Fitting a Curve, Cutting Surface, and Adjusting the Shapes of Developable Hermite Patches

Formulation of developable patches is beneficial for modeling of the plate-metal sheet in the based-metal-industries objects. Meanwhile, installing the developable patches on a frame of the items and making a hole on these objects surface still need some practical techniques for developing. For these reasons, this research aims to introduce some methods for fitting a curve segment, cutting the developable patches, and adjusting their formulas. Using these methods can design various profile shapes of rubber filer installed on a frame of the objects and create a fissure or hole on the patches' surface. The steps are as follows. First, we define the planes containing the patches' generatrixes and orthogonal to the boundary curves. Then, it fits the Hermite and Bézier curve, via arranging some control points data on these planes, to model the rubber filler shapes. Second, we numerically evaluate a method for cutting the patches with a plane and adjusting the patches' form by modifying their formula from a linear interpolation form into a combination of curve and vectors forms. As a result, it can present some equations and procedures for plotting required curves, cutting surfaces, and modifying the extensible or narrowable shape of Hermite patches. These methods offer some advantages and contribute to designing the based-metal-sheets' object surfaces, especially modeling various forms of rubber filer profiles installed on a frame of the objects and making hole shapes on the plate-metal sheets.


Introduction
Some developable surfaces' formulas can be used to model automobile parts, ship hulls, and aircraft [1,2,3,4]. Al-Ghefari and Abdel-Baky [5] presented the procedure to construct the developable surfaces and identify these surfaces into three types, i.e., a cylinder, cone, and tangent surface. Then, Xu et al. [6] discussed the minimal surface formulation via a given boundary curve of the surface. In this case, they use a quasi-harmonic Bé zier approximation and a quasi-harmonic mask. After that, Hu et al. [7] formulated the developable Bé zier-like surfaces with Bernstein-like basis functions. Kusno [8] discussed the construction of regular developable Bé zier patches in which their boundary curves are defined by the combination of four, five, and six degrees. Then, he developed this method by applying Hermite polynomial curves [9]. Lately, Ferná ndez and Pé rez [10] introduced the technique for designing the developable surfaces using their boundary curves in the form of NURBS curves.
To model the surface parts of the based-metal-industries objects aided by the developable surfaces' formulas, it needs some practical calculations. For this reason, the paper presents a new approach for fitting, cutting, and modeling the parts of developable Hermite patches in the following steps of discussion. First, we review the numerical calculation of cubic and quintic Hermite developable patches introduced. Second, this study evaluates the method for fitting a curve segment and for modifying the developable Hermite patches. Third, we discuss a technique for cutting and amend the shapes of the pieces. Finally, the results of the study are summarized.

Formulation of Cubic and Quintic Hermite Developable Patches
In this section, we study the developable condition of the Hermite developable patches supported by two parallel planes. Then, it reviews the construction of the cubic and quintic Hermite developable patches that were introduced.
[q(u)−P(u)] with τ(u) and σ(u) two real scalars. In the application, we necessitate that the parametric functions P(u) and q(u) are respectively in the planes  1 // 2 //ZOY and It requires that τ(u) is positive constant τ. Because of this reason, it can formulate the developable condition in the form [1,8] q'(u) = τ.P '(u).
(2) When the value τ = 1, the surface shape will be a cone in which their generatrixes meet at a point. Contrary, when the value τ  1, the surface will be a cylinder, and their generatrixes will be parallel. Consequently, for any two selected generatrices, it must be coplanar.
Given two Hermite curves in the form P(u) and q(u) of equations (3), (4), and (5). These curves are respectively laid in the plane ϓ 1 , ϓ 2 , and ϓ 1 //ϓ 2 //YOZ. Based on these restrictions, we will review the construction method of the developable Hermite patches and u, v in interval 0 u,v 1. In this case, it can summarize that the formulation steps of the patch design are as follows [9].
a. Case 1: Curves P(u) and q(u) cubic of Equation (3) If P(u) and q(u) are of Equation (3), then the developable criteria (2) of both boundary curves will be in the equation q'(u) = τ(u).P′(u) or This means that q o = τ P o , q 1/2 = τ P 1/2 , q 1 = τ P 1 , q o u = τ . Due to two genaratrixes P o q o and P 1 q 1 must be laid in the same plane, this developability conditions can be stated b. Case 2: Curves P(u) and q(u) quintic of Equation (4) If P(u) and q(u) are of Equation (4), then, using the same calculation method of the case 1 will find the developable criteria (2) in the form c. Case 3: Curves P(u) and q(u) quintic of Equation (5) If P(u) and q(u) are of Equation (5), then the developable criteria (2) are in the formula 742 Fitting a Curve, Cutting Surface, and Adjusting the Shapes of Developable Hermite Patches Via Equation (7), (8), and (9), we can, generally, construct the cubic and quintic Hermite patches by using the steps: a.

Fitting a Curve Segment on Developable Hermite Patches' Boundary Curves
Given a real function of quintic polynomial 4 and R 5 (1) = R 5 . It can thus formulate the quintic Hermite polynomial curve R 5 (v) in this way with In another side, the quintic Bé zier polynomial of the control points B o , B 1 , B 2 , B 3 , B 4 , and B 5 is as follows with 0 ≤ v ≤ 1.
Concerning the application of both equations (10) and (11), in this section, the study will introduce a new approach to formulate a fitting curve segment that can apply to design a model of a rubber filler along the borders P(u) and q(u) of the developable Hermite patches L(u,v) in Equation (6). It can also be used to set the installation shape of the developable patches L(u,v) on the planes  1 // 2 // YOZ. The numerical solution method is as follows.
Consider the curve P(u) in the plane  1 , q(u) in the plane  2 . We define the unity vector 1 = [ ( ) − ( )]/| ( ) − ( )| . Meanwhile, the unity tangent vector t(u) of the boundary curve P(u) is t(u) = P u (u)/|P u (u)| for 0 ≤ u ≤ 1. Using both vectors t and u 1 , it can find a unity vector u 2 = u 1  t. Based on these triple orthonormal unity vectors [t,u 1 ,u 2 ], we will draw and evaluate a polygon shape or a curve in the plane [u 1 ,u 2 ] that can be moved (swabbed) orthogonally along the curve P(u) to model the cross-section profile curves of the rubber filler. For this purpose, it can arrange the control points' coordinate frame and apply the equations (10,11) with steps in this way.

Cutting Surface and Adjusting the Shapes of Developable Hermite Patches
Consider a developable Hermite patch L(u,v) of Equation (6) with the endpoints of their boundary curves [P o ,P 1 ] and [q o ,q 1 ] in the plane  1 // 2 //YOZ, respectively.
We determine two alternative points R = (1-x) P o +x q o and S = (1-y) P 1 +y q 1 with 0 x,y 1, then define a plane T(s,t) that passes to the line RS and perpendicular to the plane [P o P 1 ,q o q 1 ] as shown in Figure 3a. Due to the plane T(s,t) cuts the developable patch L(u,v), the problem that will be discussed is to calculate the surface part of the patch that is limited by the plane T(s,t) and ϓ 1 . The solution method is as follows.
If Using the dot and cross multiplication of the vector algebra operations [11,15], for each value ui, it can, therefore, compute the parameter values  In industrial applications, the surface form of the developable patches in Equation (6) sometimes needs to be modified in shape or measure. This surface area sometimes goes beyond the boundary curve, or even it must lay in the interior between both boundary curves of the patches. For this reason, based on their boundary curves P(u) and q(u), we adapt Equation (6)

Conclusions
Using the presented method of fitting a curve segment on developable Hermite patches' boundary curves can design the various rubber filler's cross-section profile curves. The process of cutting and adjusting the developable patches' shapes that were introduced will offer some advantages to model the surface parts of the based-metal-industries objects, for example, in making a hole on the plates or modifying the form of plate sheets. Hereafter, the exciting thing to develop is how to model the developable surfaces when their boundary curves are not laid in the planes.