REVIEWThe two-revolution cyclical surface <strong>of</strong> <strong>the</strong> type P g 2(u,v) is cre<strong>at</strong>edby transl<strong>at</strong>ion <strong>of</strong> <strong>the</strong> circle c 2 (R 2 , r 2 ) along <strong>the</strong> curve k g 2 <strong>at</strong> anangular velocity w 2 m 1 v, where <strong>the</strong> circle is always in <strong>the</strong> plane(n 2 , b 2 ) if <strong>the</strong> index g t, or in <strong>the</strong> plane (t 2 , b 2 )ifg n, or in<strong>the</strong> plane (t 2 , n 2 )ifg b and its centre is <strong>the</strong> point R 2 ∈ k g 2. Wewill cre<strong>at</strong>e it so th<strong>at</strong> <strong>the</strong> circle c 02 determined by <strong>the</strong> vector functionc 02 (u) (0, r 2 cos u, r 2 sin u, 1) if g t, or c 02 (u) (r 2 cos u,0, r 2 sin u, 1) if g n, or c 02 (u) (r 2 cos u, r 2 sin u, 0, 1) if g bwe will transform into <strong>the</strong> circle c 2 in <strong>the</strong> coordin<strong>at</strong>e system (R 2 ,t 2 , n 2 , b 2 ) using <strong>the</strong> m<strong>at</strong>rix M 2 (w 2 ) by equ<strong>at</strong>ions (5) (Fig. 4). Thevector function <strong>of</strong> <strong>the</strong> cyclical surface <strong>of</strong> <strong>the</strong> type P g 2(u,v) isP g 2(u,v) r 2 (v) c 02 (u) . M 2 (w 2 ),u ∈ 0, 2π. v ∈ 0, 2π. (9)Fig. 2 Cre<strong>at</strong>ion <strong>the</strong> curves k 1 , k g 2, k g 3 h and <strong>the</strong>ir trihedronswhere r 2 (v) is <strong>the</strong> vector function <strong>of</strong> <strong>the</strong> curve k g 2 determinedby (6).Fig. 3 Combin<strong>at</strong>ions <strong>of</strong> <strong>the</strong> curves k 1 , k t 2, k t 3 t , k 1 , k n 2, k n 3 n and k 1 , k b 2, k b 3 b .3. Vector functions <strong>of</strong> cyclical surfaces <strong>of</strong> <strong>the</strong> typeP 1 (u,v), P g 2(u,v), P g 3 h (u,v)The cyclical surface <strong>of</strong> <strong>the</strong> type P 1 (u,v) is cre<strong>at</strong>ed by transl<strong>at</strong>ion<strong>of</strong> <strong>the</strong> circle c 1 (R 1 , r 1 ) in <strong>the</strong> plane (n 1 , b 1 ) along <strong>the</strong> curvek 1 <strong>at</strong> an angular velocity w 1 v. We will cre<strong>at</strong>e it so th<strong>at</strong> <strong>the</strong> circlec 01 determined by <strong>the</strong> vector function c 01 (u) (0, r 1 cos u, r 1 sin u,1) will be transformed into <strong>the</strong> circle c 1 in <strong>the</strong> coordin<strong>at</strong>e system(R 1 , t 1 , n 1 , b 1 ) using <strong>the</strong> m<strong>at</strong>rix M 1 (w 1 ) expressed by equ<strong>at</strong>ions (5)(Fig. 4).isThe vector function <strong>of</strong> <strong>the</strong> cyclical surface <strong>of</strong> <strong>the</strong> type P 1 (u,v)P 1 (u,v) r 1 (v) c 01 (u) . M 1 (w 1 ),u ∈ 0, 2π. v ∈ 0, 2π. (8)where r 1 (v) is <strong>the</strong> vector function <strong>of</strong> <strong>the</strong> curve k 1 determined byequ<strong>at</strong>ion (1). This surface is surface <strong>of</strong> torus.The three-revolution cyclical surface <strong>of</strong> <strong>the</strong> type P g 3 h (u,v) iscre<strong>at</strong>ed by transl<strong>at</strong>ion <strong>of</strong> <strong>the</strong> circle c 3 (R 3 , r 3 ) along <strong>the</strong> curvek 3 <strong>at</strong> an angular velocity w 3 m 2 m 1 v, where <strong>the</strong> circle is alwaysin <strong>the</strong> plane (n 3 , b 3 ) if <strong>the</strong> index h t, or in <strong>the</strong> plane (t 3 , b 3 ) ifh n, or in <strong>the</strong> plane (t 3 , n 3 ) if h b and its centre is <strong>the</strong> pointR 3 ∈ k g 3 h . We will cre<strong>at</strong>e it so th<strong>at</strong> <strong>the</strong> circle c 03 determined <strong>the</strong> byvector function c 03 (u) (0, r 3 cos u, r 3 sin u, 1) if h t, orc 03 (u) (r 3 cos u, 0, r 3 sin u, 1) if h n, or c 03 (u) (r 3 cos u, r 3sin u, 0, 1) if h b we will transform into <strong>the</strong> circle c 3 in <strong>the</strong> coordin<strong>at</strong>esystem (R 3 , t 3 , n 3 , b 3 ) using <strong>the</strong> m<strong>at</strong>rix M 3 (w 3 ) by equ<strong>at</strong>ions(5). The vector function <strong>of</strong> <strong>the</strong> cyclical surface <strong>of</strong> <strong>the</strong> typeP g 3 h (u,v) isP g 3 h (u,v) r 3 (v) c 03 (u) . M 3 (w 3 ),u ∈ 0, 2π. v ∈ 0, 2π. (10)where r 3 (v) is <strong>the</strong> vector function <strong>of</strong> <strong>the</strong> curve k g 3 h determined by(7).74 ● COMMUNICATIONS 3/2008
REVIEWFig. 4 Transform<strong>at</strong>ion <strong>of</strong> <strong>the</strong> circle c 0i into <strong>the</strong> circle c i4. Classific<strong>at</strong>ion <strong>of</strong> cyclical surfaces <strong>of</strong> <strong>the</strong> typeP g 2(u,v), P g 3 h (u,v)The two-revolution cyclical surface <strong>of</strong> <strong>the</strong> type P g 2(u,v) can beclassified according to <strong>the</strong> index g:Table 1: Classific<strong>at</strong>ion <strong>of</strong> cyclical surfaces <strong>of</strong> <strong>the</strong> type P g 2(u,v)g t n bP t2(u,v) P n 2(u,v) P b 2(u,v)The three-revolution cyclical surface <strong>of</strong> <strong>the</strong> type P g 3 h (u,v) canbe classified according to <strong>the</strong> index g and h:Table 2: Classific<strong>at</strong>ion <strong>of</strong> cyclical surfaces <strong>of</strong> <strong>the</strong> type P g 3 h (u,v)g/h t n btP t3 t (u,v) P t3 n (u,v) P t3 b (u,v)n P n 3 t (u,v) P n 3 n (u,v) P n 3 b (u,v)5. Illustr<strong>at</strong>ions <strong>of</strong> cyclical surfaces <strong>of</strong> <strong>the</strong> typeP 1 (u,v), P g 2(u,v), P g 3 h (u,v)In Fig. 5 <strong>the</strong>re are displayed three combin<strong>at</strong>ions <strong>of</strong> cyclicalsurfaces <strong>of</strong> <strong>the</strong> type P 1 (u,v), P t 2(u,v), P t 3t (u,v) in fig. a), P 1 (u,v),P t 2(u,v), P t 3n (u,v) in fig.b), P 1 (u,v), P t 2(u,v), P t 3b (u,v) in fig. c).In Fig. 6 <strong>the</strong>re are displayed three combin<strong>at</strong>ions <strong>of</strong> cyclicalsurfaces <strong>of</strong> <strong>the</strong> type P 1 (u,v), P n 2(u,v), P n 3t (u,v) in fig. a), P 1 (u,v),P n 2(u,v), P n 3n (u,v) in fig. b), P 1 (u,v), P n 2(u,v), P n 3b (u,v) in fig. c).In Fig. 7 <strong>the</strong>re are displayed three combin<strong>at</strong>ions <strong>of</strong> cyclicalsurfaces <strong>of</strong> <strong>the</strong> type P 1 (u,v), P b 2(u,v), P b 3t (u,v) in fig. a), P 1 (u,v),P b 2(u,v), P b 3n (u,v) in fig.b), P 1 (u,v), P b 2(u,v), P b 3b (u,v) in fig. c).The surfaces mentioned above can be used in design practiceas constructive or ornamental structural components.The paper was supported by <strong>the</strong> grant VEGA 1 / 4002 / 07.b P b 3 t (u,v) P b 3 n (u,v) P b 3 b (u,v)a) b) c)Fig. 5 Combin<strong>at</strong>ions <strong>of</strong> cyclical surfaces for g tCOMMUNICATIONS 3/2008 ●75
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References[1] JOVANOVIC, R.: Tensio
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Fig. 3 Geometrical parameters of ch
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Fig. 13 10. eigenvector when the ei
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and D R _ y,wi=0 if y 0 (12) Dampe
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Fig. 22 Comfort for passengers for
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3. Intermodal trainsIntermodal trai
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Fig. 2 Consignment reloading proces
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- Page 24 and 25: Juraj Gerlici - Tomas Lack *MODIFIE
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- Page 30 and 31: [2] GERLICI, J., LACK, T.: Methods
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- Page 44 and 45: REVIEWlines with numerous curves th
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- Page 48 and 49: REVIEWBojan Cene - Aleksandar Rados
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- Page 55 and 56: REVIEWRadomir Brkic - Zivoslav Adam
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- Page 59 and 60: REVIEWJan Krmela *COMPUTATIONAL MOD
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- Page 75 and 76: BOOK REVIEWKavicka, A., Klima, V.,