The Development of Log Aesthetic Patch and Its Projection onto the Plane

: In this work, we introduce a new type of surface called the Log Aesthetic Patch (LAP). This surface is an extension of the Coons surface patch, in which the four boundary curves are either planar or spatial Log Aesthetic Curves (LACs). To identify its versatility, we approximated the hyperbolic paraboloid to LAP using the information of lines of curvature (LoC). The outer part of the LoCs, which play a role as the boundary of the hyperbolic paraboloid, is replaced with LACs before constructing the LAP. Since LoCs are essential in shipbuilding for hot and cold bending processes, we investigated the LAP in terms of the LoC’s curvature, derivative of curvature, torsion, and Logarithmic Curvature Graph (LCG). The numerical results indicate that the LoCs for both surfaces possess monotonic curvatures. An advantage of LAP approximation over its original hyperbolic paraboloid is that the LoCs of LAP can be approximated to LACs, and hence the ﬁrst derivative of curvatures for LoCs are monotonic, whereas they are non-monotonic for the hyperbolic paraboloid. This conﬁrms that the LAP produced is indeed of high quality. Lastly, we project the LAP onto a plane using geodesic curvature to create strips that can be pasted together, mimicking hot and cold bending processes in the shipbuilding industry.


Introduction
The introduction of Bezier curves and surfaces representation was a significant breakthrough of Computer Aided Geometric Design (CAGD), which was later extended to B-spline representation [1,2]. Eventually, B-splines and NURBS became the de facto standard for computer graphics packages, computer-aided design (CAD), and computer-aided manufacturing (CAM) [3]. The complex form of curvature of these curves, which are not suitable for direct manufacturing, led to the introduction of a variety of efficient fairing algorithms to reduce the oscillation in the curvature profile of these curves [4][5][6]. However, Cornu spirals, or clothoids, are members of spiral curves that have monotonic curvature profiles by nature [7][8][9].
The Log Aesthetic Curve (LAC) is a type of curve that possesses a monotonic curvature profile, hence suiting the aesthetic design environment. The research on LACs has been active since Miura [10] introduced a linear Logarithmic Curvature Graph (LCG) as its fundamental equation. The LAC equation can be used to represent various spirals, e.g., clothoid, logarithmic spiral, circle involute, and Nielsen's spiral. LCG can also be used as a shape interrogation tool to investigate the characteristics of arbitrary curves [11]. Yoshida In 2021, Gobithaasan et al. [18] applied LACs to draw LA surfaces of revolution and LA swept surfaces, and the LoCs of these surfaces are indeed the LAC itself. This motivated us to design free-from surfaces by applying LACs or LASCs as boundary curves for the Coons patch. In this paper, we use LACs or LASCs to form a Coons-like patch, denoted as the Log Aesthetic Patch, or LAP in short. Next, we approximated the hyperbolic paraboloid using the LAP and analyzed the characteristics of LoCs on the resulting LAP. Finally, we employed an algorithm to project the LAP onto a plane for the fabrication mimicking the shipbuilding process.
The rest of the paper is arranged as follows. Sections 2-4 describe the literature review of this work; Section 2 introduces the LAC and LASC; Section 3 reviews the differential geometry of the surface and the development of the surface onto a plane; and Section 4 describes the fundamental equation of the Coons patch. Section 5 describes the development of the LAP and presents a numerical example with the development of a surface by mapping onto a plane before finally elaborating on the conclusion.

LAC and LASC
This section describes the general equations of the LAC and LASC.
Equation (1) represents the LCG as a linear function, where α is the slope of LCG, s is the arc length of the curve, ρ is the radius of curvature, and C 1 is a constant [12]. Equation (2) represents the LTG function, where β is the slope of LTG, s is the arc length of a curve, µ is the radius of torsion and C 2 is a constant [13]. If we differentiate and simplify Equations (1) and (2), we obtain: where Λ = e −C 1 and Ω = e −C 2 are shape parameters. Integrating Equations (3) and (4) yields: where ρ 0 is the initial radius of curvature and µ 0 is the initial radius of torsion. It is important to note that Yoshida et al. [13] set µ 0 = ν. Hence, the curvature and torsion of a curve are shown below: We can substitute Equation (3) to dθ ds = 1 ρ , where θ is the tangential angle [12], to obtain: Integrating Equation (9) yields: Let C(s) = (x(s), y(s), z(s)) be an arc length of the parameterized space curve; t(s) is the unit tangent vector, n(s) is the unit normal vector, and b(s) is the unit binormal vector. The Frenet-Serret formulas in terms of arc length parameterized can be represented as follows: Assume that ϕ(u) = ds du ; the Frenet-Serret formula in terms of the parameter u is defined as [28]: The LASC is a curve in three-dimensional space that can be drawn by applying curvature (Equation (7)) and torsion (Equation (8)) to Equation (11). Since the LAC is a curve in two-dimensional space, we can draw this curve by applying curvature (Equation (7)) and assigning torsion, τ(s) = 0, in Equation (11). The details on how the LAC and LASC can be drawn and controlled interactively are fully discussed in [12,13].

Lines of Curvature
Let a parametric surface R(u, v) = (x(u, v), y(u, v), z(u, v)); the first and second fundamental equation of the surface are [25,29]: where (17) represent the Gaussian curvature, mean curvature, and principle curvature, respectively.
By solving the following initial value problems numerically, the LoC on a surface can be computed as [25]: Mathematics 2022, 10, 160 where the non-zero factor (η and µ) can be obtained by normalizing the first fundamental equation, as shown below: The sign of dt ds can be determined from Equation (20) [29]. LoCs can be generated by solving initial value problems (Equations (18) and (19)) using various types of numerical approaches, such as the Runge-Kutta method.
Let c(s) = R(u(s), v(s)) represent a curve on a surface; the derivatives of the curve on the surface can be obtained using the chain rule as follows [25]: In 2014, Joo et al. [25] proposed a novel method to compute the curvature and torsion of LoCs. Equation (24) represents the proposed method for computing u , v , and the geodesic curvature, κ g .
where U = N × t, The curvature of the LoC can be calculated using principal curvature and geodesic curvature, as shown below: Note that the classical method for computing geodesic curvature is given by [25]: where , . Mathematics 2022, 10, 160 6 of 17 The torsion and first derivative of LoC curvature on a surface can be computed using the equation below: where t = c (s), Further details of LoCs can be obtained from [25] and references therein.

The Projection of a Surface onto a Plane
Geodesic curvature is an important tool for developing a surface on a plane. Joo et al. [25] proposed six steps to project a surface onto a plane as follows: 1. Generate two LoCs with larger curvature magnitude named C1 3D and C2 3D along the surface (refer to Figure 1a). 2. Generate n number of LoCs with smaller curvature magnitude (named Di 3D , i = 1, . . . , n), starting from C1 3D and stopping at C2 3D . The stopping points on C2 3D are labelled as Pi 3D , i = 1, . . . , n. 3. Project C1 3D and Di 3D onto a plane isometrically, and represent them as C1 2D and Di 2D . Note that the starting angles of Di 2D from C1 2D are 90 • (refer to Figure 1b). 4. Transform C2 3D onto a plane isometrically from the end point of Dj 2D with an angle of 90 • , and represent it as C2j 2D . The corresponding points of Pi 3D on C2j 2D are indicated as Pi 2D . 5. Compute the sum of gaps δ ij between the endpoints of Dj 2D and Pi 2D where δ j = ∑ δ ij . 6. Select the connection j with minimum δ j .
A curve on a surface can be projected onto a plane by applying geodesic curvature along the curve to the Frenet-Serret formula t = κ g n. The plane curve is calculated using the classic Runge-Kutta method to solve the Frenet-Serret formula by assigning the torsion as 0. For an example, we used Joo et al.'s [25] method to reconstruct the hyperbolic paraboloid onto a plane, as shown in Figure 1. Figure 1c is the planar hyperbolic paraboloid by [25], while Figure 1d is the planar hyperbolic paraboloid computed on GPU using Mathematica.
A curve on a surface can be projected onto a plane by applying geodesic curvature along the curve to the Frenet-Serret formula ′ = . The plane curve is calculated using the classic Runge-Kutta method to solve the Frenet-Serret formula by assigning the torsion as 0. For an example, we used Joo et al.'s [25] method to reconstruct the hyperbolic paraboloid onto a plane, as shown in Figure 1. Figure 1c is the planar hyperbolic paraboloid by [25], while Figure 1d is the planar hyperbolic paraboloid computed on GPU using Mathematica.

Efficient LoC Computation
This paper used CUDA programming to perform LoC calculations on GPU. Mathematica 11.0 was used to compute the performance metrics of CPU and GPU. Hence, we computed the significant error of LoCs on the hyperbolic paraboloid in CPU and GPU to validate our method. The significant error of the LoC computation is 3.869 × 10 −7 (all the function types are "Float"). If we change all the function types to "Double", the significant error is reduced to 6.02988 × 10 −16 . Readers are referred to [18] for the details of LA surface's LoC computation using GPU, where the computation time is greatly reduced.

Efficient LoC Computation
This paper used CUDA programming to perform LoC calculations on GPU. Mathematica 11.0 was used to compute the performance metrics of CPU and GPU. Hence, we computed the significant error of LoCs on the hyperbolic paraboloid in CPU and GPU to validate our method. The significant error of the LoC computation is 3.869 × 10 −7 (all the function types are "Float"). If we change all the function types to "Double", the significant error is reduced to 6.02988 × 10 −16 . Readers are referred to [18] for the details of LA surface's LoC computation using GPU, where the computation time is greatly reduced.

Coons Patch
Assume that the four parametric curves defined in c 0 (u), Three surfaces are defined by linear interpolation [20]: A bilinear Coons patch CP(u, v) is defined over the parameter domain containing the unit square (u, v) ∈ [0,1] × [0,1]: In this paper, we defined the Log Aesthetic Patch (LAP) as a surface using Equations (28)-(31) with four parametric curves either in the form of LAC or LASC. To show its versatility, we approximated the hyperbolic paraboloid using LAP by replacing the LoCs of the hyperbolic paraboloid with LACs/LASCs. This approach is in line with Joo et al.'s [25] idea, where they considered LoCs the boundaries of a developable surface. The first step is to replace the four connected LoCs of the hyperbolic paraboloid with LACs or LASCs, depending on the type of boundary curves we are dealing with. On the basis of these four boundary curves, an LAP surface is constructed using the Coons patch equations, as stated in (28)-(31). In the first step, we obtain four LoCs from a hyperbolic paraboloid, which are connected like a closed fence, as shown in Figure 2.
In this paper, we defined the Log Aesthetic Patch (LAP) as a surface using Equations (28)-(31) with four parametric curves either in the form of LAC or LASC. To show its versatility, we approximated the hyperbolic paraboloid using LAP by replacing the LoCs of the hyperbolic paraboloid with LACs/LASCs. This approach is in line with Joo et al.'s [25] idea, where they considered LoCs the boundaries of a developable surface. The first step is to replace the four connected LoCs of the hyperbolic paraboloid with LACs or LASCs, depending on the type of boundary curves we are dealing with. On the basis of these four boundary curves, an LAP surface is constructed using the Coons patch equations, as stated in (28)-(31). In the first step, we obtain four LoCs from a hyperbolic paraboloid, which are connected like a closed fence, as shown in Figure 2. Next, we calculated the four endpoints (the intersection of LoCs) and their maximum and minimum principal direction with their corresponding vectors. Then, LACs or LASCs were generated using two endpoints and their respective tangent vector (maximum or minimum principal direction). Using the given LAC or LASC shape parameters ( , , 0 , and Ω), we could compute the LAC/LASC that meets the constraints using the bisection method [12] and the modified Nelder and Mead downhill simplex method [13]. We then scaled the LACs and converted them back to the position of the point shown in Figure 3.  Next, we calculated the four endpoints (the intersection of LoCs) and their maximum and minimum principal direction with their corresponding vectors. Then, LACs or LASCs were generated using two endpoints and their respective tangent vector (maximum or minimum principal direction). Using the given LAC or LASC shape parameters (α, β, ρ 0 , and Ω), we could compute the LAC/LASC that meets the constraints using the bisection method [12] and the modified Nelder and Mead downhill simplex method [13]. We then scaled the LACs and converted them back to the position of the point shown in Figure 3.
In this paper, we defined the Log Aesthetic Patch (LAP) as a surface using Equations (28)-(31) with four parametric curves either in the form of LAC or LASC. To show its versatility, we approximated the hyperbolic paraboloid using LAP by replacing the LoCs of the hyperbolic paraboloid with LACs/LASCs. This approach is in line with Joo et al.'s [25] idea, where they considered LoCs the boundaries of a developable surface. The first step is to replace the four connected LoCs of the hyperbolic paraboloid with LACs or LASCs, depending on the type of boundary curves we are dealing with. On the basis of these four boundary curves, an LAP surface is constructed using the Coons patch equations, as stated in (28)-(31). In the first step, we obtain four LoCs from a hyperbolic paraboloid, which are connected like a closed fence, as shown in Figure 2. Next, we calculated the four endpoints (the intersection of LoCs) and their maximum and minimum principal direction with their corresponding vectors. Then, LACs or LASCs were generated using two endpoints and their respective tangent vector (maximum or minimum principal direction). Using the given LAC or LASC shape parameters ( , , 0 , and Ω), we could compute the LAC/LASC that meets the constraints using the bisection method [12] and the modified Nelder and Mead downhill simplex method [13]. We then scaled the LACs and converted them back to the position of the point shown in Figure 3.  Note that the red curves are LAC and LASC segments, while the blue curves are original LoCs on the hyperbolic paraboloid. The transformed unit tangent, normal, and binormal vectors at the initial point are named t s (s 0 ), n s (s 0 ), and b s (s 0 ), respectively. The subscript s indicates that the vector is in terms of arc length, and the scaling ratio of LAC/LASC is denoted as m.
The Coons patch requires two general parameters to shape the surface, which are rendered in the parameter domain (u, v) ∈ [0,1] × [0,1]. Hence, we must reparametrize the curvature of LAC, the torsion of LASC, and the Frenet-Serret formula in terms of the arc length parameter to the general parameters u or v. Let the arc length s = uS T ; then we have ϕ(u) = ds du = S T , where S T is total arc length. Next, by applying s = uS T into Equations (7) and (8), we obtain the curvature of the LAC and the torsion of the LASC as well as their derivatives in terms of parameter u: . .. .
The Frenet-Serret formula of the LAC/LASC in terms of parameter u is as follows: . t u (u) = S T (κ(u)n u (u)), .
Thus, by solving the initial value problems using the classical Runge-Kutta method, the LAC and LASC segments shown in Figure 3 can be computed. Finally, an LAP can be drawn based on these four curves.

Numerical Example
The details of Figure  66676} respectively. Hence, the above information was used to find the appropriate LAC and LASC segments to replace the original LoCs of the hyperbolic paraboloid. Note that the curves from P A to P B and P A to P C are planar curves. Hence, we replaced these two curves with LACs. On the other hand, the curves from P B to P D and P C to P D are space curves. In fact, these two curves can be originated from two disjoint surfaces. For numerical illustration, we replaced these space curves with LASCs. According to [18], the LoCs of LA swept surfaces have the same monotonic curvature profile when the path curve and profile curve are both LACs with α LAC = 2. Therefore, for the LAC, we set α LAC = 2 and the initial radius of curvature ρ 0 LAC = 1 for the entire curve finding process. For the LASC, we set α LAC = 2, β LASC = 2, Ω LASC = 1, and the initial radius of curvature ρ 0 LASC = 1 for the entire curve-finding process as well. The curvature of the LAC and the torsion of the LASC were inserted into the Frenet-Serret formula and numerically solved using the classical Runge-Kutta method. The initial value of the unit tangent, normal, and binormal vectors were set as {1, 0, 0}, {0, 0, 1}, and {0, 1, 0}, respectively, and the initial coordinate was at the origin {0, 0, 0}.
For the LAC, we set torsion τ = 0 in the Frenet-Serret formula, and the curvature of the LAC can be obtained from Equation (7). Then, we used the bisection method to compute the parameter Λ LAC and the arc length s LAC . The tolerance of this method was set to 10 −15 . For the LASC, the curvature and torsion of the LASC can be obtained from Equations (7) and (8), respectively. The bisection method was used to compute the arc length s LASC , and the modified Nelder and Mead downhill simplex method was used to compute the parameters Λ LASC and υ LASC . The tolerance of these two methods was set to 10 −15 and 10 −14 , respectively. Finally, we fit the LAC segments with the shape parameter Λ LAC = 12.71843 and the arc length s LAC = 6.76436, satisfying the constraints of P A to P B and P A to P C . Meanwhile, the LASC segment with parameters Λ LASC = 6.28525, υ LASC = 7.30313, and arc length s LASC = 3.10842 satisfied the constraints of P B to P D and P C to P D . In order to simplify labels, we named the LAC or LASC from P A to P B as C AB , P A to P C as C AC , P B to P D as C BD , and P C to P D as C CD . The scaling ratios m AB , m AC , m BD , and m CD for the curves C AB , C AC , C BD , and C CD are m AB = m AC = 0.13192 and m BD = m CD = 0.38164, respectively. Because the curves are scaled, their vectors must be scaled as well. During the reverse transformation, we also transformed its tangent, normal, and binormal vectors at the origin to the original position. The scaled LAC and LASC segments that met the constraints are shown in Figure 3. Table 1 shows the Frenet-Serret equations for each curve as well as their transformed unit tangent, normal, and binormal vectors. .

Curve Frenet-Serret Equations
Note that S AB = 6.76436, S AC = 6.76436, S BD = 3.10842, and S CD = 3.10842 are the total arc lengths of each curve. In addition, the curvature κ and torsion τ in terms of parameters u or v can be obtained from Equations (32) and (35), respectively.
Next, the general equation for the LAP is shown in Section 4. We implement four LA equations (LACs and LASCs) into the Coons patch equations as follows: Hence, an LAP can be drawn using Equation (46). Figure 4a,b represents hyperbolic paraboloid and LAP, respectively. Figure 4c shows the combination of both surfaces. The zebra map on LAP shown in Figure 4d shows that it is a smooth surface.

LoCs on Surfaces
In this subsection, the properties of LoCs on the hyperbolic paraboloid and its approximated LAP are compared. Figure 5 displays the LoCs on the hyperbolic paraboloid and the approximated LAP.

LoCs on Surfaces
In this subsection, the properties of LoCs on the hyperbolic paraboloid and its approximated LAP are compared. Figure 5 displays the LoCs on the hyperbolic paraboloid and the approximated LAP.

LoCs on Surfaces
In this subsection, the properties of LoCs on the hyperbolic paraboloid and its approximated LAP are compared. Figure 5 displays the LoCs on the hyperbolic paraboloid and the approximated LAP. Even though they are visually similar, we found that the curvature profile of LoCs on LAP is different than the hyperbolic paraboloid, as shown in Figure 6. Although the curvatures of LoCs on both surfaces were monotonically decreasing, the derivative of the curvature profile of the approximated LAP was improved from non-monotonic to monotonic. However, there was not much improvement on the torsion of LoCs, but it was apparent that all the computed LoCs on approximated LAPs were indeed LACs. We can clearly see that their LCGs approximate a line with gradient 2, illustrated with a red line in Figure 6. This outcome is in line with the original setting, in which we used = 2 to generate the boundary curves.

Surface Projection onto a Plane
Joo et al. [25] mapped a surface onto a plane by implementing geodesic curvature along the LoC as the curvature of the curve on the plane. Since the boundary curve of the approximated LAP is not the LoC of the surface, the geodesic curvature of the boundary curve must be computed. The geodesic curvature of the boundary curve can be calculated using the fact that The green curves labeled as Y 1 , Y 2 , Y 3, and Y 4 and the black curves labeled as X 1 , X 2 , X 3 , and X 4 are LoCs on the hyperbolic paraboloid, whereas the blue curves are the boundaries and the LoCs of the original surface. The orange curves labeled as L 1 , L 2 , L 3 , and L 4 and purple curves labeled as J 1 , J 2 , J 3 , and J 4 are the LoCs on the approximated LAP, while the red curves are the boundaries of the surface.
Even though they are visually similar, we found that the curvature profile of LoCs on LAP is different than the hyperbolic paraboloid, as shown in Figure 6. Although the curvatures of LoCs on both surfaces were monotonically decreasing, the derivative of the curvature profile of the approximated LAP was improved from non-monotonic to monotonic. However, there was not much improvement on the torsion of LoCs, but it was apparent that all the computed LoCs on approximated LAPs were indeed LACs. We can clearly see that their LCGs approximate a line with gradient 2, illustrated with a red line in Figure 6. This outcome is in line with the original setting, in which we used α LAC = 2 to generate the boundary curves.

Surface Projection onto a Plane
Joo et al. [25] mapped a surface onto a plane by implementing geodesic curvature along the LoC as the curvature of the curve on the plane. Since the boundary curve of the approximated LAP is not the LoC of the surface, the geodesic curvature of the boundary curve must be computed. The geodesic curvature of the boundary curve can be calculated using the fact that du ds = d 2 u ds 2 = 0 or dv ds = d 2 v ds 2 = 0. By applying dv ds = d 2 v ds 2 = 0 (for curves C AB and C CD ) or du ds = d 2 u ds 2 = 0 (for curves C AC and C BD ) into Equation (26), the geodesic curvature equation of the boundary curves are shown below: At du ds = d 2 u ds 2 = 0, the geodesic curvature of curve C AB can be computed by applying v = 0, while v = 1 is used for curve C CD . The geodesic curvature of curve C AC can be calculated by setting u = 0, while u = 1 is used for curve C BD at dv ds = d 2 v ds 2 = 0. Contrarily, the geodesic curvature of the LoC can be computed using Equation (24). By solving the Frenet-Serret formula, t = κn (where n = t × {0, 0, 1}, κ = κ g and τ = 0), the LoC can be mapped onto a plane [25].

Curvature
Torsion erivative of Curvature LCG Figure 6. Curvature, torsion, and derivative of curvature of LoCs on the hyperbolic paraboloid (left) and the approximated LAP (right). The following Algorithm 1 is the algorithm for mapping an LAP onto a plane.
Step 1 Compute the boundary curves from the given G 1 data: If the LoC between two endpoints is a planar curve, the shape parameter Λ and the arc length s of the LAC must be computed. The bisection method is used to compute shape parameter Λ and arc length s of LAC as presented by Yoshida and Saito [12]. Then, scale and transform the generated LAC to the original position. If the LoC between two endpoints is a space curve, shape parameter Λ, ν, and arc length s of LASC must be computed. The bisection method and the modified Nelder and Meaddownhill simplex method are used to compute the shape parameter Λ, ν, and arc length s of hteLASC, as demonstrated by Yoshida et al. [13]. Finally, scale and transform the generated LASC to the original position.
Step 2 On the basis of the four generated LAC/LASCs boundaries, an LAP can be generated using Equation (31).
Step 3 Compute the two LoCs, denoted as C1 3D and C2 3D , and the boundary curves, denoted as B1 3D , as shown in Figure 1a.
Then, compute the intersection point's position of LoCs and the boundary curve.
Step 5 Compute the geodesic curvature along the LoCs and boundary curves.
Step 6 Draw the boundary curve onto a plane isometrically using its geodesic curvature, which is denoted as B1 2D (refer to Figure 7a).
Step 7 On the basis of the position of the intersection point in step 4, C1 3D and C2 3D are developed onto a plane isometrically (denoted as C1 2D and C2 2D as shown in Figure 7a) from a specific point on the boundary curve B1 2D . Note that the starting angles of C1 2D and C2 2D from B1 2D are 90 • .
Step 8 On the basis of the geodesic curvature of Di 3D and their stopping points Pi 3D , 2D LoCs Di 2D and their stopping points Pi 2D are computed starting from specific points on C1 2D . Then, join the stopping points Pi 2D to form a curve P 2D .
Step 9 Develop the second 3D boundary curve B2 3D (Figure 7c) onto a plane isometrically from a specific point on curve C1 2D and denote it as B2 2D (Figure 7a). Note that the starting angle of B2 2D from C1 2D is 90 • . The length of B2 2D depends on the length of B2 3D , starting at a point on C1 3D and stopping at a point on C2 3D .
Step 10 Repeat steps 4 to 9 until all the desired curves are projected onto a plane.
Step 11 Cut along the line and stick the boundaries together to create the desired LAP surface.
A simple overview of the planar curves generated by the algorithm for the approximated LAP of the hyperbolic paraboloid is shown in Figure 7. Figure 7c also shows the visualization of LoCs (purple) and boundary curves (red). The LAP surface is fabricated using paper by cutting and pasting the boundaries of the plane surface, as shown in Figure 7d. Finally, Figure 7e shows a close-up of the LoC, J 3 , which is twisted and uniformly curves along the line. The complete implementation of the CUDA coding in Mathematica for this paper is readily available on GitHub [30]. Mathematics 2022, 9,

Conclusions and Future Work
In this work, we proposed the development of the Log Aesthetic Patch (LAP) using the LAC and LASC as its boundary curves. To show its applicability, we used a hyperbolic paraboloid as a numerical example and approximated it with LAP by applying LACs and LASCs as the boundaries of the surface replacing its LoC boundaries. In comparison, the curvature profile of all the LoCs and its derivatives on the approximated LAP are always monotonic, indicating smoothness of a higher degree. The final section showed an algorithm for LAP projection onto a plane.
Our future work includes implementing this technique to steel plate fabrication in shipbuilding and investigating the generation of other types of LA surfaces.

Conclusions and Future Work
In this work, we proposed the development of the Log Aesthetic Patch (LAP) using the LAC and LASC as its boundary curves. To show its applicability, we used a hyperbolic paraboloid as a numerical example and approximated it with LAP by applying LACs and LASCs as the boundaries of the surface replacing its LoC boundaries. In comparison, the curvature profile of all the LoCs and its derivatives on the approximated LAP are always monotonic, indicating smoothness of a higher degree. The final section showed an algorithm for LAP projection onto a plane.
Our future work includes implementing this technique to steel plate fabrication in shipbuilding and investigating the generation of other types of LA surfaces.