Using the Interval Metric for Modeling Entities Geometrics in ℝ<sup>2</sup> - Case Study Interval Circumference

The study of some distances provide science a way to separate two entities. It has applications in various fields such as remote sensing, data mining, pattern recognition and multivariate data analysis and others. If the distance is a Hausdorff metric, the guarantee is that all individuals are available. With the use of the distance of Trindade et al, we intend to extend the real topology to an interval topology, since the interval distance preserves the uncertainties and exits noise in the data. The present work proposes an interval circumference using an interval distance of a point to the center (pixel), like a set of pixels obeying certain distances to the center. With the interval circumference we intend to extend the notion of open ball and the concepts of neighborhood for the construction of the interval topology. A circumference separates a space into three regions, inner region, border region and outer region, where we construct our notion of neighborhood. In this work we will explore only the geometric properties of the interval circumference, we will extrapolate the notion from point to pixel by providing a differentiated frontier region for the clustering area.

Keywords:
interval; interval distance; interval circumference; pixels

Authorship

S.S. AGUIAR ^**Corresponding author: Sergio Silva Aguiar - E-mail: felixcatus50@hotmail.com

Departamento de Ciências Exatas e Tecnologia, UESB, 45083-900, Vitória da Conquista, BA, Brazil. E-mail: felixcatus50@hotmail.comUESBBrazilVitória da Conquista, BA, BrazilDepartamento de Ciências Exatas e Tecnologia, UESB, 45083-900, Vitória da Conquista, BA, Brazil. E-mail: felixcatus50@hotmail.com

https://orcid.org/0000-0003-2072-7218

R.M.P. TRINDADE

Departamento de Ciências Exatas e Tecnologia, UESB, 45083-900, Vitória da Conquista, BA, Brazil. E-mail: roquetrindade@uesb.edu.brUESBBrazilVitória da Conquista, BA, BrazilDepartamento de Ciências Exatas e Tecnologia, UESB, 45083-900, Vitória da Conquista, BA, Brazil. E-mail: roquetrindade@uesb.edu.br

https://orcid.org/0000-0003-3863-9486

A.O. ANDRADE

Departamento de Ciências Exatas e Tecnologia, UESB, 45083-900, Vitória da Conquista, BA, Brazil. E-mail: alexsandra@uesb.edu.brUESBBrazilVitória da Conquista, BA, BrazilDepartamento de Ciências Exatas e Tecnologia, UESB, 45083-900, Vitória da Conquista, BA, Brazil. E-mail: alexsandra@uesb.edu.br

https://orcid.org/0000-0002-8964-6486

A.F. SILVA

Departamento de Ciências Exatas e Tecnologia, UESB, 45083-900, Vitória da Conquista, BA, Brazil. E-mail: arizas@gmail.comUESBBrazilVitória da Conquista, BA, BrazilDepartamento de Ciências Exatas e Tecnologia, UESB, 45083-900, Vitória da Conquista, BA, Brazil. E-mail: arizas@gmail.com

https://orcid.org/0000-0001-9560-0559

G.R. CERQUEIRA

Departamento de Ciências Exatas e Tecnologia, UESB, 45083-900, Vitória da Conquista, BA, Brazil. E-mail: goncalorenildo@gmail.comUESBBrazilVitória da Conquista, BA, BrazilDepartamento de Ciências Exatas e Tecnologia, UESB, 45083-900, Vitória da Conquista, BA, Brazil. E-mail: goncalorenildo@gmail.com

https://orcid.org/0000-0003-3996-8150

*Corresponding author: Sergio Silva Aguiar - E-mail: felixcatus50@hotmail.com

SCIMAGO INSTITUTIONS RANKINGS

Figures | Formulas

Figures (7)
Formulas (46)

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Figure 1:
Representative image of an interval point or pixel.

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Figure 2:
Representative image of interval circumference.

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Figure 3:
Representative image of interval circumference.

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Figure 4:
Representative image of the angles of the interval circumference.

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Figure 5:
Representative image of degenerate interval circumference.

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Figure 6:
Representative images of circumference interval in Case 3.

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Figure 7:
Representative images of circumference interval in Case 4.

(3.1)

(3.2)

(3.3)

Figure 1: Representative image of an interval point or pixel.

Figure 2: Representative image of interval circumference.

Figure 3: Representative image of interval circumference.

Figure 4: Representative image of the angles of the interval circumference.

Figure 5: Representative image of degenerate interval circumference.

Figure 6: Representative images of circumference interval in Case 3.

Figure 7: Representative images of circumference interval in Case 4.

M_{e i} (X, Y) = [i n f \{d_{e} (x, y) : x \in X a n d y \in Y\}; s u p \{d_{e} (x, y) : x \in X a n d y \in Y\}] .

M_{e i} (X, Y) = [Y - X; Y - X] .

M_{e i} (X, Y) = [0; Y - X] .

M_{e i} (X, Y) = [0; M a x \{X - Y, Y - X\}];

M_{e i} (X, Y) = [d_{e} (x, y); d_{e} (x, y)] .

d (P_{1}, P_{2}) = \sqrt{{(M_{e i} (P_{1 x}, P_{2 x}))}^{2} + {(M_{e i} (P_{1 y}, P_{2 y}))}^{2}},

\{\begin{cases} M_{e i} (P_{1 x}, P_{2 x}) = [P_{2 x} - P_{1 x}; P_{2 x} - P_{1 x}] \\ M_{e i} (P_{1 y}, P_{2 y}) = [P_{2 y} - P_{1 y}; P_{2 y} - P_{1 y}] \end{cases} by proposition 1

\{\begin{cases} M_{e i} (P_{1 x}, P_{2 x}) = [P_{2 x} - P_{1 x}; P_{2 x} - P_{1 x}] \\ M_{e i} (P_{1 y}, P_{2 y}) = [0; P_{2 y} - P_{1 y}] \end{cases} by propositions 1, 2

\{\begin{cases} M_{e i} (P_{1 x}, P_{2 x}) = [0; P_{2 x} - P_{1 x}] \\ m_{e i} (P_{1 y}, P_{2 y}) = [P_{2 y} - P_{1 y}; P_{2 y} - P_{1 y}] \end{cases} by propositions 1, 2

\{\begin{cases} M_{e i} (P_{1 x}, P_{2 x}) = [0; P_{2 x} - P_{1 x}] \\ m_{e i} (P_{1 y}, P_{2 y}) = [0; P_{2 y} - P_{1 y}] \end{cases} by proposition 2

\{\begin{cases} M_{e i} (P_{1 x}, P_{2 x}) = [0; m a x \{P_{1 x} - P_{2 x}, P_{2 x} - P_{1 x}\}] \\ M_{e i} (P_{1 y}, P_{2 y}) = [0; m a x \{P_{1 y} - P_{2 y}, P_{2 y} - P_{1 y}\}] \end{cases} by proposition 3

D (P) = \sqrt{D i a m^{2} (P_{x}) + D i a m^{2} (P_{y})}

P_{1} \subseteq P_{2}

M_{e i} (P_{1 x}, P_{2 x}) \leq [0; D i a m (P_{2 x})] M_{e i} (P_{1 y}, P_{2 y}) \leq [0; D i a m (P_{2 y})] .

\{\begin{cases} M_{e i} (P_{1 x}, P_{2 x}) \leq [0; D i a m (P_{2 x})] \\ M_{e i} (P_{1 y}, P_{2 y}) \leq [0; D i a m (P_{2 y})] \end{cases} \Leftrightarrow \{\begin{array}{l} P_{1 x} \subseteq P_{2 x} \\ P_{1 y} \subseteq P_{2 y} \end{array} \Leftrightarrow P_{1} \subseteq P_{2} □

M_{e i} (X, Y) = m, w h e r e X \cap Y = 0 M_{e i}^{'} (X, Y) = N, w h e r e X \cap Y = 0 .

\{\begin{cases} D i a m (M_{e i} (P_{1 x}, P_{2 x})) = D i a m (P_{1 x}) + D i a m (P_{2 x}) \\ D i a m (M_{e i} (P_{1 y}, P_{2 y})) = D i a m (P_{1 y}) + D i a m (P_{2 y}) \end{cases}

\{\begin{cases} D i a m^{2} (M_{e i} (P_{1 x}, P_{2 x})) = (D i a m (P_{1 x}) + D i a m {(P 2 x))}^{2} \\ D i a m^{2} (M_{e i} (P_{1 y}, P_{2 y})) = (D i a m (P_{1 y}) + D i a m {(P 2 y))}^{2} \end{cases}

D i a m^{2} (M_{e i} (P_{1 x}, P_{2 x})) + D i a m^{2} (m_{e i} (P_{1 y}, P_{2 y})) = D i a m^{2} (d (P_{1}, P_{2})) = D^{2} (P_{1}) + D^{2} (P_{2}) + 2 (D i a m (P_{1 x}) \cdot D i a m (P_{2 x}) + D i a m (P_{1 y}) \cdot D i a m (P_{2 y})) .

D i a m^{2} (M_{e i} (P_{1 x}, P_{2 x})) + D i a m^{2} (m_{e i} (P_{1 y}, P_{2 y})) \geq D^{2} (P_{1}) + D^{2} (P_{2}) .

\{\begin{array}{l} D i a m (m e i (P_{1 x}, P_{2 x})) < D i a m (P_{1 x}) + D i a m (P_{2 x}) \\ D i a m (m e i (P_{1 y}, P_{2 y})) < D i a m (P_{1 y}) + D i a m (P_{2 y}) \end{array} and, therefore, D i a m^{2} (d (P_{1}, P_{2})) < D^{2} (P_{1}) + D^{2} (P_{2}) + 2 Z . □

M i n \{M_{e i} (C_{x}, X)\} = m i n \{|[X - C_{x}; X - C_{x}]|\} = \{\begin{cases} X - C_{x}, i f X > C_{x} \\ C_{x} - X, i f X < C_{x} \end{cases}

M i n \{d_{e i} (C, P)\} = \sqrt{m i n^{2} \{M_{e i} (C_{x}, X)\} + m i n^{2} \{M_{e i} (C_{y}, Y)\}}

M i n \{d_{e i} (C, P)\} = \{\begin{cases} \sqrt{{(X - C_{x})}^{2} + 0^{2}} = X - C_{x} o r \\ \sqrt{{(C_{x} - X)}^{2} + 0^{2}} = C_{x} - X \end{cases}

M i n \{d_{e i} (C, P)\} = \{\begin{cases} \sqrt{0^{2} + {(Y - C_{y})}^{2}} = Y - C_{y} o r \\ \sqrt{0^{2} + {(C_{y} - Y)}^{2}} = C_{y} - Y \end{cases}

M i n \{d_{e i} (C, P)\} = \sqrt{m i n^{2} \{M_{e i} (C_{x}, X)\} + m i n^{2} \{M_{e i} (C_{y}, Y)\}},

M i n \{d_{e i} (C, P)\} = \{\begin{cases} \sqrt{{(X - C_{x})}^{2} + {(Y - C_{y})}^{2}} \\ \sqrt{{(X - C_{x})}^{2} + {(C_{y} - Y)}^{2}} \\ \sqrt{{(C_{x} - X)}^{2} + {(Y - C_{y})}^{2}} \\ \sqrt{{(C_{x} - X)}^{2} + {(C_{y} - Y)}^{2}} \end{cases}

M a x \{M_{e i} (C_{x}, X)\} = M a x \{|[X - C_{x}; X - C_{x}]|\} = \{\begin{cases} X - C_{x} \\ C_{x} - X \end{cases} M a x \{M_{e i} (C_{y}, Y)\} = m a x \{|[Y - C_{y}; Y - C_{y}]|\} = \{\begin{cases} Y - C_{y} \\ C_{y} - Y \end{cases}

M a x \{d_{e i} (C, P)\} = \{\begin{cases} \sqrt{{(X - C_{x})}^{2} + {(Y - C_{y})}^{2}} or \\ \sqrt{{(X - C_{x})}^{2} + {(C_{y} - Y)}^{2}} or \\ \sqrt{{(C_{x} - X)}^{2} + {(Y - C_{y})}^{2}} or \\ \sqrt{{(C_{x} - X)}^{2} + {(C_{y} - Y)}^{2}} \end{cases}

d_{e i} (C_{I}, P) = \sqrt{{(m_{e i} (C_{I x}, P_{x}))}^{2} + {(M_{e i} (C_{I y}, P_{y}))}^{2}},

P \cap A \Rightarrow {[R_{I}, R_{I}]}^{2} = {[0, M a x \{X - C_{I x}, C_{I x} - X\}]}^{2} + {[Y - C I y, Y - C_{I y}]}^{2}

P \cap C \Rightarrow {[R_{I}, R_{I}]}^{2} = {[0, M a x \{X - C_{I x}, C_{I x} - X\}]}^{2} + {[C I y - Y, C I y - Y]}^{2}

P \cap B \Rightarrow {[R_{I}, R_{I}]}^{2} = {[X - C_{I x}, X - C_{I x}]}^{2} + {[0, M a x \{Y - C I y, C I y - Y\}]}^{2}

P \cap D \Rightarrow {[R_{I}, R_{I}]}^{2} = {[C I x - X, C_{I x} - X]}^{2} + {[0, M a x \{Y - C I y, C I y - Y\}]}^{2}

R_{I} = \{\begin{cases} x - C_{x} \Rightarrow x = C_{x} + R_{I} (B) \\ C_{x} - x \Rightarrow x = C_{x} - R_{I} (D) \\ y - C_{y} \Rightarrow y = C_{y} + R_{I} (A) \\ C_{y} - y \Rightarrow y = C_{y} - R_{I} (C) \end{cases}

{R_{I}}^{2} = {(x - C_{x})}^{2} + {(y - C_{y})}^{2};

{R_{I}}^{2} = {(x - C_{x})}^{2} + {(C y - y)}^{2} .

{R_{I}}^{2} = {(C_{x} - x)}^{2} + {(y - C_{y})}^{2} .

{R_{I}}^{2} = {(C_{x} - x)}^{2} + {(C y - y)}^{2} .

(3.1)

M a x \{d_{e i} (C, P)\} = R_{I} = \{\begin{cases} \sqrt{{(x - C_{x})}^{2} + {(y - C_{y})}^{2}} and \\ \sqrt{{(x - C_{x})}^{2} + {(C_{y} - y)}^{2}} and \\ \sqrt{{(C_{x} - x)}^{2} + {(y - C_{y})}^{2}} and \\ \sqrt{{(C_{x} - x)}^{2} + {(C_{y} - y)}^{2}} \end{cases}

s e n θ = \frac{D i a m (C_{x})}{2 R_{I}} a n d s e n ω = \frac{D i a m (C_{y})}{2 R_{I}} . \frac{D i a m^{2} (C_{x})}{4 {R_{I}}^{2}} + \frac{D i a m^{2} (C_{y})}{4 {R_{I}}^{2}} = s e n^{2} θ + s e n^{2} ω \Rightarrow \frac{D^{2} (C)}{4 {R_{I}}^{2}} = s e n^{2} θ + s e n^{2} ω .

\frac{D (C)}{R_{I}} = 2 \sqrt{s e n^{2} θ + s e n^{2} ω} . □

(3.2)

0 \leq \frac{D i a m (C_{x})}{2 R_{I}} \leq 1 \Rightarrow R_{I} \geq \frac{D i a m (C_{x})}{2}

(3.3)

R_{I} \geq \frac{D i a m (C_{y})}{2}

2 {R_{I}}^{2} \geq \frac{d^{2} + d^{2}}{4} \Rightarrow {R_{I}}^{2} \geq \frac{D {(C)}^{2}}{4}

R_{I} \geq \frac{D (C)}{2} □

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