Developing the Developable Surfaces in a Space to the Plane using Some Triangle Pieces Kusno and Nur Hardiani Jember University, Jl. Kalimantan No. 37, Jember, Indonesia Department of Tadris Matematika, State Islam University, Mataram, Indonesia Keywords: Developing, Developable Surfaces, Space, Plane, Triangle. Abstract: This paper deals with the development in the space to the plane of the polygons, the cone, the cylindrical surfaces and the developable quartic Bézier patches in which its boundary curves are respectively parallel, and the normal vectors of the surfaces must be in the same orientation. The method is as follows, we approximate the surfaces into some triangle pieces then we transform consecutively these pieces in the plane. The result of the study shows that the use of the triangle approximation method can develop effectively these surfaces in the space to the plane. In addition, it can be applied to detect all surface measures of an object that are defined by those surface types. 1 INTRODUCTION and the cylinder defined by the linear interpolation of two parallel circles. In the third, the construction and Some methods related to the development of surface the development of developable quartic Bezier to the plane have been presented. The development of patches in a space to the plane are introduced. Finally, the pipeline surfaces can be carried out by the results will be summarized in the conclusion enumerating of two boundary curves in some section. approximation polygons (Weiss and Furtner, 1998). We can develop a surface to the plane by using the techniques of interactive piecewise flattening of 2 DEVELOPING THE TRIANGLE parametric 3-D surfaces, leading to a non-distorted AND THE POLYGON PLANE (Bennis and Gagalowicz, 1991). After that, developing an arbitrary developable surface into a SURFACE IN A SPACE TO THE flattened pattern is based on the geodesic curve length PLANE preservation and linear mapping principles → → (Clements, 1991; Gan et al., 1996). We can simulate Let a triangle plane 훥퐴퐵퐶. The vector 퐴퐵 and 퐴퐶 the physical model of transitional pipeline parts form an angle 푥° in the space orthonormal coordinate whose cross sections are plane curve and polygon and [푶, 풊, 풋, 풌]. The problem is how to develop the plane are made of unwrinkled or unstretched materials. It is 훥퐴퐵퐶 in the plane orthonormal coordinate [푶, 풊, 풋] in based on the approximation of the boundary surface → triangulation (Obradović et al., 2014). Different from which the vector 퐴′퐵′ of the side development 퐴퐵 of the previous methods, we are interested in the triangle 훥퐴퐵퐶 is align to the determined unit vector discussion about the development of the convex 푎1 (Figure 1a). polygons, the conic/cylindrical surfaces and the To develop the triangle 훥퐴퐵퐶 in space to the developable quartic Bezier patches in a space to the plane [푶, 풊, 풋] can be undertaken as follows (Gan et plane using the triangle pieces. al., 1996). → → This paper is organized in the following steps. In 1). Determine the vector퐴′퐵′ = ‖퐴퐵‖ 풂1 and the first, we talk about the development of the triangle calculate the unit vector 풂 풂 . and polygon plane surfaces in space to the plane. In 2 1 2). Evaluate the measure of angle the second, we evaluate the development of the cone 427 Kusno, . and Hardiani, N. Developing the Developable Surfaces in a Space to the Plane using Some Triangle Pieces. DOI: 10.5220/0008523104270431 In Proceedings of the International Conference on Mathematics and Islam (ICMIs 2018), pages 427-431 ISBN: 978-989-758-407-7 Copyright c 2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved ICMIs 2018 - International Conference on Mathematics and Islam → → 퐴퐶 → 푥 = arccos (퐴퐵 . → . |퐴퐶|). |퐴퐵| 3). Calculate 푂⃗⃗⃗⃗퐶⃗⃗⃗ ′ = 푂⃗⃗⃗⃗퐴⃗⃗⃗ ′ + 퐴⃗⃗⃗⃗′⃗퐶⃗⃗⃗ ′ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ = < 푥퐴′, 푦퐴′ > +(|퐴퐶| cos 푥)푎1 + (|퐴퐶| ⋅ sin 푥)푎2. (a) 4). Construct the developed triangle 훥퐴’퐵’퐶’ in the plane orthonormal coordinate [푶, 풊, 풋] by using the linear interpolation of the couple points 퐴’, 퐵’and퐶’ that are 퐴⃗⃗⃗⃗′⃗퐵⃗⃗⃗ ′ = (1 − 푢)푂⃗⃗⃗⃗퐴⃗⃗⃗ ′ + 푢 푂⃗⃗⃗⃗퐵⃗⃗⃗ ′; (1) ⃗⃗⃗⃗′⃗⃗⃗⃗ ′ ⃗⃗⃗⃗⃗⃗⃗ ′ ⃗⃗⃗⃗⃗⃗⃗ ′ 퐵 퐶 = (1 − 푢)푂퐵 + 푢 푂퐶 ; 퐶′퐴′ = (1 − 푢)푂퐶′ + 푢 푂퐴′; (b) (c) with 0 ≤ 푢 ≤ 1. Figure 1: (a) Calculation of the angle 퐶퐴퐵 in the plane [푶, 풊, 풋], (b) Development of triangle plane 퐴퐵퐶 in space to the plane [푶, 풊, 풋] by starting point at 퐴′(0,0,0), and (c) by If the positions 퐴, 퐵 and 퐶 are 퐴(−1,0,1), starting point at 퐴′(3,3,0). 퐵(2,1,4) and 퐶(0,6,3), then we will find the development of the triangle as it is shown in Figure 1b. If the positions 퐴, 퐵 and 퐶 are 퐴(−1,0,1), 퐵(4,1,2) and 퐶(0,6,3), its development is shown in Figure 1(c). Consider a piece of convex polygon plane Ꝑ of the vertices Ꝑ = [푃1, 푃2, 푃3, … , 푃푛] in space that are shown in Figure 2a. The development of the polygon Ꝑ to the plane orthonormal coordinate[푶, 풊, 풋] can be carried out as follows. (a) 1). Determine an initial point of the development 푄1 and two orthonormal unit vectors 푎1 ⊥ 푎2. 2). Determine a point 푄2 that is an image of the point 푃2 such that푄⃗⃗⃗⃗1⃗⃗⃗푄⃗⃗⃗2 = |푃⃗⃗⃗1⃗⃗푃⃗⃗⃗2 |퐚1. 3). Calculate the length 푑푖 = ⃗푃⃗⃗1⃗⃗푃⃗⃗⃗i⃗+⃗⃗⃗1 for 푖 = 2, 3, . , 푛 and evaluate the measure of the angle 푃⃗⃗⃗⃗⃗푃⃗⃗⃗ . 푃⃗⃗⃗⃗⃗푃⃗⃗⃗⃗⃗⃗⃗ (b) (c) 휃 = 푢푃 푃 푃 = arccos [ 1 2 1 푖+2 ] 푖 2 1 푖+1 Figure 2: (a) Calculation of the angles 푃 푃 푃 for 푖 = 푃⃗⃗⃗1⃗⃗푃⃗⃗⃗2 . 푃⃗⃗⃗1⃗⃗푃⃗⃗⃗푖⃗+⃗⃗⃗2 푖 1 푖+1 2, . , 푛 − 1 of the convex polygon 푃1푃2 … 푃푛 in the plane for 푖 = 1, 2, 3, . , (푛 − 2). [푶, 풊, 풋], (b) Decomposition of the polygon 푃1푃2 … 푃8 into some triangles, (c) Development of the convex polygon 4). Evaluate the points 푄푖 for 푖 = 3, 4, . , 푛 in the 푃1푃2 … 푃8 to the plane [푶, 풊, 풋]. plane[푶, 풊, 풋] as the images 푃2, 푃3, 푃4, … , 푃푛by using the formula In case of the polygon vertices P1(0,-0,7), P2(4,- 12,7), P3(8,-5,7), P4(9,0,7), P5(5,3,7), P6(0,4,7), P7(- 푄⃗⃗⃗⃗1⃗⃗푄⃗⃗⃗ 푖 = 푂⃗⃗⃗⃗⃗푄⃗⃗⃗1 + 푑푖 [cos휃푖−2퐚ퟏ + sin휃푖−2퐚ퟐ]. 3,3,7), P8(-4,0,7) and P9(-3,-5,7) that are lied in the 5). Using equation (1), construct the polygon of plane z =7, the development of the polygon in plane development 푄1푄2푄3 … 푄푛 in the plane [푶, 풊, 풋]. can be shown in Figure 2b. If the polygon vertices P1(0,-10,18), P2(4,-12,16), P3(8,-5, 5), P4(7,0,1), P5(4,3,1), P6(0,4,4), P7(-3,3,8), P8(-4,0,12) and P9(- 3,-5,16) are determined in the planex + y + z - 8= 0, then the result of development is shown in Figure 2c. 428 Developing the Developable Surfaces in a Space to the Plane using Some Triangle Pieces In the more general cases, if we are given a series surface S(u,v) to the plane [O,i,j] can be carried out of 푛 consecutive triangles plane in space [P1P2P3, by using the triangles approximation as follows P2P3P4, ... , PnP(n+1)P(n+2)] that are defined by n+2 (Figure 4a,b). (푖−1) points P1, P2, P3, ..., P(n+2), then the development of 1) Determine the (n +1) parameter values ui = the triangles to the plane [푶, 풊, 풋] can be realized 푛 respectively by equation (1). To justify the method, (2) of i = 1, 2, 3, ..., (n+1) to define the 2n when we give the points data of triangle pieces in triangle plane pieces [P1Q1P2, P2Q1Q2, P2Q2P3, ..., PnQnPn+1, Pn+1QnQn+1]. The point Pi is space P1(3,0,3), P2(0,0,4), P3(3,3,2), P4(0,4,4), defined by C1(ui) and the point Qi is defined by P5(1,8,5) and P6(1,9,5), we will find its development in the plane that are shown in Figure 3a. If the triangle C2(ui) for i = 1, 2, 3, ..., n+1 with Pn+1 = P1 and Qi+1 = Q1. pieces are defined by the points P1(6,0,3), P2(0,1,2), P3(3,3,3), P4(0,4,2), P5(1,8,3), P6(1,11,5), P7(5,8,4), , , 2) Calculate the length Qi Pi Qi Pi1 QiQi1 and P8(1,12,4), P9(10,12,3) and P10(7,14,4), then its development in the plane are shown in Figure 3b. the measure of consecutive angles PiQi Pi1 and Pi1QiQi1 for i = 1, 2, 3, ..., n. 3) In the plane [O,i,j], determine an initial point of development S1 and two orthonormal vectors a1a2. Using the triangle development method in section 2 and the determined initial point S1 can be developed respectively and consecutively the triangles [P1Q1P2, P2Q1Q2, P2Q2P3,..., PnQnPn+1, (a) Pn+1QnQn+1] in space to the plane [O,i,j] that are [R1S1R2, R2S1S2, R2S2R3, ..., RnSnRn+1, R(n+1)SnS(n+1)]. If C1(u) = <2cos u + 1, 2 sin u - 3, 4> and C2(u) = <4cos u + 1, 4 sin u - 3, 2>, then the development of the conic surface S(u,v) into 8 triangles to the plane [O,i,j] is shown in Figure 4c. In the Figure 4d, we present a cone approximated by 32 triangles. The (b) Figure 4e show the development of the conic surface into 16 triangles to the plane [O,i,j].