis. If 𝐴^′𝐵^′ is the image of 𝐴𝐵 under the rotations with the angle 𝜃, then𝑑_𝑔(𝐴^′,𝐵^′)=cos²_𝑔𝛼+sin²_𝑔𝛼cos²_𝑔(𝛼+𝜃)+sin²_𝑔(𝛼+𝜃)⋅𝑑_𝑔(𝐴,𝐵). (2.3)Proof. We can translate the line segment 𝐴𝐵 to the line segment 𝑂𝑋 such that𝑑_𝑔(𝐴,𝐵)=𝑑_𝑔(𝑂,𝑋)=𝑘 (2.4)since all the translations preserve the 𝑑_𝑔-distance. 𝑋=(𝑘⋅cos_𝑔𝛼,𝑘⋅sin_𝑔𝛼) and under the rotation 𝜃, 𝑂𝑋 transforms to 𝑂𝑋^′, 𝑋^′=(𝑘⋅cos_𝑔(𝛼+𝜃),𝑘⋅sin_𝑔(𝛼+𝜃)). If we calculate 𝑑_𝑔(𝑂,𝑋^′)=𝑘^′ by using 𝑑_𝐸(𝑂,𝑋)=𝑑_𝐸(𝑂,𝑋^′), we obtain𝑘^′=𝑘⋅cos²_𝑔𝛼+sin²_𝑔𝛼cos²_𝑔(𝛼+𝜃)+sin²_𝑔(𝛼+𝜃). (2.5)Corollary 2.3. If the segment 𝐴𝐵 is parallel to the 𝑥-axis, then𝑑_𝑔(𝐴^′,𝐵^′𝑑)=^𝑔(𝐴,𝐵)cos²_𝑔𝜃+sin²_𝑔𝜃. (2.6)Proof. In this situation, since 𝛼=0, the proof is obvious.

3. Norm in (ℝ²,𝑑_𝑔)

In this section, we will define the norm in (ℝ²,𝑑_𝑔) by using the standard inner product in ℝ² as in [8] .Definition 3.1. Let 𝐴=(𝑥,𝑦) be any vector in (ℝ²,𝑑_𝑔). Then the norm of 𝐴 is defined by‖𝐴‖_𝑔=⟨𝐴,𝐴⟩cos²_𝑔𝛼+sin²_𝑔𝛼, (3.1)where 𝛼 is the reference angle of 𝐴.Corollary 3.2. ‖𝐴‖_𝑔=𝑑_𝑔(𝑂,𝐴).Proof. If 𝐴=(𝑥,𝑦)=(𝑑_𝑔(𝑂,𝐴)cos_𝑔𝛼,𝑑_𝑔(𝑂,𝐴)sin_𝑔𝛼) is used,𝑑_𝐸(𝑂,𝐴)=cos²_𝑔𝛼+sin²_𝑔𝛼⋅𝑑_𝑔(𝑂,𝐴). (3.2)If we use 𝑑_𝐸(𝑂,𝐴)=⟨𝐴,𝐴⟩ and Definition 3.1,‖𝐴‖_𝑔=𝑑_𝑔(𝑂,𝐴) (3.3)is obtained.

As in Euclidean geometry, it is easily seen that the norm function ‖⋅‖_𝑔 satisfies the following properties. So, the following proposition is given without proof. Proposition 3.3. Let 𝐴=(𝑥₁,𝑦₁), 𝐵=(𝑥₂,𝑦₂), and 𝐶=(𝑥₃,𝑦₃) be any three vectors in (ℝ²,𝑑_𝑔) and 𝑟∈ℝ. Then
(i)‖𝐴‖_𝑔≥0 and (‖𝐴‖_𝑔=0⇔𝐴=0),(ii)‖𝑟𝐴‖_𝑔=|𝑟|⋅‖𝐴‖_𝑔,(iii)‖𝐴+𝐵‖_𝑔≤‖𝐴‖_𝑔+‖𝐵‖_𝑔,(iv)‖𝐴−𝐵‖_𝑔≥‖𝐴‖_𝑔−‖𝐵‖_𝑔,(v)‖𝐴−𝐵‖_𝑔≤‖𝐴−𝐶‖_𝑔+‖𝐶−𝐵‖_𝑔.

It is important that Schwarz's inequality is valid for a metric. For example, matched filters are found in nearly every communication device, such as cell phones and television broadcasts. The detector utilizes Cauchy-Schwarz's inequality to compare functions. When the inner product of the functions is “large,” then there are significant similarities between the two signals. This detector will help us better select which tile is appropriate for a small section of the final image. If a quantized block of the target image is mostly one color on one side but another color on the opposite side, then the matched filter detector will find a tile with similar attributes.

The following proposition shows that Schwarz's inequality is valid for some restricted cases of 𝑑_𝑔. Proposition 3.4 (Schwarz's inequality). If 𝐴=(𝑥₁,𝑦₁) and 𝐵=(𝑥₂,𝑦₂) are any two vectors in (ℝ²,𝑑_𝑔) and 𝑏≥1, then|⟨𝐴,𝐵⟩|≤‖𝐴‖_𝑔⋅‖𝐵‖_𝑔. (3.4)Proof. (i) If 𝐴=0, then ⟨𝐴,𝐵⟩=0 and |⟨𝐴,𝐵⟩|=0. Furthermore, ‖𝐴‖_𝑔=0 and ‖𝐴‖_𝑔⋅‖𝐵‖_𝑔=0.
(ii) Let 𝐴≠0, then we know that|⟨𝐴,𝐵⟩|=|𝑥₁𝑥₂+𝑦₁𝑦₂|≤|𝑥₁𝑥₂|+|𝑦₁𝑦₂|,‖𝐴‖_𝑔⋅‖𝐵‖_𝑔=(𝑎max{|𝑥₁|,|𝑦₁|}+𝑏min{|𝑥₁|,|𝑦₁|})⋅(𝑎max{|𝑥₂|,|𝑦₂|}+𝑏min{|𝑥₂|,|𝑦₂|})=𝑎²|𝑥₁𝑥₂|+𝑎𝑏(|𝑥₁𝑦₂|+|𝑦₁𝑥₂|)+𝑏²|𝑦₁𝑦₂|. (3.5)(3.6)
In this case, there are four subcases. (1)|𝑥₁|≥|𝑦₁| and |𝑥₂|≥|𝑦₂|.(2)|𝑥₁|≥|𝑦₁| and |𝑥₂|≤|𝑦₂|.(3)|𝑥₁|≤|𝑦₁| and |𝑥₂|≥|𝑦₂|.(4)|𝑥₁|≤|𝑦₁| and |𝑥₂|≤|𝑦₂|.
Here, subcase (1) is proved; the others can be similarly proved.
If the right-hand sides of (3.4) and (3.5) are observed,𝑎𝑏|𝑥₁𝑦₂|+𝑎𝑏|𝑦₁𝑥₂|+𝑎²|𝑥₁𝑥₂|+𝑏²|𝑦₁𝑦₂|≥|𝑥₁𝑥₂+𝑦₁𝑦₂| (3.7)is valid since 𝑏≥1,|𝑥₁|≥|𝑦₁|, and |𝑥₂|≥|𝑦₂|. This completes the proof.

Now, we will give the area of the triangle 𝐴𝐵^Δ𝐶.Theorem 3.5. Let 𝐴𝐵^Δ𝐶 be any triangle in (ℝ²,𝑑_𝑔) and let 𝜃 be the angle between 𝐴𝐶 and 𝐵𝐶. Then the area 0𝑥0𝑓100 of the triangle 𝐴𝐵^Δ𝐶 is10𝑥0𝑓100=2⋅‖𝐴𝐶‖_𝑔⋅‖𝐵𝐶‖_𝑔⋅𝑔sin𝜃. (3.8)Proof. One can take that 𝐶 is the origin since all the translations preserve 𝑑_𝑔-distance. One can easily obtain the following relation between 𝑔sin𝜃 and sin𝜃:𝑔sin𝜃=‖𝑂𝐴‖⋅‖𝑂𝐵‖‖𝑂𝐴‖_𝑔⋅‖𝑂𝐵‖_𝑔⋅sin𝜃. (3.9)
From (3.9) and 20𝑥0𝑓100=‖𝑂𝐴‖⋅‖𝑂𝐵‖⋅sin𝜃 for 0≤𝜃≤𝜋,10𝑥0𝑓100=2⋅‖𝐴𝐶‖_𝑔⋅‖𝐵𝐶‖_𝑔⋅𝑔sin𝜃. (3.10)

4. Application

In Latin, tessella was a small cubical piece of clay, stone, or glass used to make mosaics as mentioned in [9, 10]. The word “tessella” means “small square” (from “tessera,” square, derived from the Greek word for “four”). It also corresponds to the everyday term “tiling” which refers to the applications of tessellation, often made of glazed clay.

Certain shapes of tiles, most obviously rectangles, can be replicated to cover a surface with no gaps. These shapes are said to tessellate (from the Latin tessella, “tile”). Over the centuries, many ceramic engineers and artists have employed plane tiling in their work. They used tiles for floors and walls because they are durable, waterproof, and beautiful. If you are mathematically inclined, no doubt you see some mathematics in the tiling. There are some examples where plane tiling on floors, walls, and in paintings underlies proofs of some well-known (and some not so well-known) theorems. However, tilings provide “picture proofs” for many theorems. They also illustrate inequalities such as the arithmetic mean-geometric mean inequality. Squares and rectangles are some of the most popular tiling used by engineers and artists to tile the plane. The design of geometric shapes, which fit together to cover a surface without gaps or overlapping, has a long history. Now, we will give a plane tiling example satisfying Schwarz's inequality in (ℝ²,𝑑_𝑔) as an application.

Suppose that the road is tiled with bricks of two different sizes as illustrated in Figure 1. Let 𝐴=(𝑥₁,𝑦₁) and 𝐵=(𝑥₂,𝑦₂) be any two vectors in (ℝ²,𝑑_𝑔) and the side lengths of two bricks are |𝑥₁|,|𝑥₂| and |𝑦₁|,|𝑦₂|, respectively; let 𝑑_𝑔 satisfy the condition 𝑏≥1 such that 𝐶=(𝑥₂,𝑦₁) and 𝐷=(𝑥₁,𝑦₂). Then the following tiling forms part of a proof of some special cases of Schwarz's inequality in two dimensions.

Figure 1 

Acknowledgment

The authors would like to thank Professor Rüstem Kaya for his kind help and valuable suggestions.