Helical-One, Two, Three-Revolutional CyclicalSurfaces

Transcription

1 Gloal Journal of Science Frontier Research Mathematics and Decision Sciences Volume 3 Issue 4 Version. Year Tpe : Doule Blind eer Reviewed International Research Journal ulisher: Gloal Journals Inc. (USA Online ISSN: & rint ISSN: B T. Olejníková Technical Universit of Košice Astract - This paper descries method for modelling of helical-n-revolutional cclical surfaces. The ais of the cclical surface Φ is the heli s created revolving the point aout n each other revolving aes o n (n,2,3, that move together with Frenet-Serret moving trihedron along the clindrical heli s. articular evolutions are determined its angular velocit and orientation. The moving circle along the heli s or s, where its center lies on the heli and circle lies in the normal plane of the heli creates the cclical surface. Kewords : cclical surface, heli, frenet-serret moving trihedron, transformation matrices. GJSFR-F Classification : MSC 2: 5B2 Helical-One, Two, Three-Revolutional CclicalSurfaces Strictl as per the compliance and regulations of :. T. Olejníková. This is a research/review paper, distriuted under the terms of the Creative Commons Attriution- Noncommercial 3. Unported License permitting all non commercial use, distriution, and reproduction in an medium, provided the original work is properl cited.

2 Helical-One, Two, Three-Revolutional Cclical Surfaces T. Olejníková Astract - This paper descries method for modelling of helical-n-revolutional cclical surfaces. The ais of the cclical surface Φ is the heli s created revolving the point aout n each other revolving aes o n (n,2,3, that move together with Frenet-Serret moving trihedron along the clindrical heli s. articular evolutions are determined its angular velocit and orientation. The moving circle along the heli s or s, where its center lies on the heli and circle lies in the normal plane of the heli creates the cclical surface. Kewords : cclical surface, heli, frenet-serret moving trihedron, transformation matrices. I. Introduction Let thre-dimensional Euclidean space E 3 is determined Cartesian coordinate sstem (,,,. In this space is given clindrical heli s with ais identical with coordinate ais determined vector function (Fig. (,,, ( a cos mv, a sin mv, v, r, v, 2π, ( s s s where parameter a is radius of the heli, is the reduced pitch, determined orientation of the heli, ( + for right-handed and for left-handed revolution, m is numer of pitches. Let (, n,, t e Frenet-Serret moving trihedron of the clindrical heli s represented regular square matri n n n M t t t, (2 where the matri elements are the coordinates of unit vectors of the principle normal n, inormal and tangent t of the heli s in the point s,,, t in the coordinate sstem ( ( t t, t,, (3, (,, (4 n n, n ( t n,. (5 Gloal Journal of Science Frontier Research F Volume XIII 47 Author : Department of Applied Mathematics Institute of Construction, Technolog and Management Facult of Civil Engineering, Technical Universit of Košice, Slovakia. tatiana.olejnikova@tuke.sk Gloal Journals Inc. (US

3 Transformations of revolutions aout coordinate aes,, are represented matices ϕ, ψ ϕ, ψ T ϕ, ψ, where ϕ is angle and ψ is orientation of the revolution, T (, T (, ( transformation of translation is represented matri ( ± d, ± d, ± d ( ± d, ± d, ± d is translation vector determined its coordinates (6, (7: T, where cosϕ T ( ϕ, ψ, ( ϕ, ψ ψ sin ϕ ψ sin ϕ cosϕ cosϕ ψ sin ϕ T ψ sin ϕ cosϕ, (6 Gloal Journal of Science Frontier Research F Volume XIII 48 ψ sin ϕ cosϕ T ( ϕ, ψ, ( ± d, ± d, ± d cosϕ ψ sin ϕ The moving circle c (, r T. (7 ± d ± d ± d along the heli s, where its center lies in the normal plane determined principal normal n and inormal of the heli in the point s creates the cclical surface Φ. The vector function of this surface is ( u, v r + c ( u M., u, 2π, v, 2π, (8 where r is vector function of the heli s epressed in equation (, of the coordinate sstem (, n,, t into coordinate sstem (,, ( u ( r cos u, r sin u,,, u, 2π M is transformation matri, (2 and c is vector function of the circle c determined its center and radius r (Fig.2. In Fig.3 there are displaed two screws of the right-handed cclical surface Φ together with the clindrical surface on which heli s is wound. s s Fig. : Heli s with Trihedron Fig. 2 : Cclical Surface Φ Fig. 3 : Surface Φ and Clinder c Φ Φ II. Cclical Helical Surface Created One Revolution The heli,,, aout the ais o connected to the moving trihedron of the heli s, is represented vector function s created revolution of the point ( r (,,. T ( m v, M, r. (9 + Gloal Journals Inc. (US

4 and cclical surface Φ created in a similar wa as surface Φ vector function (, v r c ( u M +, u, 2π, v, 2π, ( u. where r is vector function of the heli s epressed in equation (9, M ( v is transformation matri of the coordinate sstem (, n,, t into coordinate sstem (,,, (, c ( u ( r cos u, r sin u,,, u, 2π is vector function of the circle c determined center s and radius r n n t t n M ( v. ( t Elements of this matri are coordinates of unit vectors of the principle normal n, inormal and tangent t of the heli s in the point s in the coordinate sstem (, n,, t r ( t, t, t t (2,, r ( r, (3 n n n ( t n,,. (4 a Revolution aout tangent t of the heli s The heli created the revolution of the point aout the ais o t is epressed vector function (9, in which matri T ( mv, T ( mv,. In Fig.4 is displaed cclical surface Φ, whose ais is heli s with parameters m 2, + and surface Φ, whose ais is heli s created aout tangent t of the heli s with parameters m m,. revolution of the point ( d,,, Φ Φ Φ 3 Φ 2 Φ 4 Φ 8 + Gloal Journal of Science Frontier Research F Volume XIII 49 Fig. 4 : Cclical Surfaces Φ, Φ Fig. 5 : 4 Surfaces i Φ Fig. 6 : Left-handed Surfaces Φ, i Φ Gloal Journals Inc. (US

5 In Fig.5 are displaed k 4 surfaces i Φ, whose aes are heli s, i,..., k created revolution of the points i ( d cosiα, d sin iα,,, α 2π/ k aout tangent t of the heli s with parameters m 4m, +, in Fig.6 are displaed the same surfaces with altered orientation of the revolution m 2,, m m,. 4 Revolution aout principal normal n of the heli s i Gloal Journal of Science Frontier Research F Volume XIII 5 Fig. 7 : Normal Surface Φ Fig. 8 : 4 Surfaces i Φ Fig. 9 : Normal Surfaces Φ, 3 Φ The heli s created the revolution of the point aout the ais o n is epressed vector function (9, in which matri T ( mv, T ( mv,. In Fig.7 is displaed heli s with parameters m 2, + and normal surface Φ, whose ais is heli s created revolution of the point (, d,, aout normal n of the heli s with parameters m m, +, in Fig.8 are displaed k 4 normal surfaces i i Φ, whose aes are heli s, i,..., k created revolution of the points i (, d cosiα, d sin iα,, α 2π/ k aout normal n of the heli s with parameters m 7m,, in Fig.9 are displaed surfaces Φ, 3 Φ with altered orientation of the revolution ±. c Revolution aout inormal of the heli s The heli created the revolution of the point aout ais o is epressed vector function (9, in which matri T ( mv, T ( mv,. In Fig. is displaed heli s with parameters m 2, + and inormal surface Φ, whose ais is heli s created revolution of the point ( d,,, aout inormal of the heli s with parameters m m, +, in Fig. is displaed inormal surface with parameters m 8m,, in Fig.2 are surfaces Φ, 3 Φ created revolution of the points i ( d cosiα, d siniα,,, α 2π/ k aout inormal of the heli with altered orientation of the revolution. ± Gloal Journals Inc. (US

6 Fig. : Binormal Surface Φ Fig. : 4 Surfaces i Φ Fig. 2 : Binormal Surfaces Φ, 3 Φ III. The heli,,, aout ais o 2, which revolves aout the ais o identical with one edge of the moving trihedron of the heli s is represented vector function where matri 2( m2v, 2 ( m v, s Cclical Helical Surface Created Two Revolutions s created revolution of the point ( r (,,. T ( m v,. T ( m v, M r., (5 +, T represents revolution of the point aout the ais o 2 and matri T g represents revolution of the ais o 2 aout the ais o. a Revolution aout two parallel aes n o 2 d d t o o2 n o t d d d t d o 2 o n Fig. 3 Fig. 4 Fig. 5 If the heli s is created revolution of the point aout two parallel aes o 2 o and o t, where d oo 2 is the distance etween them, then (Fig.3 T ( m v T( d,,. T ( m v,. T( +,,, ( m v, T ( m v, s 2 2, d T. (6 g Gloal Journal of Science Frontier Research F Volume XIII 5 In Fig.6 is displaed this surface Φ with parameters m 8m, +, m 2 4m, 2 +. If the heli s is created revolution of the point aout parallel aes o 2 o and where d oo 2, then (Fig.4 o n, T ( m v, T(,, d. T ( m v,. T(, + d, ( m v, T ( m v, s , T. (7 g Gloal Journals Inc. (US

7 Gloal Journal of Science Frontier Research F Volume XIII 52 Fig. 6 : o 2 o, o t Fig. 7 : o 2 o, o n Fig. 8 : o 2 o, o In Fig.7 is displaed this surface Φ with parameters m 6m, +, m 2 4m, 2 +. If the heli s is created revolution of the point (,, d, aout parallel aes o 2 o and o, where d oo 2, then (Fig.5 T ( m v, T(,, d. T ( m v,. T(, + d, ( m v, T ( m v, s , T. (8 g In Fig.8 is displaed this surface Φ with parameters m m,, m m,. Revolution aout two intersecting aes ( o 2 t ( o n ( o 2 n ( o t ( o 2 n ( o Fig. 9 : Fig. 2 : Fig. 2 : In Fig.9 is displaed surface created revolution of the point ( 2,2,, perpendicular aes ( o t ( o n m 2 4m, 2, where matrices T 2( m2v, 2 T ( m2v, 2, ( mv, T ( mv, is displaed surface created revolution of the point ( 2.2,.2,, aes ( o 2 n ( o t determined parameters m 6m, +, m 2 6m, 2 + T ( m v, T ( m v, s, ( m v, T ( m v, s g2 aout mutuall 2 determined the parameters m 6m,, T, in Fig.2 aout mutuall perpendicular, and matrices T, here we see action of changing the order of g the revolutions to form of the surfaces. In Fig.2 is displaed surface created revolution of the point ( 2.5,2.5,, aout mutuall perpendicular aes ( o 2 n ( o determined parameters Gloal Journals Inc. (US

8 m 5m,, m 2 3m, 2 +, T 2( m2v, 2 T ( m2v, 2, T ( mv, T ( mv,.in Figs.22,23 is displaed surface created revolution of the point ( d,,, aout intersecting aes o 2 ( o t determined the parameters m 6m, +, m 2 6m, 2, T ( m v, T ( m v, s, T ( m v T ( α, +. T ( m v,. T ( α,. g c Revolution aout two skew aes 2 2, In Figs.24,25 is displaed surface created revolution of the point ( d,,, aout mutuall skew aes (o 2 n / ( o t m m,, m 8m, 2 2 +, determined parameters 4 +, where transformation matrices of two revolutions are T ( m v, T ( m v, s, ( m v, T(,, d. T ( m v,. T(, + d g T , In Figs.26,27 is displaed surface created revolution of the point ( d,,, aout mutuall skew aes ( o 2 t, o2 n / ( o t determined parameters m 6m, +, m 2 4m, 2, and transformation matrices T ( m v T ( α, +. T ( m v,. T ( α,, T ( m v, T ( m v, s. 2 2, n o t α d d 2 o 2 g 2 ( o t ( o 2 n / ( o t Fig. 22 : o Fig. 23 Fig. 24 : t o t o 2 Gloal Journal of Science Frontier Research F Volume XIII 53 n d d o 2 ( o 2 t, o2 n / ( o t n α d Fig. 25 Fig. 26 : Fig. 27 d 2 o Gloal Journals Inc. (US

9 IV. The heli,,, aout the ais o 3, which revolves aout the ais o 2 and this revolves aout the ais o identical with an edge of the moving trihedron of the heli s is represented vector function s created the revolution of the point ( r (,,. T ( m v,. T ( m v,. T ( m v, M r., (9 + where matri ( m v, s, T 3 3 g3 represents revolution of the point aout the ais 3 T2 m2v, 2 T m v, s represents revolution of represents revolution of the ais o 3 aout the ais 2 the ais o 2 aout the ais o. Cclical Helical Surface Created Three Revolutions o, matri ( o and matri ( g Gloal Journal of Science Frontier Research F Volume XIII 54 a Revolution aout three parallel aes In Fig.28 is displaed surface created revolution aout three parallel aes o 3 o 2 o t determined parameters m 4m, +, m 2 4m, 2 +, m 3 3m2, +, matrices T3 ( m 3v, 3 T( d2,,. T ( m3v, 3. T( + d2,,, T2 ( m 2v, 2 T( d,,. T ( m2v, 2. T( + d,,, T ( mv, T ( mv,. In Fig.29 is displaed surface created revolution aout three parallel aes o 3 o 2 o n determined parameters m 4m, +, m 2 4m, 2 +, m3 4m2, + and transformation matrices T ( mv, T ( mv, m v, T,, d. T m v,. T, + d T m v, T,, d. T m v,. T, + d. T ( ( ( (, ( ( ( ( , , Fig. 28 : o 3 o 2 o t Fig. 29 : o 3 o 2 o n Fig. 3 : o 3 o 2 o In Fig.3 is displaed surface created revolution aout three parallel aes o 3 o 2 o determined parameters m 3m, +, m 2 3m, 2 +, m 3 3m2, + and T m v, T,, d. T m v,. T, + d, transformation matrices 3 ( 3 3 ( 2 ( 3 3 (, 2 T ( m v, T(,, d. T ( m v,. T(, + d, ( m v, T ( m v, s , T. g Revolution aout three perpendicular aes In Figs.3,32,33 are displaed surfaces created revolution of the point ( d, d,, aout three perpendicular aes with common point o3 o2 o, which are identical with edges of the trihedron of the heli s, where parameters are the same m 3m, +, m 2 3m, 2 +, m m,, ut the order of the revolutions changes Gloal Journals Inc. (US

10 Fig. 3 : o n, o o t Fig. 32 : o n, o t o Fig. 33 : o, o n o t 3 2, 3 2, V. Conclusion 3 2, c Revolution aout three skew aes In Fig.34 are displaed surfaces created revolution of the point (,,, aout three skew aes o 3 / o2 / o, which are parallel with edges of the trihedron of the heli s, o 3, n o 2, o 3, t where parameters are m 4m, +, m 2 2m, 2 +, m 3 6m2, +, transformation matrices of three revolutions are T 3 ( m3v, 3 T(,, d3. T ( m3v, 3. T(,, + d3, T (, (,,. (,. (,, 2 m2v 2 T d2 T m2v 2 T + d, 2 T m v T, d,. T m v,. T, +,. ( ( ( (, d2 Fig. 34 : o 3, n o 2, o 3 t Fig. 35 : o 3, n o 2 n, o Fig. 36 : o 3, n o 2 n, o t In Figs.35,36 are displaed surfaces created revolution aout the aes o 3, n o 2 n, o or aes o 3, n o 2 n, o t. The descried method of modeling of the helical-n-revolutional cclical surfaces makes it possile to model different interest surfaces simpl changing the parameters. Gloal Journal of Science Frontier Research F Volume XIII 55 References Références Referencias [] BUDINSKÝ, B., KER, B.: Introduction to differential geometr with technical applications. SNTL ulishers of technical literature, raha, 97 [2] GRANÁT, L., SECHOVSKÝ, H.: Computer graphics. SNTL ulishers of technical literature, raha, 98 Gloal Journals Inc. (US

11 Gloal Journal of Science Frontier Research F Volume XIII 56 [3] OLEJNÍKOVÁ, T.: Two helical surfaces. In: Journal of civil engineering, Selected scientific papers, Košice, 2, Vol. 5, Issue, ISSN , pp.7-6 [4] OLEJNÍKOVÁ, T.: Composed Cclical Surfaces. Transactions of the universities of Košice, 27, Issue.3, ISSN , pp.54-6 [5] OLEJNÍKOVÁ, T.: Rope of Cclical Helical surfaces. In: Journal of civil engineering, Selected scientific papers, Košice, 22, Vol. 7, Issue 2, ISSN , pp.23-32, DOI:.2478/v , 2.ml [6] STANOVÁ, E.: Geometr and Modeling of the Oval Strand of n ( 2n n2 + Tpe. In: Journal of civil engineering, Selected scientific papers, Košice, 22, Vol. 7, Issue 2, ISSN ,pp.33-4, DOI:.2478/v , 2.ml Gloal Journals Inc. (US