ON THE MODELING OF DEVELOPABLE HERMITE PATCHES

The developable surfaces are useful in plat-metal and plywood sheet installation. For this reason, the paper discusses designing the developable Hermite patches with its boundary curves placed in different parallel planes. The steps used are in this way. Using the boundary curves data, and a developable patches’ criteria equation, we evaluate this equation to find some geometric elements of both curves that can be practically used to construct the developable pieces. Then the study makes a procedure to design the pieces. As a result, we obtain some equations and methods to model many shapes of the developable Hermite patches that can be constructed by its endpoints, intermediate points, the tangent, and acceleration vectors at the endpoints of the patches’ boundary curves. The endpoints and the intermediate control points can be applied to raise and lower the surface shape along the pieces. Moving the intermediate control points between the endpoints, and determining the tangent, and acceleration vectors are used to change the patches' surface slopes in a different form. Using this method, the surface shape of the pieces can be curved and fluctuate into more than three arches. We also numerically calculate the connection G between two developable Hermite pieces adjacent.


INTRODUCTION
The techniques for modeling the developable surfaces have already been introduced. Zhao and Wang [1] constructed the developable surface via the surface pencil passing through a given curve. This technique presented the surface with the combination of the curve, and the vectors that are defined by the Serret-Frenet frame. Meanwhile, Al-Ghefari and Abdel-Baky [2] proposed a method for modeling the developable surface and classified it in a cylinder, cone, and tangent surface. A new technique for constructing the minimal surface has been reported by Xu et al. [3].
From a given boundary of surface, they apply quasi-harmonic Bézier approximation and quasi-harmonic mask methods to define the minimal surface. Then, Xu et al. [4] presented the construction of IGA-suitable planar B-spline parameterizations from a certain complex CAD boundary. This technique requires many steps to model the surfaces. It needs a process of Bézier extraction and subdivision, a partition framework of planar domain, quadrangulation, and a optimization process. Hu et al. [5] introduced the developable Bézier-like surfaces with Bernstein-like basis functions of the multiple shape parameters. Kusno [6,7,8] discussed the construction of regular developable Bézier patches defined by their boundary curves of four, five, and six degrees. These pieces are also modeled by using Hermite spline interpolation curves, and simulated the method for cutting and adjusting the shapes of the surfaces. In recent times, the method of the developable surface construction bounded by two rational or NURBS curves is presented by Fernándes and Pérez [9]. This method imposes that the endpoints of patch' boundary curves must be coplanar, and at those endpoints position, it has to fulfill the developability condition. Evaluating these methods introduced, we get some limitations.
Applying the low degree planar curves is efficient for constructing the simple developable patches or surfaces, but it is not practical to design the developable surfaces of complex shapes.
Using the derivatives and developability condition at the endpoints of the boundary curves can locally control the form and continuity of a patch in the nearest area of the endpoints, but, it cannot straightforwardly control to create various slopes and fluctuate shapes in the middle surface part of the patch. In other words, the presented methods just can construct a surface 1147 ON THE MODELING OF DEVELOPABLE HERMITE PATCHES shape of the patches not more than two arches. To find a solution of these restrictions, this paper proposes a new approach to design the developable pieces of many arches by arranging its endpoints, intermediate points, the tangent, and acceleration vectors at the endpoints of the patches' boundary curves.

MATHEMATICAL EQUATIONS OF DEVELOPABLE SURFACES AND HERMITE CURVES
Mathematical equations of regular developable surfaces in the algebraic representation are defined as follows [6,10].
The curves f(u) and g(u) are respectively called directrix curve and generatrix (ruling).
Consider, in Definition 2.2, the ruling g(u) = q(u) − P(u) and the curve f(u) = P(u). Then, the surface ( , ) can be noted in the form of their boundary curves P(u) and q(u) as follows When the surface ( , ) is developable, then three vectors [g'(u),P'(u),g(u)] are coplanar.
This means that the vector g'(u) can be represented in linear combination of two vectors P'(u) and [q(u) − P(u)] such that q'(u) = ρ(u) P'(u) + σ(u) [q(u) − P(u)] with ρ(u) and σ(u) two real scalars. Because of application purposes, in this paper, we constrain that the curves P(u) and q(u) must be in two parallel planes ϓ1//ϓ2, correspondingly, and the criteria equation of the developable surfaces will be In this discussion, due to two curves [P(u),q(u)] are regular in the same degrees and orientation (direction), we limit that the choice of the real scalar ρ(u) is positive constant ρ for all values u. The fixed criteria ρ classify two types of developable surfaces. If the value ρ = 1, then it defines a cylinder type, if not, a cone.  1 ] of the curve is [11,12] (2.3a) H51= −6u + 15u 4 −10u 3 + 1; H52 = 6u 5 − 15u 4 + 10u 3 ; H53 = −3u 5 + 8u 4 − 6u 3 + u; If we give a tension value k1 and k2 to the tangent vectors and 1 in that order, the Hermite curve formulations of Equations (2.3) will be and 1 * = k2 1 . Therefore, it can formulate these cubic and quintic Hermite curves of Equations (2.3) in the algebraic representation Base on the developable surfaces criteria of Equation (2.2) and the patches' boundary curves P(u) and q(u) of Equations (2.5), and (2.6), in the next section, this paper will introduce a new method to design the developable Hermite pieces in the form of Equation (2.1). In these patches modeling, to apply many parameters of the equations, we use the algebraic representations rather than the geometric representations.

MAIN RESULTS OF DEVELOPABLE HERMITE PATCHES CONSTRUCTION
The developable surfaces have distinctive properties among the surface types, i.e. it can be laid flat on a plane without stretching and tearing. These surfaces are handy in plat-metal-based industries and plywood sheet installation, for example, aircrafts industries, ship hulls, and trains [13,14,15]. In the applications, those object surfaces, generally, are formed by some small surface sections (pieces) that are bounded by two curves P(u) and q(u) laid in the plane ϓ1 parallel to the plane ϓ2, respectively. For this reason, this paper discusses the construction of the developable patches via the boundary curves in form of the Hermite polynomials.
and, for the patch D5(u,v), it have to fulfill the conditions Thus, it can deduce that the patch D3(u,v) will be developable when their control points and its tangent vectors are in conditions The equations (3.2) means that // , (P1−Po)//(q1−qo), 1 // 1 , and | |/| | = |q1−qo|/|P1-Po| = | 1 |/| 1 | = ρ. It is shown in Figure 1 that, from the points data [Po, P1, , 1 ], we can calculate the control points [qo, q1, , 1 ] relative to the center point O' with the ratio value ρ by applying the triangle similarity theorem. The calculated surface shape of the cubic developable Hermite patch in Figure 1 is presented in Figure 2. In another side, the quintic Thus, designing the cubic and quintic developable Hermite patches D3(u,v) and D5(u,v) can be undertaken in the following steps: of a quintic developable Hermite patch.

Points and Tangent Vectors
To handle the plate sheets need the developable patches properties that can be adapted to the object's surface fluctuations and, locally, can be formed into many surface arches. For this purpose, in this section, first, it will introduce the formulations of quintic Hermite curves that are restricted by two intermediate points; after that, one tangent vector is posted at these intermediate points.
Second, we will present, via these curves, a method to construct the developable patches that can have these adaptive properties.
From Equation (2.2), the criteria of the developable pieces Dx5(u,v) is 52 (u) = ρ 51 (u). As a result, it can state the equation Consequently, this condition can be simplified in this manner that are shown in red color curves on Figure 5(a). In this case, when the value is be replaced by the vector =<0,60,−50>, the boundary curves Px51(u) and qx52(u) will change as exposed in black color curves in the Figure 5(a). Both data calculations build the developable patches graph, as represented in Figure 5

Connection between Two Developable Hermite Patches
In plat-metal-based industries and plywood sheet installation, the modeled object surfaces are, generally, composed of some developable patches. To obtain the best connections, continuity among these patches depends on their used continuity level. For this purpose, this section will discuss the continuous connection between two developable pieces adjacent. Thus, they must meet (3.14) P11 = Po2; q11 = qo2.
Given the cubic developable Hermite patches adjacent D1(u,v) and D2(u,v) of cylinder and cone types respecvively. The first patch D1(u,v) is constructed by the control points vectors are used to change the patches' different surface slopes. They can also be used to modify their shapes in two, three, or four surface arches. Finally, the connection G 1 between two developable Hermite pieces adjacent is determined by the existence of collinear tangent vectors at the joint points of their boundary curves. The exciting thing to discuss in the future is how to