Metric properties of fuzzy curves and fuzzy invariants

Abstract

Metric properties, i.e. properties which are invariant relative to the group E (3) of all Euclidean motions of the 3-dimensional Euclidean space of fuzzy curves are investigated in this paper. E (3)-equivalence of fuzzy curves is introduced. E (3)-equivalence of fuzzy curves is given in terms of fuzzy curvature and fuzzy torsion. All correlations between these fuzzy invariants of a fuzzy curve are described.

Keywords

Fuzzy path fuzzy curve invariant

1. Introduction

Geometric properties of fuzzy sets (in particular, connectedness, distance, relative position, convexity, area, perimeter, and diameter) and fuzzy geometric concepts (in particular, points, lines, circles and polygons) are investigated in several papers. Buckley and Eslami introduced a new concept to investigate the shapes of different fuzzy curves in [2, 3]. Metric properties (i.e. properties which are invariant relative to the group E (3) of all Euclidean motions of the 3-dimensional Euclidean space) of fuzzy sets are considered in some papers. Rosenfeld gave the definitions of diameter and perimeter of fuzzy sets in [21, 24]. Also, he obtained the distance of fuzzy sets in [22] and introduced fuzzy geometry by defining a fuzzy point, a fuzzy line, a fuzzy triangle in the fuzzy plane geometry in [23]. Using metric on fuzzy sets, different properties of sequences have been studied by Tripathy and Debanth [29], Tripathy et al. [30], Tripathy et al. [28] and others. In [7, 8], Das defined the notion of a fuzzy normed linear space and discussed some of its properties, and also defined the notion of statistically convergent and statistically null sequences with the concept of fuzzy norm. Tripathy and Das investigated different properties of the notion of convergence of series for fuzzy real numbersin [27].

Geometric properties of fuzzy sets and the idea of invariance found wide applications in several works on image processing and analysis, object representation and recognition. Chaudhuri and Rosenfeld considered the problem of defining the distance of two fuzzy sets in a metric space in [4]. Çivi, Christopher and Ercil explored the findings of the classical theory of invariants for the calculation of algebraic invariants of implicit curves and surfaces, then they applied these findings to object recognition in [6]. Also Weiss introduced model-based recognition of 3d curves. Weiss used the geometric invariants in [31]. Wu obtained a fuzzy curved search algorithm for neutral network learning in [32].

The present paper is devoted to an investigation of metric properties (euclidean invariants) of fuzzy paths and fuzzy curves in the 3-dimensional Euclidean space. Fuzzy paths and fuzzy curves were considered in [18 , 37]. The classical theory of curves, differential invariants and invariant parametrizations of curves in an n-dimensional Euclidean space can be found in [5 , 38]. The global theory of curves, global differential invariants and global invariant parametrizations of curves in an n-dimensional Euclidean space and an n-dimensional affine space can be found in [1 , 26].

Fuzzy plane geometry was introduced and curves defined by analytically in [2 , 23]. But, in this paper, we give the definition of the 3-dimensional fuzzy curves in a geometrically manner. We give the concept of fuzzy invariants of fuzzy curves, which is new in the fuzzy theory of curves. Fuzzy invariants also gives the metric properties of fuzzy curves. In this paper, to determine a fuzzy curve, we used their fuzzy invariants (fuzzy curvature, fuzzy torsion) which are significant possible inputs for a fuzzy curve. We also give the conditions of E (3)-equivalence of fuzzy curves in terms of fuzzy invariants. This is important for two fuzzy curves. Because the investigation of these curves in which cases are the same in the Euclidean space except the position is important for the theory of curves. We give the solution of this problem using fuzzy invariants. Also, we obtain these results in terms of fuzzy curvature and fuzzy torsion.

This paper is organized as follows. In Section 2, definitions of a fuzzy path, a fuzzy curve, the E (3)-equivalence of fuzzy paths, the E (3)-equivalence of fuzzy curves are introduced. Important examples of fuzzy paths are given. In Section 3, definitions of the fuzzy type of a fuzzy curve and an invariant parametrization of a fuzzy curve are given. The fuzzy type of a fuzzy curve is E (3)-invariant. All invariant parametrizations of a fuzzy curve with the fixed fuzzy type are described. In Theorem 1, the problem of the E (3)- equivalence of regular fuzzy curves is reduced to that of usual paths. This theorem is the theorem on defuzzification of the problem of the E (3)- equivalence of regular fuzzy curves. In Section 4, conditions of the global E (3)- equivalence of fuzzy curves are given in terms of the fuzzy type and the three differential invariants (curvatures) of a fuzzy curve. In Section 5, the description of the complete system of correlations between these invariants of a fuzzy curve is given.

2. Concepts of a fuzzy path and a fuzzy curve

Let $ℝ$ be the field of real numbers, $ℝ^{+} = {r \in ℝ ∣ r > 0}$ and I = (a, b) be an open interval of $ℝ$ . We consider the 3-dimensional Euclidean space as the 3-dimensional vector space $ℝ^{3}$ with the scalar product $< x, y > = \sum_{k = 1}^{3} x_{k} y_{k}$ of vectors x = (x₁, x₂, x₃) and y = (y₁, y₂, y₃).

Definition 1. (see [14]) A C^∞-map $x : I \to ℝ^{3}$ is called an I-path (a path, for short) in $ℝ^{3}$ .

Let $Q = ℝ, ℝ^{+}$ or $Q = [0, 1] \subset ℝ$ .

Definition 2. A pair (μ (t) , x (t)), where μ : I → Q and $x : I \to ℝ^{3}$ are C^∞-mappings is called a smooth Q-fuzzy I-path in $ℝ^{3}$ . The function μ (t) is called a membership function of the path x.

In the case $Q = ℝ^{+}$ or Q = [0, 1], Q-fuzzy I-path is a particular case of an $ℝ$ -fuzzy path. Therefore below we will consider only $ℝ$ -fuzzy paths. For short, $ℝ$ -fuzzy path is called fuzzy path.

Remark 1. The concept of a fuzzy path in fuzzy topological spaces was introduced in [37]. General definitions of a fuzzy path were given also in [19, 33].

Let x (t) = (x₁ (t) , x₂ (t) , x₃ (t)) be an I-path in $ℝ^{3}$ and $x^{'} (t) = (x_{1}^{'} (t), x_{2}^{'} (t), x_{3}^{'} (t))$ be the derivative of x (t). Denote the scalar product of x′ (t) with itself by <x′ (t) , x′ (t)>. Put x⁽⁰⁾ = x, x⁽¹⁾ = x′, x^(m) = (x^(m-1)) ′. For $a_{k} \in ℝ^{3}$ , k = 1, 2, 3, the determinant det(a_ij) (where a_kj are coordinates of a_k) is denoted by [a₁a₂a₃]. So [x⁽¹⁾ (t) x⁽²⁾ (t) x⁽³⁾ (t)] is the determinant of vectors x⁽¹⁾ (t) , x⁽²⁾ (t) , x⁽³⁾ (t).

Let us give some examples of membership functions of paths.

Example 1. Let $J = (0, b), b \in ℝ^{+}$ . Consider a C^∞-motion x (t) , t ∈ J, of a particle P in the three-dimensional Euclidean space with a variable mass m (t), where $m : J \to ℝ^{+}$ is a C^∞-function and m (t) is the mass of the particle P at time t ∈ J. A motion of the particle P with a variable mass is completely defined by the pair (m (t) , x (t)). Therefore we can consider a motion x (t) of the particle P with a variable mass m (t) as a fuzzy path (m (t) , x (t)), where m (t) is a membership function of the J-path x (t). Conversely, for every fuzzy path (r (t) , x (t)), where $r : J \to ℝ^{+}$ , we can consider as a motion x (t) of a particle P in three-dimensional Euclidean space with a variable mass r (t).

Example 2. Let M be a canal surface (see [9, p. 799]Gr). The canal surface M is completely defined by the pair (r (t) , γ (t)), where γ (t) is the center curve of M and r (t) is the radius of M. Therefore we can consider a canal surface M as the fuzzy path (r (t) , x (t)), where $r : J \to ℝ^{+}$ , r (t) is a membership function of the center curve x (t).

Suppose that the center curve of a canal surface is a unit-speed curve $γ : (a, b) \to ℝ^{3}$ with nonzero curvature. Then the canal surface can be parametrized by the formula (see [9, p. 800]Gr: $M (t, θ) = γ (t) + r (t) (- r^{'} T \pm \sqrt{1 - r^{'} {(t)}^{2}} (- c o s θ N + s i n θ) B),$ where T, N, B denote the tangent, normal and binormal of γ. Hence every fuzzy path (μ (t) , x (t)), where $‖ x^{'} (t) ‖ = 1$ , and $r^{'} {(t)}^{2} < 1$ , can be represented geometrically as a canal surface with the center curve x (t) and the radius function μ (t).

Example 3. Let $I = (a, b) \subset ℝ$ , x (t) be an I-path in $ℝ^{3}$ and a be a fixed point in three-dimensional Euclidean space $ℝ^{3}$ . Put r (t) = ∥x (t) - a∥ . Then the pair (r (t) , x (t)) is a fuzzy path in $ℝ^{3}$ .

Example 4. (see [1]). Let $I = (a, b) \subset ℝ$ and x be an I-path in $ℝ^{3}$ such that <x′ (t) , x′ (t) > ≠0 for all t ∈ I. For p, q ∈ I, p < q, let $l_{x} (p, q) = \int_{p}^{q} \sqrt{< x^{'} (t), x^{'} (t) >} dt .$ The limits $l_{x} (a, q) = lim_{p \to a} l_{x} (p, q) \leq + \infty$ and $l_{x} (p, b) = lim_{q \to b} l_{x} (p, q) \leq + \infty$ exist. There are four possible cases: $\begin{matrix} (i) l_{x} (a, q) < + \infty, l_{x} (p, b) < + \infty; \\ (ii) l_{x} (a, q) < + \infty, l_{x} (p, b) = + \infty; \\ (iii) l_{x} (a, q) = + \infty, l_{x} (p, b) < + \infty; \\ (iv) l_{x} (a, q) = + \infty, l_{x} (p, b) = + \infty . \end{matrix}$ Let μ_x (t) = l_x (a, t) for the cases (i), (ii) and μ_x (t) = - l_x (t, b) for the case (iii). Assume that the case (iv) holds. We choose a fixed point in every interval I of $ℝ$ and denote it by a_I. Let a_I = 0 for I = (- ∞ , ∞). We set μ_x (t) = l_x (a_I, t). In these cases, we can consider euclidean arc length parametrizations μ_x (t) of I-paths as membership functions of them.

Example 5. (see [14]). Let $I = (a, b) \subset ℝ$ and x (t) be an I-path in $ℝ^{3}$ such that $[x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)] \neq 0$ for all t ∈ I, where $I = (a, b) \subset ℝ$ . For p, q ∈ I, p < q, let $l_{x} (p, q) = \int_{p}^{q} {| [x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)] |}^{\frac{1}{6}} dt$ . As in Example 4, let μ_x (t) = l_x (a, t) for the cases (i), (ii), μ_x (t) = - l_x (t, b) for the case (iii) and μ_x (t) = l_x (a_I, t) for the case (iv). In these cases, we can consider equi-affine arc length parametrization μ_x (t) of the I-path x (t) as a membership function of it.

Example 6. (see [20]). Let $I = (a, b) \subset ℝ$ and x (t) be an I-path in $ℝ^{3}$ such that [x⁽¹⁾ (t) x⁽²⁾ (t) x⁽³⁾ (t)] ≠ 0 and [x (t) x⁽¹⁾ (t) x⁽²⁾ (t)] ≠ 0 for all t ∈ I, where $I = (a, b) \subset ℝ$ . For p, q ∈ I = (a, b) , p < q, let $l_{x} (q, p) = \int_{q}^{p} {| \frac{[x^{'} (t) x^{(2)} (t) x^{(3)} (t)]}{[x (t) x^{'} (t) x^{(2)} (t)]} |}^{\frac{1}{3}} dt$ As in Example 4, let μ_x (t) = l_x (a, t) for the cases (i), (ii), μ_x (t) = - l_x (t, b) for the case (iii) and μ_x (t) = l_x (a_I, t) for the case (iv). In these cases, we can consider centro-affine arc length parametrizations μ_x (t) of the I-path x (t) as membership functions of it.

Let $I_{1} = (a, b) \subset ℝ$ and $I_{2} = (c, d) \subset ℝ$ .

Definition 3. (see [14]) An I₁-path x (t) and an I₂-path y (r) in $ℝ^{3}$ is called C^∞-equivalent if there exists a C^∞ -diffeomorphism φ : I₂ → I₁ such that φ′ (r) >0 and y (r) = x (φ (r)) for all r ∈ I₂. A class of C^∞-equivalent paths in $ℝ^{3}$ is called a curve in $ℝ^{3}$ (see [15, p. 9]). A path x ∈ α is called a parametrization of a curve α.

Example 7. Let $x : (0, 2 π) \to ℝ^{3}$ be defined by x (t) = (cost, sint, 3) and let $y : (0, 1) \to ℝ^{3}$ be defined by y (r) = (cos2πr, sin2πr, 3). For the C^∞-diffeomorphism φ : (0, 1) → (0, 2π), φ (r) =2πr, since x (φ (r)) = y (r), the paths x and y are C^∞-equivalent.

Definition 4. A fuzzy I₁-path (μ (t) , x (t)) and a fuzzy I₂-path (ν (r) , y (r)) in $ℝ^{3}$ are called C^∞-equivalent if there exists a C^∞ -diffeomorphism φ : I₂ → I₁ such that φ′ (r) >0, y (r) = x (φ (r)) and ν (r) = μ (φ (r)) for all r ∈ I₂. A class of C^∞-equivalent fuzzy paths is called a fuzzy curve in $ℝ^{3}$ . A fuzzy path (μ, x) ∈ α is called a parametrization of a fuzzy curve α.

Example 8. If we use the paths x and y in Example 6 and membership functions in Example 4, these fuzzy paths are C^∞-equivalent.

Let $O (3, ℝ)$ be the group of all 3 × 3-orthogonal real matrices. Then E (3)= ${F : ℝ^{3} \to ℝ^{3} ∣ Fx = gx + b$ , $g \in O (3, ℝ)$ , $b \in ℝ^{3}}$ is the group of all Euclidean motions of the 3-dimensional Euclidean space, where gx is the multiplication of the matrix g and the column vector $x \in ℝ^{3}$ .

Let x (t) be an I-path. Then Fx (t) is an I-path in $ℝ^{3}$ for all F ∈ E (3).

Definition 5. (see [14]) I-paths x (t) and y (t) in $ℝ^{3}$ are called E (3)-equivalent if there exists F ∈ E (3) such that y (t) = Fx (t). In this case, we shall write $x \overset{E (3)}{\sim} y$ .

Example 9. Let x (t) = (t, t², t³), y (t) = (t - 1, t² - 2, t³ + 3) and Fx = gx + b, where $g = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix})$ and $b = (\begin{matrix} - 1 \\ - 2 \\ 3 \end{matrix})$ . Since y (t) = Fx (t), it is obtained $x \overset{E (3)}{\sim} y$ .

Definition 6. Fuzzy I-paths (μ (t) , x (t)) , (ν (t) , y (t)) are called E (3)-equivalent if μ (t) = ν (t) and there exists F ∈ E (3) such that y (t) = Fx (t) for all t ∈ I. In this case, it is denoted by $(μ, x) \overset{E (3)}{\sim} (ν, y)$ .

Example 10. If we use the paths x and y in Example 9 and the membership functions in Example 4, then we have $(μ, x) \overset{E (3)}{\sim} (ν, y)$ .

Let α = {h_τ, τ ∈ Q} be a curve in $ℝ^{3}$ , where h_τ is a parametrization of α. Then Fα = {Fh_τ, τ ∈ Q} is a curve in $ℝ^{3}$ for all F ∈ E (3).

Definition 7. (see [14]) Curves α and β in $ℝ^{3}$ are called E (3)-equivalent if β = Fα for some F ∈ E (3).

Example 11. Let us use the curves α and β which are the equivalence classes of the paths in Example 9, then α and β are E (3)-equivalent.

Let α ={ (μ_τ, h_τ) , τ ∈ Q } be a fuzzy curve in $ℝ^{3}$ , where (μ_τ, h_τ) is a parametrization of α. Then Fα ={ (μ_τ, Fh_τ) , τ ∈ Q } is a fuzzy curve in $ℝ^{3}$ for all F ∈ E (3).

Definition 8. Fuzzy curves α and β in $ℝ^{3}$ are called E (3)-equivalent if β = Fα for some F ∈ E (3). In this case, it is denoted by $α \overset{E (3)}{\sim} β$ .

3. Invariant parametrizations of fuzzy curves and the theorem on defuzzification

For a fuzzy I-path (μ (t) , x (t)), put $m (μ) = inf_{t \in I} μ (t) and M (μ) = sup_{t \in I} μ (t) .$ Let (μ (t) , x (t)), (ν (t) , y (t)) be fuzzy I-paths and $(μ, x) \overset{E (3)}{\sim} (ν, y)$ . Then m (μ) = m (ν) and M (μ) = M (ν). Thus the numbers m (μ) , M (μ) are E (3)-invariants of a fuzzy I-path (μ (t) , x (t)).

Let α be a fuzzy curve. Then m (μ) = m (ν) and M (μ) = M (ν) for all (μ, x) , (ν, y) ∈ α. For (μ, x) ∈ α, the numbers m (μ) and M (μ) are denoted by m (α) and M (α), respectively. The following inequalities -∞ ≤ m (α) < + ∞ , - ∞ < M (α) ≤ + ∞ are obvious.

Definition 9. A fuzzy I-path (μ (t) , x (t)) is called regular if μ′ (t) ≠0 for all t ∈ I, where μ′ (t) is the derivative of the function μ (t).

Example 12. A fuzzy I-path (μ (t) , x (t)) as in Example 4 is a regular fuzzy I-path.

If (μ, x) is a regular fuzzy path and a fuzzy path (ν, y) is C^∞-equivalent to (μ, x), then (ν, y) is also regular. A fuzzy curve α is called regular if it contains a regular path. Let (μ, x) be a regular fuzzy path.

Definition 10. In the case μ′ (t) >0, the pair (m (μ) , M (μ)) is called the fuzzy type of (μ, x). In the case μ′ (t) <0, the pair (M (μ) , m (μ)) is called the fuzzy type of (μ, x). The fuzzy type of (μ, x) is denoted by T (μ).

Example 12. Let (μ (t) , x (t)) be the fuzzy I-path in Example 4. Let x (t) = (e^t, 2, 0) and I = (0, ∞). In that case, <x′ (t) , x′ (t) > = e^2t and it is obtained T (μ) = (0, ∞).

It is obvious that T (μ) = T (ν) for all (μ, x) , (ν, y) ∈ α. The fuzzy type T (μ) of a fuzzy path (μ, x), where (μ, x) ∈ α, is called the fuzzy type of the fuzzy curve α and denoted by T (α). Thus T (α) = (m (α) , M (α)) for the case μ′ (t) >0 and T (α) = (M (α) , m (α)) for the case μ′ (t) <0. Let α and β be fuzzy curves and $α \overset{E (3)}{\sim} β$ . Then T (α) = T (β). Thus T (α) is an E (3)-invariant of a fuzzy curve α.

According to Definition 10, the fuzzy type of a fuzzy path (μ, x) is a pair (c, d) such that c ≠ d and -∞ ≤ c, d ≤ + ∞. It is easy to see that, for every pair (c, d), where c, d are arbitrary numbers such that c ≠ d and -∞ ≤ c, d ≤ + ∞, there exists a fuzzy path (μ, x) of the fuzzy type T (μ) = (c, d).

Now we define the invariant parameter of a regular fuzzy path in $ℝ^{3}$ . Let I = (a, b) and (μ, x) be a regular fuzzy I-path in $ℝ^{3}$ . We define the fuzzy arc length function s_μ (t) for each fuzzy type as follows. We consider all possible cases for the fuzzy type of the form T (μ) = (m (μ) , M (μ)):

in the case m (μ)≠ - ∞, put s_μ (t) = μ (t) - m (μ);

in the case m (μ) = -∞ , M (μ) ≠ + ∞, put s_μ (t) = μ (t) - M (μ);

Let m (μ) = -∞ , M (μ) = + ∞. We choose a fixed point in every interval I = (a, b) of $ℝ$ and denote it by a_I. Let a_I = 0 for I = (- ∞ , + ∞). We put s_μ (t) = μ (t) - μ (a_I).

Now we consider all possible cases for the fuzzy type of the form T (μ) = (M (μ) , m (μ)):

in the case m (μ)≠ - ∞, put s_μ (t) = - (μ (t) - m (μ));

in the case m (μ) = -∞ , M (μ) ≠ + ∞, put s_μ (t) = - (μ (t) - M (μ));

Let m (μ) = -∞ , M (μ) = + ∞. We put s_μ (t) = - (μ (t) - μ (a_I)).

By the definition of the function s_μ (t), we have

s_{μ}^{'} (t) > 0

for all t ∈ I and every regular fuzzy path (μ, x). Therefore the inverse function of s_μ (t) exists. Let us denote it by t_μ (s).The domain of the function t_μ (s) denote by D (μ). By

s_{μ}^{'} (t) > 0

for all t ∈ I, we obtain

t_{μ}^{'} (s) > 0

for all s ∈ D (μ). The description of all possible domains D (μ) is following:

for the case γ₁, D (μ) = (0, M (μ) - m (μ));

for the case γ₂, D (μ) = (- ∞ , 0);

for the case γ₃, D (μ) = (- ∞ , + ∞);

for the case δ₁, D (μ) = (- (M (μ) - m (μ)) , 0);

for the case δ₂, D (μ) = (0, + ∞);

for the case δ₃, D (μ) = (- ∞ , + ∞);

It is obvious that D (μ) = D (ν) for all (μ, x) , (ν, y) ∈ α. The interval D (μ), where (μ, x) ∈ α, is denoted by D (α).

Proposition 1. Let (μ, x) and (ν, y) be E (3)- equivalent regular fuzzy I-paths in $ℝ^{3}$ . Then s_μ (t) = s_ν (t) for all t ∈ I and t_μ (s) = t_ν (s) for all s ∈ D (μ) = D (ν).

Proof. It follows from Definition 6 and definitions of functions s_μ (t) and t_μ (s).□Let g (t) be a C^∞-function on the interval I and φ : J = (c, d) → I is a C^∞-diffeomorphism. Denote the function g (φ (r)) by g (φ).

Proposition 2. Let (μ, x) be a regular fuzzy I-path in $ℝ^{3}$ and φ : J = (c, d) → I is a C^∞-diffeomorphism such that φ′ (r) >0 for all r ∈ J. Then

for T (μ), satisfying the conditions T (μ) ≠ (- ∞ , + ∞) and T (μ) ≠ (+ ∞ , - ∞), equalities s_μ(φ) (r) = s_μ (φ (r)) and φ (t_μ(φ) (s)) = t_μ (s) hold.

for T (μ), satisfying the conditions T (μ) = (- ∞ , + ∞) or T (μ) = (+ ∞ , - ∞), equalities s_μ(φ) (r) = s_μ (φ (r)) + s₀ and φ (t_μ(φ) (s + s₀)) = t_μ (s) hold, where s₀ = μ (a_I) - μ (φ (a_J)).

Proof. (i). We consider the case γ₁. Using the equality m (μ (φ)) = m (μ), we obtain s_μ(φ) (r) = μ (φ (r)) - m (μ (φ)) = μ (φ (r)) - m (μ) = s_μ (φ (r)). A proof of the equality φ (t_μ(φ) (s)) = t_μ (s) follows from the equality s_μ(φ) (r) = s_μ (φ (r)). Proofs in cases γ₂, δ₁, δ₂ are similar.

(ii). We consider the case γ₃: T (μ) = (- ∞ , + ∞). We have s_μ(φ) (r) = μ (φ (r)) - μ (φ (a_J)) = μ (φ (r)) - μ (a_I) + μ (a_I) - μ (φ (a_J)) = s_μ (φ (r)) + s₀, where s₀ = μ (a_I) - μ (φ (a_J)). A proof in the case δ₃ is similar. □

Let α be a regular fuzzy curve, (μ, x) ∈ α. Then (μ (t_μ (s)) , x (t_μ (s)) is a parametrization of α.

Definition 11. The parametrization (μ (t_μ (s)) , x (t_μ (s)) of a regular fuzzy curve α is called an invariant parametrization of α.

Example 13. Given an interval I = (0, ∞) and a I-path $x (t) = (\cos \frac{3 t}{5}, \sin \frac{3 t}{5}, \frac{4 t}{5})$ . If we use the membership function in Example 4, we have that the parametrization (μ (t) , x (t)) is an invariant parametrization.

We denote the set of all invariant parametrizations of α by Ip (α). Every (ν, y) ∈ Ip (α) is a fuzzy J-path, where J = D (α).

Proposition 3. Let α be a regular fuzzy curve of the type T (α) = (m (α) , M (α)) (μ, x) ∈ α and (μ (t) , x (t)) is a fuzzy J-path, where J = D (α). Then the following conditions are equivalent:

(μ (t) , x (t)) is an invariant parametrization of α;

μ′ (t) =1 for all t ∈ D (α);

s_μ (t) = t for all t ∈ D (α).

Proof. (i) → (ii). Let (μ, x) ∈ Ip (α). Then there exists (ν, y) ∈ α such that μ (s) = ν (t_ν (s)) and x (s) = y (t_ν (s)) for all s ∈ D (α). Put φ (s) = t_ν (s). Since s_ν (t_ν (s)) = s for all s ∈ D (α), using Proposition 2, we obtain s_μ (s) = s_ν(φ) (s) = s_ν (φ (s)) + s₀ = s_ν (t_ν (s)) + s₀ = s + s₀, where s₀ = 0 or s₀ is as in Proposition 2

(ii). Since s₀ does not depend on s, $\frac{{ds}_{μ} (s)}{ds}$ =μ′ (s) =1. Hence μ′ (s) =1 for all s ∈ D (α).

(ii) → (iii). Let μ′ (s) =1 for all s ∈ D (α). By the definitions (γ₁) , (γ₂) , (γ₃) of s_μ (t), we have $\frac{{ds}_{μ} (s)}{ds} = μ^{'} (s)$ . Hence $\frac{{ds}_{μ} (s)}{ds} = 1$ for all s ∈ D (α). Therefore s_μ (s) = s + c for some $c \in ℝ$ . In the case T (μ) ≠ (- ∞ , + ∞), conditions s_μ (s) = s + c and s_μ (s) ∈ D (α) for all s ∈ D (α) implies c = 0, that is, s_μ (s) = s. In the case T (μ) = (- ∞ , + ∞) or T (μ) = (+ ∞ , - ∞), equalities s_μ (s) = μ (s) - μ (a_J) = μ (s) - μ (0) = s + c implies 0 = μ (0) - μ (0) = s_μ (0) = c. Hence s_μ (s) = s.

(iii) → (i). The equality s_μ (s) = s implies t_μ (s) = s. Therefore x (s) = x (t_μ (s)) and μ (s) = μ (t_μ (s)) for all s ∈ D (α). Hence (μ (s) , x (s)) ∈ Ip (α).□

Proposition 4. Let α be a regular fuzzy curve of the type T (α) = (M (α) , m (α)) (μ, x) ∈ α and (μ (t) , x (t)) is a fuzzy J-path, where J = D (α). Then the following conditions are equivalent:

(μ (t) , x (t)) is an invariant parametrization of α;

μ′ (t) = -1 for all t ∈ D (α);

s_μ (t) = - t for all t ∈ D (α).

Proof. The proof is similar to the proof of Proposition 3.□

Proposition 5. Let α be a regular fuzzy curve such that T (α) ≠ (- ∞ , + ∞) and T (α) ≠ (+ ∞ , - ∞). Then there exists the unique invariant parametrization of α.

Proof. Let (μ, x) , (ν, y) ∈ α, where x is a J₁-path and y is a J₂-path. Then there exists a C^∞-diffeomorphism φ : J₂ → J₁ such that φ′ (r) >0 ν (r) = μ (φ (r)) and y (r) = x (φ (r)) for all r ∈ J₂. Using Proposition 2 (i), we obtain $ν (t_{ν} (s)) = ν (t_{μ (φ)} (s)) = μ (φ (t_{μ (φ)} (s)) = μ (t_{μ} (s))$ and $y (t_{ν} (s)) = x (φ (t_{ν} (s)) = x (φ (t_{μ (φ)} (s))) = x (t_{μ} (s)) .$ Hence (ν (t_ν (s)) , y (t_ν (s))) = (μ (t_μ (s)) , x (t_μ (s))).□

Let α be a regular fuzzy curve and T (α) = (- ∞ , + ∞) or T (α) = (+ ∞ , - ∞). Then it is easy to see that the set Ip (α) is not countable.

Proposition 6. Let α be a regular fuzzy curve, T (α) = (- ∞ , + ∞) and (μ, x) ∈ Ip (α). Then $Ip (α) = {(ν, y) : ν (s) = μ (s + c), y (s) = x (s + c), c \in ℝ}$ .

Proof. Let (μ, x) , (ν, y) ∈ Ip (α). Then there exist (ρ, h) , (τ, k) ∈ α such that x (s) = h (t_ρ (s)) , μ (s) = ρ (t_ρ (s)) , y (s) = k (t_τ (s)) , ν (s) = τ (t_τ (s)), where h is a J₁-path and k is a J₂-path. Since (ρ, h) , (τ, k) ∈ α there exists φ : J₂ → J₁ such that φ′ (r) >0, τ (r) = ρ (φ (r)) and k (r) = h (φ (r)) for all r ∈ J₂. Using Proposition 2 (ii), we obtain y (s) = k (t_τ (s)) = h (φ (t_τ (s))) = h (φ (t_ρ(φ) (s))) = h (t_ρ (s - s₀)) = x (s - s₀) and ν (s) = τ (t_τ (s)) = ρ (φ (t_τ (s))) = ρ (φ (t_ρ(φ) (s))) = ρ (t_ρ (s - s₀)) = μ (s - s₀). Hence y (s) = x (s + c) and ν (s) = μ (s + c), where c = - s₀.

Let (μ, x) ∈ Ip (α) and c ∈ (- ∞ , + ∞). We prove that (μ (ψ) , x (ψ)) ∈ Ip (α), where ψ (s) = s + c. By Proposition 3, μ′ (s) =1 and s_μ (s) = s. Put z (s) = x (ψ (s)) , ζ (s) = μ (ψ (s)). Since ψ is a C^∞-diffeomorphism of (- ∞ , + ∞) onto (- ∞ , + ∞), we have (ζ, z) = (μ (ψ) , x (ψ)) ∈ α. Using Proposition 2 and s_μ (s) = s, we get s_ζ (s) = s_μ(ψ) (s) = s_μ (ψ (s)) + s₁ = (s + c) + s₁. By the definition of the function s_ζ (s), we have s_ζ (s) = ζ (s) - ζ (0) = μ (ψ (s)) - μ (ψ (0)). Then μ (ψ (s)) - μ (ψ (0)) = s + c + s₁. For s = 0, we have 0 = μ (ψ (0)) - μ (ψ (0)) = c + s₁. Hence s₁ = - c. Then s_ζ (s) = s. By Proposition 3, (ζ, z) ∈ Ip (α).□

Proposition 7. Let α be a regular fuzzy curve, T (α) = (+ ∞ , - ∞) and (μ, x) ∈ Ip (α). Then $Ip (α) = {(ν, y) : ν (s) = μ (s + c), y (s) = x (s + c), c \in ℝ}$ .

Proof. The proof is similar to the proof of Proposition 6.□

Theorem 1. Let α, β be regular fuzzy curves, T (α) = T (β) and (μ, x) ∈ Ip (α) , (ν, y) ∈ Ip (β). Then:

for T (α) = T (β) ≠ (- ∞ , + ∞) and T (α) = T (β) ≠ (+ ∞ , - ∞), $α \overset{E (3)}{\sim} β$ if and only if $(μ, x) \overset{E (3)}{\sim} (ν, y)$ ;

for T (α) = T (β) = (- ∞ , + ∞) or T (α) = T (β) = (+ ∞ , - ∞), $α \overset{E (3)}{\sim} β$ if and only if $(μ, x) \overset{E (3)}{\sim} (ν (ψ_{c}), y (ψ_{c})$ for some c ∈ (- ∞ , + ∞), where ψ_c (s) = s + c.

Proof. (i). Let $α \overset{E (3)}{\sim} β$ and (μ₁, h) ∈ α. Then there exists F ∈ E (3) such that β = Fα. This implies (μ₁, Fh) ∈ β. Since (μ, x) ∈ Ip (α) and (μ₁, h) ∈ α, using Propositions 2 and 5, we get μ (s) = μ₁ (t_{μ
₁} (s)), x (s) = h (t_{μ
₁} (s)). According to (ν, y) ∈ Ip (β) and (μ₁, Fh) ∈ β, we obtain similarly ν (s) = μ₁ (t_{μ
₁} (s)) , y (s) = (Fh) (t_{μ
₁} (s)). Using Propositions 1 and 2, we get Fx (s) = F (h (t_{μ
₁} (s))) = (Fh) (t_{μ
₁} (s)) = y (s). We have μ (s) = μ₁ (t_{μ
₁} (s)), ν (s) = μ₁ (t_{μ
₁} (s)) = μ (s). Thus $(μ, x) \overset{E (3)}{\sim} (ν, y)$ . Conversely, let $(μ, x) \overset{E (3)}{\sim} (ν, y)$ , that is, there exists F ∈ E (3) such that Fx = y and μ (s) = ν (s). This means that $α \overset{E (3)}{\sim} β$ .

(ii). Let $α \overset{E (3)}{\sim} β$ . Then there exist J-paths (μ₁, h) ∈ α, (μ₂, k) ∈ β and F ∈ E (3) such that k (t) = Fh (t) and μ₂ (t) = μ₁ (t). Then s_{μ
₂} (t) = s_{μ
₁} (t) and t_{μ
₂} (s) = t_{μ
₁} (s). We have k (t_{μ
₂} (s)) = k (t_{μ
₁} (s)) = (Fh) (t_{μ
₁} (s)). By Proposition 6, x (s) = h (t_{μ
₁} (s + s₁)) , μ (s) = μ₁ (t_{μ
₁} (s + s₁)) , y (s) = k (t_{μ
₂} (s + s₂)) , ν (s) = μ₂ (t_{μ
₂} (s + s₂)) , for some s₁, s₂ ∈ (- ∞ , + ∞). Therefore y (s - s₂) = Fx (s - s₁) and μ (s - s₁) = μ₁ (t_{μ
₁} (s)) = μ₂ (t_{μ
₁} (s)) = μ₂ (t_{μ
₂} (s)) = ν (s - s₂). This implies that $(μ, x) \overset{E (3)}{\sim} (ν (ψ_{c}), y (ψ_{c}))$ , where ψ_c (s) = s + c and c = s₁ - s₂. Conversely, let $(μ, x) \overset{E (3)}{\sim} (ν (ψ_{c}), y (ψ_{c}))$ , for some c ∈ (- ∞ , + ∞), where ψ_c = s + c. Then there exists F ∈ E (3) such that y (s + c) = Fx (s) and μ (s + c) = μ (s). Since (ν (s + c) , y (s + c)) ∈ β, this means that $α \overset{E (3)}{\sim} β$ .□

Theorem 1 reduces the problem of the E (3)- equivalence of regular fuzzy curves to that of usual paths. Thus this theorem is a theorem on defuzzification of the problem E (3)-equivalence of regular fuzzy curves (here we use the concept of defuzzification in the introduction of [17]).

4. Conditions of E (3)-equivalence of fuzzy curves

Definition 12. An I-path x (t) in $ℝ^{3}$ is called non-singular if [x⁽¹⁾ (t) x⁽²⁾ (t) x⁽³⁾ (t)] ≠ 0 for all t ∈ I. A regular fuzzy curve α is called non-singular if it contains a fuzzy I-path (μ (t) , x (t)), where x is a non-singular path.

Let x (t) be a J-path in $ℝ^{3}$ and k ≥ 1, l ≥ 1. We note that <x^(k) (t) , x^(l) (t)> is an E (3)-invariant that is < (Fx) ^(k) (t) , (Fx) ^(l) (t)> = < x^(k) (t) , x^(l) (t) > for all F ∈ E (3).

Theorem 2. Let α, β be non-singular fuzzy curves in $ℝ^{3}$ and (μ, x) ∈ Ip (α) , (ν, y) ∈ Ip (β). Then:

for T (α) = T (β) ≠ (- ∞ , + ∞) and T (α) = T (β) ≠ (+ ∞ , - ∞), $α \overset{E (3)}{\sim} β$ if and only if $< x^{(i)} (s), x^{(i)} (s) > = < y^{(i)} (s), y^{(i)} (s) >$ (1) for all s ∈ D (α) and i = 1, 2, 3;

for T (α) = T (β) = (- ∞ , + ∞) or T (α) = T (β) = (+ ∞ , - ∞), $α \overset{E (3)}{\sim} β$ if and only if there exists c ∈ (- ∞ , + ∞) such that <x⁽ⁱ⁾ (s) , x⁽ⁱ⁾ (s)> = < y⁽ⁱ⁾ (s + c) , y⁽ⁱ⁾ (s + c) > for all s ∈ D (α) and i = 1, 2, 3.

Proof. (i). Let $α \overset{E (3)}{\sim} β$ . Then according to Theorem 1(i), $x \overset{E (3)}{\sim} y$ . Since <x⁽ⁱ⁾ (s) , x⁽ⁱ⁾ (s)> is E (3)-invariant for all i = 1, 2, 3, $x \overset{E (3)}{\sim} y$ implies equalities (1). Conversely, suppose that the conditions (1) are valid. We prove that $x \overset{E (3)}{\sim} y$ . Using equalities (1) and $< x^{(1)}, x^{(2)} > = \frac{1}{2} < x^{(1)}, x^{(1)} >^{'}, < x^{(2)}, x^{(3)} > = \frac{1}{2} < x^{(2)}, x^{(2)} >^{'}$ , we obtain equalities $< x^{(i)} (s), x^{(j)} (s) > = < y^{(i)} (s), y^{(j)} (s) >$ (2) for all s ∈ D (α) and i, j = 1, 2, 3.

Put A_x (t) = ||x⁽¹⁾ (t) x⁽²⁾ (t) x⁽³⁾ (t) ||. Since x is a non-singular path, we have ${detA}_{x} (t) = [x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)] \neq 0 .$ By detA_x (t) ≠0, the matrix $A_{x}^{- 1} (t)$ exists for all t ∈ I. It is easy to see that $A_{x}^{- 1} (t) A_{x}^{'} (t) = | | c_{ij}^{x} (t) | |_{i, j = 1, 2, 3}$ , where $c_{11}^{x} (t) = c_{12}^{x} (t) = c_{22}^{x} (t) = c_{31}^{x} (t) = 0, c_{21}^{x} (t) = c_{32}^{x} (t) = 1$ and $\begin{matrix} c_{13}^{x} (t) = \frac{[x^{(4)} (t) x^{(2)} (t) x^{(3)} (t)]}{[x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)]} \\ = \frac{[x^{(4)} (t) x^{(2)} (t) x^{(3)} (t)] [x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)]}{[x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)] [x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)]}, \end{matrix}$ (3) $\begin{matrix} c_{23}^{x} (t) = \frac{[x^{(1)} (t) x^{(4)} (t) x^{(3)} (t)]}{[x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)]} \\ = \frac{[x^{(1)} (t) x^{(4)} (t) x^{(3)} (t)] [x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)]}{[x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)] [x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)]}, \end{matrix}$ (4) $\begin{matrix} c_{33}^{x} (t) = \frac{[x^{(1)} (t) x^{(2)} (t) x^{(4)} (t)]}{[x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)]} \\ = \frac{[x^{(1)} (t) x^{(2)} (t) x^{(4)} (t)] [x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)]}{[x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)] [x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)]} \end{matrix}$ (5) for all t ∈ I.

Lemma 1. Let y₁, y₂, y₃, z₁, z₂, z₃ be column-vectors in $ℝ^{3}$ , where k, l = 1, 2, 3. Then $[y_{1} y_{2} y_{3}] [z_{1} z_{2} z_{3}] = det {∥ < y_{k}, z_{l} > ∥}_{k, l = 1, 2, 3} .$ (6)Proof. Let ∥y₁y₂y₃∥ and ∥z₁z₂z₃∥ be matrices with the column-vectors y₁, y₂, y₃ and z₁, z₂, z₃, respectively. Then following matrix equality ${‖ y_{1} y_{2} y_{3} ‖}^{*} ‖ z_{1} z_{2} z_{3} ‖ = {‖ 〈 y_{i}, z_{j} 〉 ‖}_{i, j = 1, 2, 3}$ is obvious, where ∥y₁y₂y₃∥ ^* is the transpose matrix of ∥y₁y₂y₃∥. This equality implies the following equality for determinants $\begin{matrix} [y_{1} y_{2} y_{3}] [z_{1} z_{2} z_{3}] = det ∥ y_{1} y_{2} y_{3} ∥ det ∥ z_{1} z_{2} z_{3} ∥ \\ = det {∥ < y_{i}, z_{j} > ∥}_{i, j = 1, 2, 3} . \\ □ \end{matrix}$ Applying Lemma 1 to vectors x⁽¹⁾ (t), x⁽²⁾ (t), x⁽³⁾ (t), x⁽⁴⁾ (t), we obtain the following equalities: ${[x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)]}^{2} =$ (7) $\begin{matrix} = \det | | < x^{(k)} (t), x^{(l)} (t) > | |_{k = 1, 2, 3; l = 1, 2, 3}, \\ {[x^{(4)} (t) x^{(2)} (t) x^{(3)} (t)]}^{2} = \end{matrix}$ (8) $\begin{matrix} = \det | | < x^{(k)} (t), x^{(l)} (t) > | |_{k = 4, 2, 3; l = 1, 2, 3}, \\ {[x^{(1)} (t) x^{(4)} (t) x^{(3)} (t)]}^{2} = \end{matrix}$ (9) $\begin{matrix} = \det | | < x^{(k)} (t), x^{(l)} (t) > | |_{k = 1, 4, 3; l = 1, 2, 3}, \\ {[x^{(1)} (t) x^{(2)} (t) x^{(4)} (t)]}^{2} = \end{matrix}$ (10) $= \det | | < x^{(k)} (t), x^{(l)} (t) > | |_{k = 1, 2, 4; l = 1, 2, 3},$ Using equalites (2) , (7 - 10) and (3 -5), we have equalities $c_{i j}^{x} (t) = c_{i j}^{y} (t)$ for all t ∈ I and i, j = 1, 2, 3. Therefore $A_{x}^{- 1} (t) {A^{'}}_{x} (t) = A_{y}^{- 1} (t) {A^{'}}_{y} (t)$ . Then we have $\begin{matrix} (A_{y} A_{x}^{- 1})^{'} & = A_{y}^{'} A_{x}^{- 1} + A_{y} (A_{x}^{- 1})^{'} \\ = A_{y}^{'} A_{x}^{- 1} + A_{y} (- A_{x}^{- 1} A_{x}^{'} A_{x}^{- 1}) \\ = A_{y} (A_{y}^{- 1} A_{y}^{'} - A_{x}^{- 1} A_{x}^{'}) A_{x}^{- 1} = 0 . \end{matrix}$ Thus $A_{y} (t) A_{x}^{- 1} (t)$ does not depend on t. Put $g = A_{y} (t) A_{x}^{- 1} (t)$ . Since detA_x (t) ≠0 and detA_y (t) ≠0 we have detg ≠ 0 and A_y (t) = gA_x (t). Then $y^{'} (t) = g x^{'} (t)$ . We prove that g ∈ O (3). Let $A_{x}^{★} (t)$ be the transpose matrix of A_x (t). We have $A_{x}^{★} (t) A_{x} (t) = | | < x^{(k)} (t), x^{(l)} (t) > | | = | | < y^{(k)} (t), y^{(l)} (t) > | | = A_{y}^{★} (t) A_{y} (t)$ . From this equality and A_y (t) = gA_x (t), we obtain g^★g = e, where e is the unit matrix. Thus g ∈ O (3). From y′ (t) = gx′ (t) we have y (t) = gx (t) + b for some $b \in ℝ^{3}$ .

Thus we have $x \overset{E (3)}{\sim} y$ . Using Theorem 1, we obtain $α \overset{E (3)}{\sim} β$ .

(ii). Similarly, using the claim (ii) of Theorem 1, we obtain a proof of the claim (ii) of our theorem.□

Remark 2. We note that the system (1) is the generating system of the differential field of all E (3)-invariant rational functions of a path in $ℝ^{3}$ (see [13, p.109]). The system T (α) , < x⁽¹⁾ (t) , x⁽¹⁾ (t) > , < x⁽²⁾ (t) , x⁽²⁾ (t) > , < x⁽³⁾ (t) , x⁽³⁾ (t) > is called the complete system of E (3)-invariants of a fuzzy curve α.

Proposition 8. Let x (t) be a path in $ℝ^{3}$ such that [x⁽¹⁾ (t) x⁽²⁾ (t) x⁽³⁾ (t)] ≠0 for all t ∈ I. Then <x⁽¹⁾ (t) , x⁽¹⁾ (t) > ≠0, |x⁽¹⁾ (t) × x⁽²⁾ (t) | ≠ 0 and det|| < x^(k) (t) , x^(l) (t) > ||_k,l=1,2 ≠ 0 for all t ∈ I.

Proof. By equality (7), [x⁽¹⁾ (t) x⁽²⁾ (t) x⁽³⁾ (t)] ≠0 implies det|| < x^(k) (t) , x^(l) (t) > ||_k,l=1,2 ≠ 0. The inequality [x⁽¹⁾ (t) x⁽²⁾ (t) x⁽³⁾ (t)] ≠0 means that vectors x⁽¹⁾ (t) , x⁽²⁾ (t) , x⁽³⁾ (t) are linearly independent for all t ∈ I. This implies <x⁽¹⁾ (t) , x⁽¹⁾ (t) > ≠0 and |x⁽¹⁾ (t) × x⁽²⁾ (t) | ≠ 0 for all t ∈ I.□

Proposition 9. Let x (t) be a path in $ℝ^{3}$ such that [x⁽¹⁾ (t) x⁽²⁾ (t) x⁽³⁾ (t)] ≠0 for all t ∈ I and κ [x] (t) is the curvature of a path x (t) (see [9], p. 192). Then $\begin{matrix} κ^{2} [x] (t) = \\ = \frac{< x^{(2)} (t), x^{(2)} (t) > < x^{(1)} (t), x^{(1)} (t) >}{< x^{(1)} (t), x^{(1)} (t) >^{3}} \\ - \frac{< x^{(1)} (t), x^{(2)} (t) >^{2}}{< x^{(1)} (t), x^{(1)} (t) >^{3}} . \end{matrix}$ (11)Proof. By [9, Theorem 8.8], $κ [x] (t) = \frac{| x^{(1)} (t) \times x^{(2)} (t) |}{< x^{(1)} (t), x^{(1)} (t) >^{\frac{3}{2}}}$ . Using this equality and the following equality ([9], p. 183, (8.2)) $\begin{matrix} {| x^{(1)} (t) \times x^{(2)} (t) |}^{2} = \\ = < x^{(2)} (t), x^{(2)} (t) > < x^{(1)} (t), x^{(1)} (t) > \\ - < x^{(1)} (t), x^{(2)} (t) >^{2}, \end{matrix}$ (12) we obtain the equality (11).□

Proposition 10. Let x (t) be a path in $ℝ^{3}$ such that [x⁽¹⁾ (t) x⁽²⁾ (t) x⁽³⁾ (t)] ≠0 for all t ∈ I and τ [x] (t) is the torsion of a path x (t) (see [9], p. 192). Then $τ^{2} [x] (t) =$ (13) $= \frac{\det | | < x^{(k)} (t), x^{(l)} (t) > | |_{k, l = 1, 2, 3}}{M^{2}},$ (14) where $\begin{matrix} M = < x^{(2)} (t), x^{(2)} (t) > < x^{(1)} (t), x^{(1)} (t) > \\ - < x^{(1)} (t), x^{(2)} (t) >^{2} . \end{matrix}$ Proof. By [9, Theorem 8.8], $τ [x] (t) = \frac{x^{(1)} (t) \times x^{(2)} (t) \cdot x^{(3)} (t)}{{| x^{(1)} (t) \times x^{(2)} (t) |}^{2}}$ . Using this equality, equality (12), equality (7) and the following equality $\begin{matrix} (x^{(1)} (t) \times x^{(2)} (t) \cdot x^{(3)} (t))^{2} = \\ = {[x^{(1)} (t) x^{(2)} (t) x^{(3)} (t)]}^{2} \\ = \det | | < x^{(k)} (t), x^{(l)} (t) > | |_{k, l = 1, 2, 3}, \end{matrix}$ we obtain equality (13).□

Now we give conditions of E (3)-equivalence of fuzzy curves in terms of the curvature and the torsion.

Theorem 3. Let α, β be non-singular fuzzy curves in $ℝ^{3}$ and (μ, x) ∈ Ip (α) , (ν, y) ∈ Ip (β). Then:

for T (α) = T (β) ≠ (- ∞ , + ∞) and T (α) = T (β) ≠ (+ ∞ , - ∞), $α \overset{E (3)}{\sim} β$ if and only if $\begin{matrix} < x^{(1)} (s), x^{(1)} (s) > = < y^{(1)} (s), y^{(1)} (s) >, \\ κ^{2} [x] (s) = κ^{2} [y] (s), τ^{2} [x] (s) = τ^{2} [y] (s) \end{matrix}$ (15) for all s ∈ D (α);

for T (α) = T (β) = (- ∞ , + ∞) or T (α) = T (β) = (+ ∞ , - ∞), $α \overset{E (3)}{\sim} β$ if and only if there exists c ∈ (- ∞ , + ∞) such that <x⁽¹⁾ (s) , x⁽¹⁾ (s)> = < y⁽¹⁾ (s + c) , y⁽¹⁾ (s + c) >, κ² [x] (s) = κ² [y] (s + c), τ² [x] (s) = τ² [y] (s + c) for all s ∈ D (α).

Proof. (i). Let $α \overset{E (3)}{\sim} β$ . By Theorem 2, conditions (1) are valid. From conditions (1), using Propositions 9 and 10, we obtain equalities (14). Conversely, suppose that conditions (14) are valid. We prove that $x \overset{E (3)}{\sim} y$ . Since α, β are non-singular fuzzy curves in $ℝ^{3}$ and (μ, x) ∈ Ip (α) , (ν, y) ∈ Ip (β), we have [x⁽¹⁾ (s) x⁽²⁾ (s) x⁽³⁾ (s)] ≠0 and [y⁽¹⁾ (s) y⁽²⁾ (s) y⁽³⁾ (s)] ≠0 for all s ∈ D (α). This inequalities imply <x⁽¹⁾ (s) , x⁽¹⁾ (s) > ≠0, <y⁽¹⁾ (s) , y⁽¹⁾ (s) > ≠0, |x⁽¹⁾ (s) × x⁽²⁾ (s) | ≠ 0 and |y⁽¹⁾ (s) × y⁽²⁾ (s) | ≠ 0. The equality $< x^{(1)}, x^{(1)} > = < y^{(1)}, y^{(1)} >$ in (14) implies $\begin{matrix} < x^{(1)}, x^{(2)} > = \frac{1}{2} < x^{(1)}, x^{(1)} >^{(1)} \\ = \frac{1}{2} < y^{(1)}, y^{(1)} >^{(1)} = < y^{(1)}, y^{(2)} > . \end{matrix}$ By (11), equalities <x⁽¹⁾, x⁽¹⁾ > = < y⁽¹⁾, y⁽¹⁾ > , <x⁽¹⁾, x⁽²⁾> = < y⁽¹⁾, y⁽²⁾ > and κ² [x] (s) = κ² [y] (s) imply <x⁽²⁾, x⁽²⁾> = < y⁽²⁾, y⁽²⁾ >, $< x^{(2)}, x^{(3)} > = \frac{1}{2} < x^{(2)}, x^{(2)} >^{(1)} = \frac{1}{2} < y^{(2)}, y^{(2)} >^{(1)} = < y^{(2)}, y^{(3)} >$ . By (13), equalities <x⁽¹⁾, x⁽¹⁾ > = < y⁽¹⁾, y⁽¹⁾ > , < x⁽¹⁾, x⁽²⁾ > = < y⁽¹⁾, y⁽²⁾ > , <x⁽²⁾, x⁽²⁾ > = < y⁽²⁾, y⁽²⁾ > , <x⁽²⁾, x⁽³⁾> = < y⁽²⁾, y⁽³⁾ > and τ² [x] (s) = τ² [y] (s) imply <x⁽³⁾, x⁽³⁾ > = < y⁽³⁾, y⁽³⁾ > . Now, using Theorem 2, we obtain $α \overset{E (3)}{\sim} β$ .(ii). A proof of the assertion (ii) of our theorem is similar.□

Theorem 4. Let u (t) and v (t) be non-singular I-paths (that is [u⁽¹⁾ (t) u⁽²⁾ (t) u⁽³⁾ (t)] ≠ 0, [v^(t)v⁽²⁾ (t) v⁽³⁾ (t)] ≠0 for all t ∈ I), μ_u (t) and μ_v (t) are as in Example 4, α is the fuzzy curve containing (μ_u (t) , u (t)) and β is the fuzzy curve containing (μ_v (t) , v (t)). Assume that (ν_x (s) , x (s)) ∈ Ip (α), (ν_y (s) , y (s)) ∈ Ip (β) and T (α) = T (β). Then:

for T (α) = T (β) ≠ (-∞, + ∞) and T (α) = T (β) ≠ (+ ∞ , - ∞), $α \overset{E (3)}{\sim} β$ if and only if κ² [x] (s) = κ² [y] (s) , τ² [x] (s) = τ² [y] (s) for all s ∈ D (α);

for T (α) = T (β) = (- ∞ , + ∞) or T (α) = T (β) = (+ ∞ , - ∞), $α \overset{E (3)}{\sim} β$ if and only if there exists c ∈ (- ∞ , + ∞) such that κ² [x] (s) = κ² [y] (s + c), τ² [x] (s) = τ² [y] (s + c) for all s ∈ D (α).

Proof. (i). Prove that α is a regular fuzzy curve that is (μ_u (t)) ′ ≠ 0 for all t ∈ I. The condition [u⁽¹⁾ (t) u⁽²⁾ (t) u⁽³⁾ (t)] ≠ 0 for all t ∈ I implies <u⁽¹⁾ (t) , u⁽¹⁾ (t) > ≠0 for all t ∈ I. Hence $μ_{u}^{'} (t) = (\int_{a}^{t} < u^{(1)} (t), u^{(1)} (t) >^{\frac{1}{2}})^{'} = < u^{(1)} (t), u^{(1)} (t) >$ >0 for all t ∈ I. This means that α is a regular fuzzy curve. A proof of the regularity of β is similar. Since [u⁽¹⁾ (t) u⁽²⁾ (t) u⁽³⁾ (t)] ≠ 0 and [v^(t)v⁽²⁾ (t) v⁽³⁾ (t)] ≠ 0 for all t ∈ I, α and β are non-singular fuzzy curves. By Proposition 3, we have $\begin{matrix} ν_{x}^{'} (s) & = (\int_{a}^{s} < x^{(1)} (t), x^{(1)} (t) >^{\frac{1}{2}})^{'} \\ = < x^{(1)} (s), x^{(1)} (s) > = 1 \end{matrix}$ and $\begin{matrix} ν_{y}^{'} (s) & = (\int_{a}^{s} < y^{(1)} (t), y^{(1)} (t) >^{\frac{1}{2}})^{'} \\ = < y^{(1)} (s), y^{(1)} (s) > = 1 \end{matrix}$ for all s ∈ D (α). Since <x⁽¹⁾ (s) , x⁽¹⁾ (s) > = < y⁽¹⁾ (s) , y⁽¹⁾ (s) > =1, the assertion (i) of our theorem follows from Theorem 3(i).

(ii). Similarly, the assertion (iv) follows from Theorem 3(ii).□

Remark 3. Theorem 4 is a global variant of the classical theorem on E (3)-equivalence of curves in $ℝ^{3}$ . The system <x⁽¹⁾ (s) , x⁽¹⁾ (s) > , κ² [x] (s) , τ² [x] (s) is called the system of curvatures of a fuzzy curve (μ_x (s) , x (s)).

5. Correlations between elements of the complete system of invariants of a fuzzy curve

For a non-singular path x (t), the matrix $A_{x}^{★} (t) A_{x} (t)$ is positive definite. This gives some correlations between

$\begin{matrix} < x^{'} (t), x^{'} (t) >, < x^{(2)} (t), x^{(2)} (t) >, \\ < x^{(3)} (t), x^{(3)} (t) > \end{matrix}$ and their derivatives. We prove that an arbitrary correlation between $\begin{matrix} T (α), < x^{'} (t), x^{'} (t) > \\ < x^{(2)} (t), x^{(2)} (t) >, < x^{(3)} (t), x^{(3)} (t) > \end{matrix}$ and their derivatives is a consequence of the same correlations.

Let (a, b) be a pair of numbers such that a ≠ b and -∞ ≤ a, b ≤ + ∞. We define the interval D (a, b) for every pair (a, b) as follows:

in the case a < b, a≠ - ∞, put D (a, b) = (0, b - a);

in the case a = -∞, b≠ + ∞, put D (a, b) = (- ∞ , 0);

in the case a = -∞, b =+ ∞, put D (a, b) = (- ∞ , + ∞);

in the case a > b, b≠ - ∞, put D (a, b) = (- (a - b) , 0);

in the case b = -∞, a≠ + ∞, put D (a, b) = (0, + ∞);

in the case a =+ ∞, b = -∞, put D (a, b) = (- ∞ , + ∞).

Theorem 5. Let (a, b) be a pair such that a ≠ b and -∞ ≤ a, b ≤ + ∞. Assume that f₁ (s) , f₂ (s) , f₃ (s) are C^∞-functions on I = D (a, b) such that the matrix ||f_kl (s) ||_k,l=1,2,3 is positive definite for all s ∈ D (a, b), where f₁₁ (s) = f₁ (s) , f₂₂ (s) = f₂ (s) , f₃₃ (s) = f₃ (s) , $f_{12} (s) = f_{21} (s) = \frac{1}{2} f_{11}^{(1)} (s),$ $f_{23} (s) = f_{32} (s) = \frac{1}{2} f_{22}^{(1)} (s)$ . Then there exists a non-singular fuzzy curve α such that T (α) = (a, b) and <x^(j) (s) , x^(j) (s) > = f_j (s), j = 1, 2, 3, for some x ∈ Ip (α) and all s ∈ D (a, b).

Proof. First we prove an existence of such path x.

Lemma 2. Let B (t) = ||b_kl (t) ||_k,l=1,2,3 and Q (t) = ||q_kl (t) ||_k,l=1,2,3 be matrix C^∞functions on I such that:

Q^* (t) = Q (t), where Q^* (t) is the transpose matrix of Q (t);

Q′ (t) = B^★ (t) Q (t) + Q (t) B (t), where Q′ (t) is the derivative of Q (t);

the matrix Q (t) is positive definite for some t₀ ∈ I.

Then there exists 3 × 3-matrix C^∞-function A (t) on I such that:

A (t) = ||y (t) y⁽¹⁾ (t) y⁽²⁾ (t) || for some I-path y (t) in $ℝ^{3}$ ;

detA (t₀) ≠0, where t₀ is as in the condition ρ₃;

A′ (t) = A (t) B (t);

A^★ (t) A (t) = Q (t).

Proof. From the theory of linear differential equations, we obtain that there exists a solution of the equation (σ₃) such that detA (t₀) ≠0, where t₀ is as in the condition ρ₃. Then det (A^★ (t₀) A (t₀)) ≠0. The condition (ρ₃) implies that detQ (t₀) ≠0. Let $GL (3, ℝ)$ be the group of all nondegenerate real 3 × 3-matrices. By det (A^★ (t₀) A (t₀)) ≠0, the condition (ρ₁) and the condition (ρ₃), there exists $g \in GL (3, ℝ)$ such that (g^★) ^-1 (A^★ (t₀)) ^-1Q (t₀) A^-1 (t₀) g^-1 = E, where E is the unit matrix. Then we have A^★ (t₀) g^★gA (t₀) = Q (t₀). The matrix function gA (t) is also a solution of the equation (σ₃). The matrix function H (t) = A^★ (t) g^★gA (t) satisfies the following conditions: $H^{★} (t) = H (t), H' (t) = B^{★} (t) H (t) + H (t) B (t)$ for all t ∈ I. But these conditions are also valid for the function Q (t). This and H (t₀) = Q (t₀), in view of the theorem on existence and uniqieness of a solution of linear differential equations, implies that H (t) = Q (t) for all t ∈ I. The lemma is proved.□

Put $\begin{matrix} f_{14} (s) = f_{41} (s) = f_{13}^{'} (s) - f_{23} (s), \\ f_{24} (s) = f_{42} (s) = f_{23}^{'} (s) - f_{33} (s), \\ f_{34} (s) = f_{43} (s) = \frac{1}{2} f_{33}^{'} (s) . \end{matrix}$ (16) We consider matrices Q (s) = ||f_kl (s) ||_{k=1,2,3;l=1,2,3}, Q₁ (s) = ||f_kl (s) ||_{k=1,2,3;l=4,2,3}, Q₂ (s) = ||f_kl (s) ||_{k=1,2,3;l=1,4,3}, Q₃ (s) = ||f_kl (s) ||_{k=1,2,3;l=1,2,4}. By the supposition of our theorem, detQ (s) ≠0 for all s ∈ D (a, b). Put B (s) = ||b_kl (s) ||_k,l=1,2,3, where b₁₁ (s) = b₁₂ (s) = b₂₂ (s) = b₃₁ (s) =0, b₂₁ (s) = b₃₂ (s) =1 for all s ∈ D (a, b) and $b_{j 3} (s) = \frac{{detQ}_{j} (s)}{detQ (s)}, j = 1, 2, 3$ . It is obvious that the system (b₁₃ (s) , b₂₃ (s) , b₃₃ (s)) is the unique solution of the following system of algebraic linear equations: $f_{k 1} (t) b_{13} + f_{k 2} (t) b_{23} + f_{k 3} (t) b_{33} = f_{k 4} (t),$ (17) for k = 1, 2, 3. It is easy to see that equalities (15 - 17) imply the equality Q′ (t) = B^★ (t) Q (t) + Q (t) B (t). Thus the matrices Q (t) and B (t) satisfy the conditions of Lemma 2. By Lemma 2, there exists a matrix A (t) = ||y (t) y⁽¹⁾y⁽²⁾ (t) || such that A′ (t) = A (t) B (t) , A^★ (t) A (t) = Q (t).Then, using A^★ (t) A (t) = || < y⁽ⁱ⁾ (t) , y^(j) (t) > ||, we obtain <y⁽ⁱ⁾ (t) , y^(j) (t) > = f_i+1j+1 (t) for all i, j = 0, 1, 2. Put $x (t) = \int_{t_{0}}^{t} y (t) dt$ . Then <x⁽ⁱ⁾ (t) , x⁽ⁱ⁾ (t) > = f_ii (t), i = 1, 2, 3. Since [x⁽¹⁾ (t) x⁽²⁾ (t) x⁽³⁾ (t)] = detA^★ (t) A (t) = detQ (t) ≠0 for all t ∈ I, the path x(t) is non-singular. An existence of a path x is proved.

Now we prove an existence of a membership function μ (t) with the domain I = D (a, b) and of the fuzzy type T (μ) = (a, b). Let (a, b) be a pair of numbers such that a ≠ b and -∞ ≤ a, b ≤ + ∞. In the case a< b, a ≠ - ∞, put μ (t) = t + a for all t ∈ D (a, b) = (0, b - a). In the case a = -∞ , b ≠ + ∞, put μ (t) = t + b for all t ∈ D (a, b) = (- ∞ , 0). In the case a = -∞ , b = + ∞, put μ (t) = t for all t ∈ D (a, b) = (- ∞ , + ∞). In the case a> b, b ≠ - ∞, put μ (t) = - t + b for all t ∈ D (a, b) = (- (a - b) , 0). In the case b = -∞ , a ≠ + ∞, put μ (t) = - t + a for all t ∈ D (a, b) = (0, + ∞). In the case a =+ ∞ , b = - ∞, put μ (t) = - t for all t ∈ D (a, b) = (- ∞ , + ∞). Then, according to Propositions 3 and 4, the fuzzy path (μ (t) , x (t)) is an invariant parametrization. It satisfies the conditions of our theorem. Hence the proof is completed.□

Theorem 6. Let (a, b) be a pair such that a ≠ b and -∞ ≤ a, b ≤ + ∞. Assume that g₁ (s) , g₂ (s) , g₃ (s) are C^∞-functions on I = D (a, b) such that: g₁ (s) >0, g₂ (s) >0, g₃ (s) >0 for all s ∈ D (a, b). Then there exists a non-singular fuzzy curve α such that T (α) = (a, b) and <x⁽¹⁾ (s) , x⁽¹⁾ (s) > = g₁ (s), κ² [x] (s) = g₂ (s), τ² [x] (s) = g₃ (s) for all s ∈ D (a, b).

Proof. We define the function f_ij (s) _i,j=1,2,3 as follows. Put f₁₁ (s) = g₁ (s) , f₁₂ (s) = $f_{21} (s) = \frac{1}{2} g_{1}^{'} (s),$ $f_{22} (s) = \frac{g_{1}^{3} (s) g_{2} (s) + \frac{1}{4} (g_{1}^{'} (s))^{2}}{g_{1} (s)},$ (18) $f_{13} (s) = f_{31} (s) = f_{12}^{'}$ $(s) - f_{22} (s), f_{23} (s) = f_{32} (s) = \frac{1}{2} f_{22}^{'} (s)$ . These equalities imply that $\frac{f_{22} (s) f_{11} (s) - f_{12}^{2} (s)}{g_{1}^{3} (s)} = g_{2} (s) .$ Since g₂ (s) >0, we obtain that $f_{22} (s) f_{11} (s) - f_{12}^{2} (s) > 0$ for all s ∈ D (a, b). We define f₃₃ (s) as the solution of the following equation $\frac{\det ∥ f_{ij} (s) ∥_{i, j = 1, 2, 3}}{(f_{22} (s) f_{11} (s) - f_{12}^{2} (s))^{2}} = g_{3} (s) .$ (19) We have $\begin{matrix} f_{33} (s) & = \frac{g_{3} (s) (f_{22} (s) f_{11} (s) - f_{12}^{2} (s))}{(f_{22} (s) f_{11} (s) - f_{12}^{2} (s))^{2}} \\ - \frac{f_{31} (s) (f_{12} (s) f_{23} (s) - f_{13} (s) f_{22} (s))}{(f_{22} (s) f_{11} (s) - f_{12}^{2} (s))^{2}} \\ + \frac{f_{32} (s) (f_{11} (s) f_{23} (s) - f_{13} (s) f_{21} (s))}{(f_{22} (s) f_{11} (s) - f_{12}^{2} (s))^{2}} . \end{matrix}$ (20) Since g₁ (s) >0, g₂ (s) >0, g₃ (s) >0, we have $f_{11} (s) > 0, f_{11} (s) f_{22} (s) - f_{12}^{2} (s) > 0$ for all s ∈ D (a, b). From the equation (19), we obtain that det ∥f_ij (s)∥_i,j=1,2,3 > 0. These inequalities means that the the matrix ∥f_ij (s)∥_i,j=1,2,3 is positive definite. Hence, according to Theorem 5, there exists a fuzzy curve α such that T (α) = (a, b) and $< x^{(1)} (s), x^{(1)} (s) > = f_{jj} (s), j = 1, 2, 3,$ (21) where f₁₁ (s) = g₁ (s), f₂₂ (s) has the form (18) and f₃₃ (s) has the form (20). Using equalities (21), Propositions 9 and 10, we obtain equalities $\begin{matrix} < x^{(1)} (s), x^{(1)} (s) > = g_{1} (s), \\ κ^{2} [x] (s) = g_{2} (s), \\ τ^{2} [x] (s) = g_{3} (s) . \\ □ \end{matrix}$ Remark 4. In the case g₁ (s) =1 for all s ∈ D (a, b), Theorem 6 is a global variant of the classical theorem on the correlation between functions κ² [x] (s) and τ² [x] (s) of curves in $ℝ^{3}$ .

6. Conclusion

Fuzzy theory has many applications. One of them is decision making. The decision making can be regarded as the precess of choosing the optimal alternatives from the feasible alternatives. Also, multi-attribute decision making is a major part of decision making, which in turn place an important role in management science, operations research, industrial engineering and so on. Some papers on this field have important solution which solves the real world problems that under considerations practically. In [11], the authors elaborate some reviews to decision making methods based on the classes of SRF-sets and SFR-sets of hybrid soft models. Also, in [36], in order to solve some complicated problems, two novel decision making models that are stated in turns of novel and flexible generalized IF rough set models are established. By proposing two models that benefit from some novel fuzzy rough set models, multi-attribute decision making problems are solved in [12].

Novel and flexible fuzzy rough set models are also considered in [35]. For the purpose of defining these models, it is employed a fuzzy implication operator and a triangular norm. With these adoptable tools, four kinds of fuzzy β-coverings based fuzzy rough set models are designed. Finally, two novel methodologies to solve multi-attribute decision making problems evolution of fuzzy information are proposed, which are reasonable. Also, in [34], covering based multi granulation fuzzy rough set models from fuzzy β-neighbourhood is introduced. By using different implicators and t-norms, the corresponding axiomatic characterization of covering based optimistic, pessimistic and variable precision multi granulation fuzzy rough set models are investigated. Based on the theoretical analysis for the covering based multi granulation fuzzy rough set models, solutions to problems in multi-attribute group decision making by means of two kinds of decision making methods are respectively established.

Our work may have contribution to these practices in the following way. In the above examples, a fuzzy curve can be taken in cases where the continuity of the sample space data is required. In decision making problems, fuzzy invariants could be used to investigate the results of two different methods.

Footnotes

Acknowledgements

The authors would like to thank the anonymous reviewers for their helpful and constructive comments that greatly contributed to improving the final version of the paper. They would also like to thank the Editors for their generous comments and support during the review process. Finally, the second author would like to thank TUBITAK (The Scientific and Technological Research Council of Turkey) for their financial supports during his doctorate studies.

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