Hyperboloid Offset Surface in the Architecture and Construction Industry and Shopping Centres

Open Eng. 2019; 9:404–413

Research Article

Edwin Koźniewski and Anna Borowska* Hyperboloid offset surface in the architecture and construction industry https://doi.org/10.1515/eng-2019-0051 and shopping centres. Nowadays, lattice domes with elab- Received May 21, 2019; accepted Jul 02, 2019 orate shapes (Golden Terraces in Warsaw) and multilayer (retractable) roofs of large objects (National Stadium in Abstract: In this paper the issue of approximation of the hy- Singapore) are used. perboloid offset surface off(S(t, v); d) at distance d by the hyperboloid surface S1(ϕ, v) is considered. The problem of determining various surfaces approximating the hyperboloid offset surface off (S(t, v); d) is important due to the applications of the hyperboloid as a mathematical model for miscellaneous objects in the architecture and construction industry. The paper presents the method of determining the angles and coordinates of points of various surfaces approximating the hyperboloid of revolution. A two- sheet hyperboloid offset surface can be used for modelling double-layer domes. A one-sheet hyperboloid offset surface was used to model the reinforced structure of the cooling tower.

Keywords: hyperbola offset curve, one-sheet hyperboloid, two-sheet hyperboloid

1 Introduction

Figure 1: Kobe Port Tower in Japan (cf. [1]) A one-sheet hyperboloid is often used as a model for various objects in the construction industry. It is a doubly ruled surface. Thanks to this, one-sheet hyperboloid shaped constructions can be built with, for example straight steel elements that form a strong structure. Such a clever design guarantees lower costs than other technical solutions. Examples of such constructions are cooling towers (Ja- worzno, Cracow in Poland), Kobe Port Tower in Japan, Newcastle International Airport in England, Cathedral of Brasilia in Brazil, the Canton Tower in China and many other structures. A two-sheet hyperboloid can be used to model domes and other roofs which cover stadiums, halls,

*Corresponding Author: Anna Borowska: Faculty of Computer Science, Bialystok University of Technology, ul. Wiejska 45a, 15-351 Bialystok, Poland; Email: [email protected] Edwin Koźniewski: Faculty of Civil and Environmental Engineer- Figure 2: National Stadium in Singapore (cf. [2]) ing, Bialystok University of Technology, ul. Wiejska 45e, 15-351 Bialystok, Poland; Email: [email protected]

Open Access. © 2019 E. Koźniewski and A. Borowska, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License Hyperboloid offset surface in the architecture and construction industry Ë 405

To design structures based on the hyperboloid S(t, v), the coordinates of points of other surfaces that approximate the hyperboloid S(t, v) are necessary. The hyperboloid S(t, v) can be approximated by means of the offset surfaces, a hyperboloid of similar parameters or another surface with the desired properties. The paper provides a convenient methodology for acquiring the angles and coordinates of points of various surfaces that approximate the one-sheet (or two-sheet) hyperboloid surface. Section 3 suggests the construction of a rein- Figure 3: The arrangement of points and curves forced hyperboloid-shaped double-layer dome. Section 4 presents a proposal for a reinforced cooling tower structure. Sections 2, 3.1, 3.2 and 4.1 provide mathematical for- (cf. [7], p. 98), mulas defining the angles and coordinates of points of c ϕ x a ϕ d z b ϕ different surfaces approximating the hyperboloid. The in- h1( ): = cosh( ) − , = sinh( ), (3) teresting offset surfaces (offset curves) are described in ch2(ϕ): x = a cosh(ϕ) + d, z = b sinh(ϕ),

[3–5]. Section 5 contains examples of the applications of off(ch; d) = ch(t) ± dnver(t). the given geometric method. Definition 1 (horizontal offset thickness) Let the points P,

P1, PP1 (zP1 = zPP1) and curves ch, ch1, off(ch; d) be de- 2 Mathematical formulas fined as above (see Figure 3). Horizontal offset thickness measured at height z = zP1 (zP1 is arbitrary but fixed) is defined as the length of the section |P1PP1| = dP1PP1 ≥ d. Let c(t) = (x(t), y(t))(t ∈ [α, β]) be a parametric represen- x t y tation of a planar curve (we write down functions ( ), (t) Definition 2 (cf. [7], p. 97) as xt, yt). The normal vector to the curve c(t) at the point   ′ ′ t −t t −t P(xt , yt) is as follows n = [−yt , xt]. The unit normal vector cosh(t) = (e + e ) 2, sinh(t) = (e − e ) 2. at the point P(xt , yt) is of the form (cf. [6], p. 335) √︁ y′ x′ x′ 2 y′ 2 nver = [− t , t] ( t) + ( t) . (1) 3 Double-layer lattice domes For a smooth planar curve c, we define an offset curve In this section, we suggest the use of the two-sheet ro- cd at distance d in the following way. On each curve nor- tational hyperboloid surface to form double-layer lattice mal, we mark the two points that are at distance d from domes. Such roofs of large buildings (stadiums, halls, the curve c. The set of all of these points forms the offset ′ ′′ shopping centres) made of metal bars can be light con- cd = (cd ∪ cd ) (cf. [6], p. 335). The offset cd(t) at distance d structions with considerable spread. The bar structures of to c(t) is obtained as (cf. [6], p. 335) cd(t) = c(t) ± dnver(t). ′ ′′ the low-profile single-layer domes are particularly suscep- The curve c and its offset curves cd and cd are not always tible to stability loss [8]. Therefore, a more advantageous of the same type. solution is the construction of double-layer structures con- Let us assume that P is any point on the hyperbola ch nected by bars [9]. We provide a convenient method for de- and l is the normal line to ch at the point P. Points P1 and termining the angles and coordinates of points of various P2 lie on the normal l at distance d from P. Q1, Q2 are the in- surfaces (also the offset surfaces) approximating the two- tersection points of the normal line l with hyperbolas ch1 sheet hyperboloid surface. and ch2 respectively. Non-zero distances dP1Q1 = |P1Q1| and dP2Q2 = |P2Q2| mean that hyperbolas ch1 and ch2 do not keep a constant distance d relative to the basic hyper- 3.1 The coordinates of points P1 and P2 bola ch (cf. Figure 3). The ch(t), ch (ϕ) and off (ch; d) curves shown in Fig- 1 Let us take the parametric equations of the hyperbola ure 3 are defined as follows Ch(t): x = a sinh(t), z = −bcosh(t). We can assume that t ≥ 0, because the graph of the curve Ch(t) is symmetrical ch(t): x = a cosh(t), z = b sinh(t) (2) about the Z axis. The coordinates of points P1 and P2 lying 406 Ë E. Koźniewski and A. Borowska

on the normal l to the hyperbola Ch(t) (at the point P(xt, = a sinh(ϕ) / • a cosh(t) zt)) and distant from P by the length d were determined a2 sinh(ϕ) cosh(t) + b2 cosh(ϕ) sinh(t) = using the following equation of the offset curves off(Ch(t); = sinh(t) cosh(t)(a2 + b2) − db sinh(t). d)

off(Ch(t); d):[X, Z] = (4) Hence and from definition 2 we have d[b sinh(t), a cosh(t)] = [xt , zt] ± √︁ 2 ϕ −ϕ  2 ϕ −ϕ  a2cosh2(t) + b2sinh2(t) a cosh(t)(e − e ) 2 + b sinh(t)(e + e ) 2 = = sinh(t) cosh(t)(a2 + b2) − db sinh(t) for xt = a sinh(t), zt = −bcosh(t). From here we obtain the P P coordinates of points 1 and 2 Hence

db sinh(t) 2ϕ ϕ xP1 = xt − √︁ , (5) Ae + Be + C = 0, a2cosh2(t) + b2sinh2(t) da t where cosh( ) 2 2 zP1 = zt − √︁ , A = a cosh(t) + b sinh(t), 2 2 a2cosh (t) + b2sinh (t) B = −2 sinh(t)(cosh(t)(a2 + b2) − db), db t C b2 t a2 t x x sinh( ) = sinh( ) − cosh( ). P2 = t + √︁ , ϕ 2 2 2 Let us denote E = e . Then for ∆ = B − 4AC > 0 we have a2cosh (t) + b2sinh (t) √  √  E1 = (−B + ∆) 2A and E2 = (−B − ∆) 2A. Hence ϕ1 = da cosh(t) zP2 = zt + √︁ ln E1 and ϕ2 = ln E2. Finally we obtain the parameter k for 2 2 2 2 a cosh (t) + b sinh (t) the point Q1 (︂ )︂ x a t z b t b cosh(ϕ ) 1 for t = sinh( ), t = − cosh( ). k = k = 1 − 1 + , (7) 1 da cosh(t) a cosh(t) (︂ (︂ )︂)︂ k1db ϕ1 = ln E1 = arsinh sinh(t) 1 − . 3.2 The coordinates of the point Q1 a

Lemma 1 (cf. [5], Lemma 3, p. 46) Let us take the parametric equations of the hyperbola √︁ k b2 2 t a2 2 t Ch1(ϕ): x = a sinh(ϕ), z = −b cosh(ϕ) − d. The coordi- Let us denote P1 = 1 sinh ( ) + cosh ( ). nates of the point Q1 (the intersection of the normal line l (a) The coordinates of the points P, P1 and Q1 can be to the hyperbola Ch(t) at the point P(xt , zt) with the hyper- determined using the parametric equations (6) of bola Ch1(ϕ)) were determined as follows. We can assume the normal line l to the hyperbola Ch(t) at the point that t, ϕ ≥ 0. The normal vector to the curve Ch(t) at the P(xt , zt) (we also write down P(xP , zP)) for the pa- point P is of the form n = [b sinh(t), a cosh(t)]. The para- rameter k equal respectively k = 0, k = kP1 (cf. (5)) metric equations of the normal line l to the hyperbola Ch(t) and k = k (cf. (7)). at the point P are as follows (cf. [10], p. 140) 1 (b) dPQ1 ≤ d iff k1 ≤ kP1. x = sinh(t)(a − kdb), z = − cosh(t)(b + kda). (6) The property (b) results from the fact that points P, P1 and Q lie on the normal l. Let us set the parameter k giving the intersection points of 1 Figure 4 shows the two-sheet rotational hyperboloid the line l with the hyperbola Ch1(ϕ). Sh(t, v) (the red surface), the two-sheet rotational hyper- S ϕ v x = sinh(t)(a − kdb) = a sinh(ϕ), boloid h1( , ) (the bright blue surface) and the hyperboloid offset surface off(Sh; d) at distance d (the dark blue z = − cosh(t)(b + kda) = −(b cosh(ϕ) + d). surface). Let us determine the parameter k from the second equation Sh(t, v): x = (a sinh(t)) cos(v), and put it in the first equation. y = (a sinh(t)) sin(v), z = −b cosh(t), (︂ )︂ b cosh(ϕ) 1 k = − 1 + Sh (ϕ, v): x = (a sinh(ϕ)) cos(v), da cosh(t) a cosh(t) 1 y = (a sinh(ϕ)) sin(v), z = −b cosh(ϕ) − d, b2 b2 cosh(ϕ) sinh(t) bd sinh(t) a sinh(t) + sinh(t) − − = a a cosh(t) a cosh(t) off(Sh; d): x = sinh(t)(a − dbkP1) cos(v), Hyperboloid offset surface in the architecture and construction industry Ë 407

y = sinh(t)(a − dbkP1) sin(v),

z = − cosh(t)(b + dakP1), √︁ 2 2 2 2 where kP1 = 1 b sinh (t) + a cosh (t). Figure 5 shows a fragment of the double-layer lattice dome of the shape based on the two-sheet rotational hyperboloid. The black lattice is shaped by the hyperboloid Sh(t, v) (a = 30, b = 40), the blue lattice is formed by the offset surface off(Sh; d)(d = 6) and the pink lattice is a part

Figure 4: Fragment of the (red) two-sheet hyperboloid Sh(t, v), the (bright blue) two-sheet hyperboloid Sh1(ϕ, v) and the (blue) hyperboloid offset surface off(Sh; d) Figure 6: Fragment of the double-layer lattice dome of the shape based on the two-sheet rotational hyperboloid

of the hyperboloid Sh1(ϕ, v). The bars connecting the two lattice surfaces are perpendicular to the outer surface. The given formulas allow designers to use any lattice pattern.

4 Cooling towers

A cooling tower is a device used to cool industrial water in those energy and industrial plants that do not have the pos- sibility to make use of water from a river, lake or sea. Cool- ing towers are equipped with a very high (usually) reinforced concrete chimney. The chimney wall is a thin shell in the shape (from outside and inside) of the one-sheet rotational hyperboloid. This chimney construction is rigid and resistant to bending. Such a design guarantees the pos- sibility of obtaining a large diameter and height. Thanks to the high chimney and the heated water, the so-called chimney effect is obtained. Figure 7 shows the model of the chimney wall. The one-sheet hyperboloid surface sh(t, v) modelling the outer part of the chimney wall is coloured bright blue. The one-

sheet hyperboloid surface sh1(ϕ, v) modelling the inner Figure 5: Fragment of the double-layer lattice dome of the shape part of the chimney wall is coloured red. The offset surface based on the two-sheet rotational hyperboloid off(sh; d) at distance d to the outer part of the chimney wall 408 Ë E. Koźniewski and A. Borowska

tant from P by the length d. db cosh(t) xP1 = xt − √︁ , (8) a2sinh2(t) + b2cosh2(t) da sinh(t) zP1 = zt + √︁ , a2sinh2(t) + b2cosh2(t) db cosh(t) xP2 = xt + √︁ , a2sinh2(t) + b2cosh2(t) da sinh(t) zP2 = zt − √︁ a2sinh2(t) + b2cosh2(t)

for xt = a cosh(t), zt = b sinh(t). By conducting analogous calculations as in 3.2 we get the coordinates of the point Q1 (and the point Q2) (the intersection of the normal line l to the hyperbola ch(t) at the

point P(xt , zt) with the hyperbola ch1(ϕ) (with the hyperbola ch2(ϕ))).

Lemma 2 (a) The coordinates of the point Q1 can be de- Figure 7: Wall model of a cooling tower termined using the parametric equations

x = cosh(t)(a − kdb), z = sinh(t)(b + kda) (9) is coloured dark blue. The use of the offset surface (i.e. hori- l c t zontal offset thickness) could reinforce the structure ofthe of the normal line to the hyperbola h( ) at the point P x z k chimney wall. The wall would be more resistant to wind ( t , t) for the following parameter (︂ )︂ force. The surfaces sh(t, v) and sh1(ϕ, v) were cut out to a cosh(ϕ1) 1 k = k1 = 1 − + (10) show the next layer (see Figure 7). db cosh(t) b cosh(t) (︂ (︂ k da )︂)︂ ϕ E t 1 sh(t, v): x = a cosh(t) cos(v), 1 = ln 1 = arsinh sinh( ) 1 + b , y = a cosh(t) sin(v), z = b sinh(t), √ where E1 = (−B + ∆)/2A for sh1(ϕ, v): x = (a cosh(ϕ) − d) cos(v), A = a2 sinh(t) + b2 cosh(t), y = (a cosh(ϕ) − d) sin(v), z = b sinh(ϕ), B = −2 sinh(t)(cosh(t)(a2 + b2) + da), off(sh; d) = sh(t, v) + dnver(t, v), C = a2 sinh(t) − b2 cosh(t), ∆ = B2 − 4AC. where nver(t, v) is the unit normal vector to the surface (b) The coordinates of the point Q2 can be determined sh(t, v) at any point. using the parametric equations

x = cosh(t)(a + kdb), z = sinh(t)(b − kda) (11) 4.1 The coordinates of points P1, P2, Q1 and of the normal line l to the hyperbola ch(t) at the point Q2 P(xt , zt) for the following parameter k (︂ )︂ P P P Q Q a cosh(ϕ ) 1 Let the points , 1, 2, 1, 2 and curves (hyperbolas) k = k = 1 − 1 + (12) 1 db cosh(t) b cosh(t) ch(t), ch1(ϕ), ch2(ϕ) be defined as in section 2 (see Figure 3, (︂ (︂ k da )︂)︂ (2), (3)). We can assume that t, ϕ ≥ 0, because the graph of ϕ E t 1 1 = ln 1 = arsinh sinh( ) 1 − b , the curve ch(t) (also ch1(ϕ), ch2(ϕ)) is symmetrical about √ the X axis. By conducting analogous calculations as in 3.1 where E1 = (−B + ∆)/2A for we get the coordinates of points P1 and P2 lying on the nor- A a2 t b2 t mal l to the hyperbola ch(t) (at the point P(xt , zt)) and dis- = sinh( ) + cosh( ), B = −2 sinh(t)(cosh(t)(a2 + b2) − da), C = a2 sinh(t) − b2 cosh(t), ∆ = B2 − 4AC. Hyperboloid offset surface in the architecture and construction industry Ë 409

The measurements of the shape of the outer surface of the Rybnik chimney made in 1991 and 2010 showed the execution of the object with numerous deviations from the ideal geometry of the one-sheet rotational hyperboloid. The results obtained in 2010 gave extreme deviation values (to the interior of the shell −1.124m, outside the shell +0.788m) (cf. [11]). The measurements made in the last 19 years have shown a very disturbing increment of displacements in extreme cases reaching values of 0.3-0.4m (with a tendency to move towards the interior in the lower part of the shell and outside in the upper one). The tests showed a clear threat to the load-bearing capacity and stability of the object (cf. [11]). The method for determining the wall thickness of the cooling tower has been given. The thickness of the chimney wall is measured along the normal line l to the external surface of the chimney passing through the point P. More specifically, it was assumed that the external surface (the inner surface) of the chimney was shaped as Figure 8: Diagram of the Rybnik chimney (cf. [11, 12]) the hyperboloid surface SRh(t, v) (respectively as the hy-

perboloid surface SRh1(ϕ, v)). The formula specifying dis- 4.2 Variability of the thickness of the tance dPQ1 = |PQ1| (measured along the normal l to the cooling tower shell surface SRh(t, v), between the point P and the point Q1 (the intersection of the normal l with the hyperboloid sur- The deviation of the chimney wall thickness from the hor- face SRh1(ϕ, v))) was given. Because both hyperboloids izontal offset thickness has been analyzed. The structure are surfaces of revolution, they were replaced with hyper- under investigation was the chimney of the Rybnik cooling bolas ch(t) and ch1(ϕ) on the XZ plane. Because the hyper- tower. The outer side of the chimney of the Rybnik cooling bolas graphs are symmetrical about the X axis, it was as- t ϕ tower has the shape of the hyperboloid SRh(t, v). Parame- sumed that , ≥ 0. Points and curves are defined as in ters of the Rybnik chimney are shown in Figure 8 (a = 26m, Figure 3. b = 60.83m, constant shell thickness d = 0.14m, radius Test 1 A numerical analysis was carried out for the prob- of the bottom base R2 = 47.5m, radius of the upper base lem described above (see Table 1). R1 = 27.4m, chimney height z = 93 + 20m). Figure 8 also d shows the variability of the chimney shell thickness. The The distances PQ1 were measured for points P x z z first ring from the bottom is 0.6m thick. Next, on the 13me- ( P , P), where P = 0, 10, ..., 100m. The distance d z tres section, the thickness of the shell decreases to 0.18m PQ1 decreases with the increase of P and at the height z d d according to the hyperbolic function. On the next 36 me- P = 100m PQ1 = 0.939 . tres section the thickness decreases linearly to 0.14m. At In order to strengthen the chimney wall (at the design the next stage (59m) the thickness is constant, equal to stage) in the part where the thickness of the shell is small- 0.14m. Over the last 5m the thickness increases linearly est, it is proposed to design a wall with the horizontal off- to 0.25m. We assume that the inside of the chimney (on set thickness. Such a construction will be more resistant the 59 metres section) has the shape of the hyperboloid to wind force. For this purpose, it is enough to shape the inside of the chimney using the hyperboloid offset surface SRh1(ϕ, v). off(SRh; d) SRh(t, v): x = 26 cosh(t) cos(v), off(SRh; d): x = cosh(t)(a − dbkP1) cos(v), y = 26 cosh(t) sin(v), z = 60.83 sinh(t), y = cosh(t)(a − dbkP1) sin(v), SRh1(ϕ, v): x = (26 cosh(ϕ) − d) cos(v), z = sinh(t)(b + dakP1), y = (26 cosh(ϕ) − d) sin(v), √︁ 2 2 2 2 z = 60.83 sinh(ϕ). where kP1 = 1 a sinh (t) + b cosh (t). 410 Ë E. Koźniewski and A. Borowska

Table 1: Coordinates of points P and Q1, the angle t (for the point P) and ϕ (for the point Q1), coeflcient k and distance dPQ1 measured at successive heights of the Rybnik chimney

xP / zP t / ϕ k / dPQ1 xQ1 / zQ1 ∘ xP=26 t=0 k=0.016439 xQ1=25.86 ∘ zP=0 ϕ=0 dPQ1=0.14(100%) zQ1=0 ∘ xP=26.3490 t=9 22’37" k=0.016144 xQ1=26.2096 ∘ zP=10 ϕ=9 23’10" dPQ1=0.1397(99.8%) zQ1=10.0097 ∘ xP=27.3692 t=18 30’51" k=0.015343 xQ1=27.2317 ∘ zP=20 ϕ=18 31’50" dPQ1=0.1388(99.1%) zQ1=20.0184 ∘ xP=28.9900 t=27 13’17" k=0.014235 xQ1=28.8548 ∘ zP=30 ϕ=27 14’34" dPQ1=0.1376(98.3%) zQ1=30.0255 ∘ xP=31.1175 t=35 23’1" k=0.013017 xQ1=30.9849 ∘ zP=40 ϕ=35 24’29" dPQ1=0.1363(97.3%) zQ1=40.0311 ∘ xP=33.6559 t=42 57’22" k=0.011828 xQ1=33.5255 ∘ zP=50 ϕ=42 58’55" dPQ1=0.1351(96.5%) zQ1=50.0354 ∘ xP=36.5196 t=49 56’39" k=0.010736 xQ1=36.3911 ∘ zP=60 ϕ=49 58’12" dPQ1=0.1341(95.8%) zQ1=60.0385 ∘ xP=39.6380 t=56 22’56" k=0.009766 xQ1=39.5112 ∘ zP=70 ϕ=56 24’27" dPQ1=0.1332(95.1%) zQ1=70.0409 ∘ xP=42.9559 t=62 19’4" k=0.008918 xQ1=42.8304 ∘ zP=80 ϕ=62 20’31" dPQ1=0.1325(94.6%) zQ1=80.0427 ∘ xP=46.4303 t=67 48’5" k=0.008180 xQ1=46.3059 ∘ zP=90 ϕ=67 49’28" dPQ1=0.1320(94.3%) zQ1=90.0440 ∘ xP=50.0288 t=72 52’56" k=0.007538 xQ1=49.9053 ∘ zP=100 ϕ=72 54’16" dPQ1=0.1315(93.9%) zQ1=100.0451

Test 2 A numerical analysis was carried out for the fol- surface off(SRh; d)). The distance dP1PP1 increases with in- lowing problem. At the following heights zPP1 = 0, 10, creasing zP1. ..., 100m of the chimney (see the point PP1(xPP1, zPP1) in Figure 3) the horizontal offset thickness of the chimney wall was determined. I.e. for consecutive values 5 Applications of the given method zP1 = zPP1 the coordinates of points P1, PP1 and distance dP PP = |P PP | were calculated. Because the hy- 1 1 1 1 The given geometric method can be used, among other perboloid SRh(t, v) and the offset off(SRh; d) are surfaces things, for modelling hyperboloid objects in architecture of revolution, they were replaced with hyperbola ch(t) and and civil engineering design. Its advantages will be pre- offset off(ch; d) on the XZ plane. Because the graphs of sented using examples of domes and elevations designed both these curves are symmetrical about the X axis, it was with a hyperboloid shape. assumed that t, ϕ ≥ 0. Points and curves are defined as in The geometry of the base surface of the dome (roof) Figure 3. has a decisive influence on the majority of its construction Method. For each value i = 0, 10, ..., 100 such an an- features: load-bearing capacity, rigidity, simplicity of execution, aesthetics. Therefore, at present mainly geodesic gle t was determined that zP1(t) = i. Because zP1(t) = lattice domes are used (low dead weight of the structure zPP1(t), the coordinates of the point PP1 can be determined with relatively high load capacity [13]) (cf. [14]). Since as follows xPP1 = a cosh(α), zPP1 = b sinh(α), where the offset surfaces of the sphere are spheres, it iseasy α = arsinh(zP1/b). Additionally, the point Q1(xQ1, zQ1) was determined for the angle ϕ (cf. (10)). Table 2 also gives to obtain surfaces approximating the base sphere. How- ever, for more representative architectural objects, design- the distance dPQ1 ≤ d (when the inner wall is shaped ers propose more glamorous roofs (Sydney Opera House, as the surface SRh1) and the horizontal offset thickness Wanda Metropolitano (Madrit), Yas Island Marina Hotel dP1PP1 ≥ d (when the inner wall is shaped as the offset Hyperboloid offset surface in the architecture and construction industry Ë 411

Table 2: Coordinates of points P, PP1, P1 and Q1, the angle t (for the point P), ϕ (for the point Q1), α (for the point PP1) and distances dPQ1 and dP1PP1 measured at successive heights of the Rybnik chimney

xP/xPP1/xP1/xQ1 zP/zPP1/zP1/zQ1 t / ϕ / α dPQ1 / dP1PP1 ∘ xP=26 zP=0 t=0 dPQ1=0.14 ∘ xPP1=26 zPP1=0 ϕ=0 (100%) ∘ xP1=25.86 zP1=0 α=0 dP1PP1=0.14 xQ1=25.86 zQ1=0 ∘ xP=27.3668 zP=19.9815 t=18 29’51" dPQ1=0.1388 ∘ xPP1=27.3692 zPP1=20 ϕ=18 30’50" (99.1%) ∘ xP1=27.2280 zP1=20 α=18 30’51" dP1PP1=0.1412 xQ1=27.2292 zQ1=19.9998 ∘ xP=31.1100 zP=39.9680 t=35 21’30" dPQ1=0.1363 ∘ xPP1=31.1175 zPP1=40 ϕ=35 22’58" (97.3%) ∘ xP1=30.9737 zP1=40 α=35 23’1" dP1PP1=0.1438 xQ1=30.9773 zQ1=39.9991 ∘ xP=36.5075 zP=59.9598 t=49 55’2" dPQ1=0.1341 ∘ xPP1=36.5196 zPP1=60 ϕ=49 56’35" (95.8%) ∘ xP1=36.3734 zP1=60 α=49 56’39" dP1PP1=0.1462 xQ1=36.3790 zQ1=59.9983 ∘ xP=42.9405 zP=79.9549 t=62 17’31" dPQ1=0.1325 ∘ xPP1=42.9559 zPP1=80 ϕ=62 18’59" (94.6%) ∘ xP1=42.8080 zP1=80 α=62 19’4" dP1PP1=0.1479 xQ1=42.8150 zQ1=79.9976 ∘ xP=50.0113 zP=99.9520 t=72 51’32" dPQ1=0.1315 ∘ xPP1=50.0288 zPP1=100 ϕ=72 52’51" (93.9%) ∘ xP1=49.8798 zP1=100 α=72 52’56" dP1PP1=0.1490 xQ1=49.8878 zQ1=99.9971

(Abu Dhabi), Mercedes Benz Stadium (Atlanta), Khalifa In- three-dimensional mock-up made by the artist. There are ternational Stadium (Qatar). a lot of programs (Digital Project, Rhino-Grasshopper, CA- In order to shape a lattice dome (or a roof) it is nec- TIA, Pro / ENGINEER) that contain parametric design tools essary to define (in a mathematical way) the surface on and offer designers tools for creating parametric scripts which the coating will be stretched. The proposed method (cf. [16, 17]). Mainly thanks to them, it was possible to provides simple mathematical formulas for angles and co- design objects such as Sagrada Familia (Barcelona), Bei- ordinates of points of various surfaces approximating the jing National Stadium (China) (cf. [16]). The reader can see hyperboloid of revolution. Additionally, it allows the de- the parametric script for positioning hyperbolas and their signer to adjust the height of the dome in order to ensure its placement on a given substrate written (by Burry M.) in stability. The given method can be used in CAD (Computer- Python programming language (cf. [16, 18]). (B) The pos- Aided Design). (A) Thanks to the given formulas, any ele- sibility of (easy) precise dimensioning of individual ele- ment (e.g. a dome, its fragment) of hyperboloid shape can ments of a structure with complex geometry can be used be designed in any programming language (e.g. Cpp) and for prefabrication using CNC machine (Computerized Nu- saved in DXF format. The DXF file can be imported into Au- merical Control). More precisely, for a 3D model designed toCAD to get a DWG file (a standard file format for CAD). In in CAD a designer can obtain (using CAM software (Com- this way, the designer can obtain (cf. [15]) in a very short puter Aided Manufacturing)) instructions for a CNC ma- time (at very little cost) a set of complicated digital mod- chine [19]. CNC machining allows fast, precise and repeat- els (for different parameters). If the object consists of many able execution of elements with a complex shape. Hence, hyperboloid fragments, in order to make a digital mock- more and more architectural companies are interested in up of the whole object, the designer can "assemble" cre- designing objects with a curvilinear geometry. The advan- ated digital submodels (fragments) instead of scanning a tages of assembling prefabricated structures are precise ex- 412 Ë E. Koźniewski and A. Borowska ecution, easier transport and assembly. The hyperboloid to strengthen the wall of the cooling tower (at the design surface ensures repeatability of elements (e.g. triangular stage). The new mathematical models have been tested panels) at the same height of the coating. and proven in real cases, such as the Rybnik cooling tower The given method simplifies the analysis of the approx- chimney, providing real parameters. imation of the hyperboloid offset surface off(S(t, v); d) by A double-layer hyperboloid-shaped lattice dome was the hyperboloid surface S1(ϕ, v). For illustration, the wall proposed as a roof for a large object (e.g. shopping cen- thickness of the cooling tower formed by two one-sheet tre). The connection by means of bars of two layers with rotational hyperboloids SRh(t, v) and SRh1(ϕ, v) was an- an approximate shape is intended to strengthen the roof alyzed (cf. section 4.2). The distance between the two hy- structure. The bar structures of the low-profile single-layer perboloid surfaces was measured (along the normal line domes are particularly susceptible to stability loss (cf. [8]). to the surface SRh(t, v)). It was shown that the deviation The main advantage of the given method is the sim- from the constant thickness d = 0.14m increases with the plification of the design process as well as time andcost increase of the coordinate zP. For the cooling tower Rybnik, saving. To design any hyperboloid shaped dome, a student the largest deviation is 3% (dPQ1 = 0.1358m) and takes needs a compiler for a programming language (cost=0) place for zP = 44m (=59m-15m (cf. Figure 8)). and AutoCAD (for students). In a similar way, the thickness of the dome formed by the two-sheet rotational hyperboloids Sh(t, v) and Acknowledgement: The research presented in this

Sh1(ϕ, v) (cf. 3.2) was tested (for the parameter values paper was founded by the BST S/WI/1/2014 and a = 30, b = 40, d = 0.4m). The maximum devi- WZ/WBiIŚ/6/2019. ation from the intended thickness d = 0.4m (for the dome with a base diameter equal to 30m (40m, 50m, 60m) and a height equal to 4.7214m (8.0740m, 12.0684m, References 16.5686m)) is 14.00% (19.50%, 23.86%, 27.18%). The test b was repeated for = 50. Then the maximum deviation [1] Figure 1. https://commons.wikimedia.org/wiki/File:Kobe_Kobe_ from the intended thickness d (of the dome with a base di- Port_Tower_%26_Maritime_Museum_1.jpg. ameter equal to 30m (40m, 50m, 60m) and a height equal [2] Figure 2. http://chodor-projekt.net/encyclopedia/najwieksze- to 5.9017m (10.0925m, 15.0854m, 20.7107m)) was 19.67% kopuly. (26.45%, 31.53%, 35.24%). [3] Koźniewski E., Offsets in geometric creation of roof skeletons with varying slope and cut-and-fill problems in topographic pro- Similar calculations can help a designer decide jection, The Journal Biuletyn of Polish Society for Geometry and whether to use the offset surface and assemble the ob- Engineering Graphics, 2010, vol. 21, 29-35. ject from the CNC prefabricates or accept the deviation (or [4] Koźniewski M., Thickness analysis of a saddle, The Journal Bi- change the values of the hyperboloid parameters). uletyn of Polish Society for Geometry and Engineering Graphics, 2016, vol. 28, 25-32. [5] Borowska A., Approximation of the ellipse offset curves in turbo roundabouts design, The Journal Biuletyn of Polish Society for 6 Summary Geometry and Engineering Graphics, 2018, vol. 31, 43-51. [6] Pottmann H., Asperl A., Hofer M., Kilian A., Architectural Geome- try, 2007, Pennsylvania USA: Bentley Institute Press. Exton. In order to design objects in the shape of the hyperboloid [7] Leja F., Rachunek różniczkowy i całkowy [Differential and integral S(t, v), different surfaces approximating this hyperboloid calculus], 1973, Warsaw: PWN. are needed. It is necessary to ensure that the construc- [8] Bysiec D., The investigation of stability of double-layer tion has the appropriate thickness (stability). The paper octahedron-based geodesic domes, Structure and Environment, presents the method of determining the angles and coor- 2011, vol. 3, No. 3, 30-41. [9] Mirski J., The investigation of the geometric stability of the se- dinates of points of various surfaces approximating the lected group of the double-layer bar domes. The 6th Scientific- hyperboloid of revolution. Thanks to this method, the de- Technical Conference "Current Scientific and Exploratory Prob- signer can easily calculate the distance between the hyper- lems in Civil Engineering", The University of Warmia and Mazury, boloid S(t, v) and the surface approximating it along the (Olsztyn-Kortowo, Poland), 2003, 225-234. normal line l passing through the point P. [10] Grzegorczyk J., Mathematics, 1978, Warsaw: Wydawnictwa Po- litechniki Warszawskiej. The one-sheet hyperboloid is a doubly ruled surface. [11] Mazur J., Zastosowanie nowoczesnych technologii wzmacniania It means that hyperboloid shaped constructions can be obiektów przemysłwych na przykładzie wzmocnienia płaszcza built with straight steel beams that form a strong struc- żelbetowego chłodni kominowej nr 1 w EDF Rybnik SA (Elektrow- ture. The surface approximating the hyperboloid was used nia Rybnik SA) [Application of modern reinforcement technolo- Hyperboloid offset surface in the architecture and construction industry Ë 413

gies for industrial structures shown in exemplary reinforced [15] Koźniewski E., Geometria odwzorowań inżynierskich concrete lining repair at cooling tower no. 1 in EDF Rybnik SA powierzchni 05A, Scriptiones Geometrica, 2014, vol. 1, No. 5A, (Rybnik Power Plant SA)]. XXVI Konferencja Naukowo-Techniczna 1-17. https://docplayer.pl/14511873-Geometria-odwzorowan- Awarie Budowlane, (Międzyzdroje, Poland), 2013, May 21-24, inzynierskich-powierzchnie-05a.html. 1023-1032. [16] Januszkiewicz K., Projektowanie parametryczne oraz parame- [12] Owczarzy J., Kossowski J., Nieklasyczne zachowanie się modelu tryczne narzędzia cyfrowe w projektowaniu architektonicznym hiperboloidalnej chłodni kominowej pod obciążeniem osiowo [Parametric design and parametric digital tools in architectural symetrycznym [A nonclasic behaviour model of hyperbolic shell design], Architecture et Artibus, 2016, vol. 8, No. 3, 43-60. of cooling tower under axisymmetric loading], Journal of Theo- [17] Stavrić M., Stokić D., Ilić M., Architectural Scale Model in Dig- retical and Applied Mechanics, 1981, 2, 19, 225-238. ital Age – Design Process, Representation and Manufacturing, [13] Radoń U., Zastosowanie metody FORM w analizie niezawod- ECAADE Conference, (Prague, Czech Republic), 2012, September ności konstrukcji kratowych podatnych na przeskok [Applica- 12, Prague: Faculty of Architecture, 33-41. tion method FORM in reliability analysis of node snapping truss [18] Burry M., Scripting Cultures, 2013, Chichester: John Wiley & Sons structures], Monografie Studia Rozprawy nr M27, 2012, Kielce: Ltd. Wydawnictwo Politechniki Świętokrzyskiej. [19] Czech-Dudek K., Zastosowanie systemów CAD/CAM w przygo- [14] Gayakwad B.R., Hiriyur A., Patil V.V., Arjun K., Lakshmi S., Design towaniu produkcji [Application systems CAD/CAM in the prepa- considerations for a geodesic dome – a critical review, Interna- ration of production], Mechanik, 2015, 7, 149-158. tional Journal of Advances in Science Engineering and Technol- ogy, 2018, vol. 6, Iss. 1, 8-13.