In order to facilitate the wavelength measurement, the coordinates of the initial point are set as the origin, with the initial horizontal fiber direction as the x-axis, and the vertical direction as the y-axis. After force is applied in the plane, the tangent of the fiber at the initial position forms a 45° angle with the horizontal direction. FBG sensors were used for the measurements, with a spacing of 0.6 m between them (see Figure 3). The wavelength of the signal at each sensor position was first measured when the fiber is horizontal, and then measured again after the fiber is subjected to an external force. The fiber Bragg grating (FBG) sensors employed in this study had a grating length of 10 mm, providing sufficient resolution for detecting wavelength shifts while ensuring stable reflection characteristics.

In this setup, the optical fiber is attached to a flexible polycarbonate beam serving as the elastic substrate, while mechanical loading is applied through a bending device with adjustable weights. When the loading device induces bending in the substrate, the FBG region undergoes stretching or compression along with the fiber, thereby enabling the detection of local axial strain at that specific location.

The optical fiber sensor demodulation system extracts strain information, which is then used to indirectly determine the curvature. Given the relationship between curvature and strain, the relationship can be expressed as Equation 1 [19]: ε = h 2 · k (1) where h denotes the distance between each test point and the neutral axis, and k represents curvature. Based on the relationship between the central wavelength shift and the strain [19], the curvature can be expressed approximately as follows: k = c λ − λ 0 λ 0 (2) where λ 0 is the wavelength measured in the initial state of the horizontal fiber. λ is the wavelength measured after the fiber is subjected to an external force. c is a constant (4,200 m^-1) that represents a scale factor obtained via experimental calibration. In this experiment, the wavelength values before and after loading were measured for two different initial states, and the results are presented in Table 2. The wavelength data in Table 2 were obtained using a Yokogawa AQ6370D spectrum analyzer, which has a resolution bandwidth of 0.02 nm.

Cubic spline interpolation [20–22] is a piecewise interpolation method. It divides the interpolation interval into subintervals and generates a polynomial of degree at most three for each subinterval. These polynomials are constructed to satisfy continuity of the function itself as well as its first and second derivatives, ensuring smoothness and continuity at every point.

Splitting the fiber into n segments on average, i.e., there exist n + 1 splitting points satisfying S 1 < S 2 < … < S n + 1 , where S i is the distance of a point on the fiber from the starting point, then the curvature satisfies the functional relationship given in Equation 3 with the length of the fiber on the corresponding interval S i , S i + 1 , i = 1 , 2 , … , n : k S = a i + b i × S − S i + c i × S − S i 2 + d i × S − S i 3 (3) where: a i , b i , c i d i are 4 coefficients to be determined. The curve satisfies the following relationship at the i th split point: k i S i = C i k i S i + 1 = C i + 1 k i ′ S i + 1 = k i + 1 ′ S i + 1 k i ″ S i + 1 = k i + 1 ″ S i + 1 (4)

That is, the curvature satisfies the interpolating continuity condition, as well as the continuity of the first and second derivatives. This ensures that the fitted curvature curve is smooth. Equation 4, together with the boundary constraints at the beginning and end of the curve, can be used to determine the four coefficients. Here the natural boundary conditions are used as Equation 5: k ″ 0 = 0 k ″ S t o t a l = 0 (5)

As the experiments are conducted on the surface of a flexible substrate for monitoring, this study focuses on the deformation of the optical fiber caused by a force applied in a single plane, without considering torsional changes of the fiber in three-dimensional space. It is suitable for low-dimensional application scenarios, such as flexible substrate surface monitoring and simplified preliminary validation, where torsional effects are minimal.

Based on the derived curvature and the initial condition that the tangent line at the initial position of the fiber forms a 45 ° angle with the horizontal after planar stress is applied, the tangent angle recursion algorithm [23–25] is employed to obtain the coordinates of any point on the fiber deformation curve, thereby enabling the reconstruction of the curve.

For curves, the arc between two points on a curve can be approximated as a segment of a microcircular arc, provided the points are sufficiently close together. Let the curvature of the curve segment between the starting point O n and the end point O n + 1 be k n and k n + 1 , respectively. The corresponding coordinates of these points are x n , y n and x n + 1 , y n + 1 , respectively. β n and β n + 1 are the angles between the tangent lines and the x-direction of the two places O n and O n + 1 , respectively. The angle of the center of the circle corresponding to the arc can be expressed as Equation 6: Δ α n = β n + 1 − β n (6)

The chord length l n of an arc can be expressed as Equation 7: l n = 2 ⁡ sin Δ α n 2 / k n = 2 ⁡ sin ⁡ c Δ α n 2 · Δ α n k n , k n ≠ 0 l n = Δ s , k n = 0 (7) where k n is the curvature corresponding to the arc. The coordinate increment Δ x n , Δ y n is calculated from the chord length using Equation 8: Δ x = l n · ⁡ cos α n − Δ α n / 2 Δ y = l n · ⁡ sin α n − Δ α n / 2 (8)

From the recurrence formula, the new coordinates can be expressed as Equation 9: x n + 1 = x n + Δ x y n + 1 = y n + Δ y (9)

According to the definition of curvature from Equation 10: β s = ∫ k s d s (10)

The Frenet-Serret formulas constitute a system of differential equations that characterize the position and orientation of a curve in three-dimensional space [26]. These equations describe the curve’s geometry in terms of its curvature and torsion. At any point on a curve in three-dimensional space, the Frenet-Serret formulas use the tangent, normal, and binormal vectors to define a local orthonormal coordinate system moving along the curve. The translation of this coordinate system is described by Equation 11: p i + 1 = − 1 − cos θ i / k i 0 sin θ i / k i T (11)

The new coordinate system is subsequently rotated around the axis θ i and ∆ φ i + 1 , resulting in two distinct rotation matrices as given in Equations 12, 13: T B i θ i = cos θ i 0 − sin θ i 0 0 1 0 0 sin θ i 0 cos θ i 0 0 0 0 1 (12) T T i Δ φ ω 1 = cos Δ φ i + 1 − sin Δ φ i + 1 0 0 sin Δ φ i + 1 cos Δ φ i + 1 0 0 0 0 1 0 0 0 0 1 (13)

Consequently, the constant transformation matrix enables the continuous construction of a moving coordinate system. At each step, the endpoint of each micro-segment is updated and connected, yielding the fitted curve.

After obtaining the wavelength information from each sensing point, the corresponding discrete curvature values from two tests were calculated using Equation 2, and the results are presented in Table 3.

Using the curvature values calculated at each sensing point and applying cubic spline interpolation, the curvature distributions of Test 1 and Test 2 along different arc lengths are shown in Figures 4a,b. These results demonstrate that cubic spline interpolation effectively fits the discrete curvature data measured by the fiber, with curvature values along the central axis in both Tests 1 and 2 ranging from 2 to 3. This provides a smooth and physically meaningful curvature distribution that serves as a reliable basis for subsequent curve reconstruction. Additionally, Figure 4b exhibits more pronounced curvature fluctuations compared to Figure 4a, suggesting that the initial state or loading conditions have a significant influence on the fiber’s deformation response.

In the initial state, the optical fiber is aligned along the horizontal direction (x-axis), with the vertical direction as the y-axis. After deformation under the force, the tangent at the fiber’s initial point forms a 45° angle with the horizontal axis. Therefore, the initial coordinate of the deformation curve is x 0 , y 0 = 0 , 0 , and β 0 = 45 ° .

By using the curvature values at the endpoints of each arc segment obtained through cubic spline interpolation and applying the tangent angle recursion algorithm, the coordinates of any point on the fiber deformation curve can be derived. This enables the reconstruction of the curve, as shown in Figures 5a,b. It can be seen that the reconstructed curve exhibits an approximately circular shape. In Figure 5a, because the applied force is relatively small, the deformation pattern of the optical fiber remains simple and symmetric, leading to a clearly visible overlapping region. In contrast, the overlapping region in Figure 5b is larger than that in Figure 5a, as the force applied in Test 2 exceeds that in Test 1. Moreover, it can be anticipated that with further increases in the applied force, a spiral structure would emerge in the middle of the curve in a three-dimensional scenario. The geometry of the curves is governed by the curvature distribution. Regions of high curvature indicate a greater change in direction over shorter distances, suggesting stronger external forces in these regions. Conversely, regions of low curvature are closer to straight lines, reflecting either smaller external forces or the inherent properties of the material.

The reconstructed curves are continuous and smooth, without sudden changes in direction. The smooth curve generated by cubic spline interpolation ensures the physical realism of the phenomenon. In optical fiber curve reconstruction, smoothness implies continuous force interactions and material responses along the fiber, preventing abrupt transitions and helping to avoid structural instabilities in practical applications.

The parameter c is a calibration constant obtained experimentally, which establishes the relationship between the measured wavelength shift and the corresponding curvature. Although parameter c remains constant within a given experiment, its value directly influences the magnitude of the calculated curvature. Therefore, to evaluate the robustness of the reconstruction method, a sensitivity analysis of parameter c is conducted. Figure 6a illustrates that the Test 1 curve exhibits a relatively mild slope, with the average curvature displaying an approximately linear increase as the parameter c increases. The sensitivity curve of Test 2 presents a lower slope than that of Test 1, and similarly follows a linear increasing pattern. When the parameter c lies within the range of 4000 , 4400 , the curvature increases by approximately Δ k = 0.3 m − 1 , indicating a low level of sensitivity.

Figure 6b distinctly reveals the influence of the parameter c on the mean curvature response of the optical fiber sensor. The horizontal axis represents the values of the parameter c (ranging from 4,000 to 4,400 in increments of 100), while the vertical axis distinguishes between the two initial conditions of Test 1 and Test 2. Additionally, Test 2 exhibits a generally darker hue, indicating that its mean curvature values are consistently higher than those of Test 1. This reflects the differing system responses under varying initial wavelength conditions. Both bands exhibit a progressive shift from lighter to darker tones with increasing parameter values. However, the more pronounced gradient observed in Test 2 indicates that the system with an initial wavelength of 1,540 nm exhibits enhanced sensitivity to parameter variations. Notably, both color bands are situated within the mid-tone region. This offers a valuable reference for parameter calibration in practical engineering applications. These results not only verify the critical role of the parameter c , but also provide a visual framework for the optimization of the sensor and the management of associated errors in subsequent stages.

Based on the plane curve equation y = x 3 + x 0 ≤ x ≤ 1 , points are sampled at appropriately spaced arc lengths. The curvature at each sampled point is then calculated. These curvature values serve as the foundation for reconstructing the plane curve via a tangent angle recursion algorithm. The reconstructed curves are compared with the originals to verify the feasibility of the method and to analyze the sources of discrepancies.