The Pitch, the Angle of Pitch, and the Distribution Parameter of a Closed Ruled Surface

Transcription

1 Appl. Math. Inf. Sci. 9 No Applie Mathematics & Information Sciences An International Journal The Pitch the Angle of Pitch the Distribution Parameter of a Close Rule Surface Nemat Abazari Ilgin Sağer Hasan Hilmi Hacisalihoğlu Department of Mathematics Faculty of Mathematical Sciences University of Mohaghegh Arabili Arabil Iran Department of Mathematics Faculty of Science Ankara University Ankara Turkey Department of Mathematics Bilecik Seyh Eebali University Bilecik Turkey Receive: 9 Oct. 04 Revise: 9 Jan. 05 Accepte: 0 Jan. 05 Publishe online: Jul. 05 Abstract: In this paper the pitch the angle of pitch the istribution parameter of the close rule surfaces generate by tangent bi-normal normal vectors unit Darboux vector of a unit ual spherical motion are stuie on the Frenet frame. Keywors: Dual Steiner vector ual spherical motion pitch of the motion Pfaffian vector rule surface. Introuction For the analysis of spatial motions in ifferential geometry [ 6] in kinematics of the spatial mechanisms [ ] the use of ual vectors ual quaternion ual matrix algebra over the ring of ual numbers is a very irect metho. Important properties of a real vector analysis of real matrix algebra are valior the ual vectors ual matrices. The principal part of this metho is base on work by E. Stuy []. The essential iea is to replace points by straight lines as funamental builing blocks of geometric structure. The set of oriente lines in Eucliean three-imensional space E is in one-to-one corresponence with the points of a unit ual sphere in the ual space D of triples of ual numbers. The efinition of the Steiner vector for the real unit sphere [] is extene [] to the efinition of ual Steiner vector. In [] an expression of the pitch of a close rule surface is erive in terms of the elements of the ual Steiner vector. Using this expression the spatial extensions for planar [] spherical [ 4] theorems calle Steiner theorems Holitch theorems are given. The motion corresponing to the ual spherical close motion K/K on D-moule is the one-parameter motion H/H on the line-space. The close rule surface X which is rawn by the line X of H on the fixe space H the pitch the angle of pitch of the close rule surfacex are stuie in []. In this paper the pitch the angle of pitch istribution parameter of the close rule surface each of which generate by tangent bi-normal normal vectors unit Darboux vector on unit ual sphere respectively are investigate. The Pitch of a Close Rule Surface Let H H enote the fixe moving line-spaces respectively. Accoring to the result of E. Stuy unit ual spheres K K centere at a point M correspon to these spaces on D-moule respectively. Also ual-spherical motion K/K correspons to one-parameter spatial motion H/H. Let us take a line X on H. That is to say we consier a fixe point X of the unit ual sphere K. During the motion H/H the line X traces a rule surfacex which is calle the orbit surface on H. The variation of the point X accoring to the fixe sphere K i.e. the variation of the line X on H is X X where the vector Ψ ψ ε ψ Ψ Ψ Ψ with Ψ i ψ i εψi i is calle the instantaneous Pfaffian vector of the motion H/H. The rule surfacex is given by X Xt x t ε x t where X Xt is the unit ual vectorial function Corresponing author abazari@uma.ac.ir

2 80 N. Abazari et. al. : The Pitch the Angle of Pitch the Distribution... parameterize by t R. The ual curve X is the ual spherical formation of the rule surface. For the ual arc element Φ ϕεϕ on the ual spherical curve X Xt Φ X X x x ε x x is vali. efine [5] by x x x x ϕ ϕ ϕ ϕ ϕ. Definition. Let K be the moving unit ual sphere. Subject to the conition that the pitch of the motion is nonvanishing a new coorinate system is introuce by { R R P R }. This frame is calle a canonical coorinate frame. For this case Ψ Ψ R Ψ P is the instantaneous Pfaffian vector [5]. The eclaration of the variation of a point of X K accoring to the canonical coorinate frame on given [5] by X X ψ P X ψ X ψ X R X R Theorem. On the one-parameter ual motion K/K the tangent bi-normal normal vectors of a ual curve X at a point X are given by T X N B R X X R X R X R X R respectively where X X X X. R Proof. Suppose that R R where{ R R R } is R a unit ual orthogonal frame for -space D. Since X is on the unit ual sphere K we may write that X X R X R X R X T R X X X X 0 where X X X is the ual vector corresponing X X to X. The isplacements of R with respect to K K the ual moving anixe sphere respectively are given by where Ψ Ψ R Ψ ψ εψ X XR X R X R X i x i εx i i. p x x x where p ψ ψ is the pitch [5] of the motion H/H. RΩR RΩ R Ω 0 Ω Ω Ω 0 Ω Ω Ω 0 Ω 0 Ω Ω Ω 0 Ω. Ω Ω 0 The Distribution Parameter of a Close Rule Surface On the one-parameter ual spherical motion the fixe point X K constructs a ual curve on K. The tangent bi-normal normal of the ual curve X at a point X are X Ψ X T X X X f X B X f X N B T respectively. Then the isplacements of X with respect to K K are given by X X T RX T RX T X T ΩR X T RX T R T X T Ω R respectively. Since Ω Ω are anti-symmetric matrixes we have Ω T Ω Ω T Ω. For any fixe vector X we get X 0 X 0. Therefore from equations we get X T X T Ω 0 X T X T Ω X T Ω T

3 Appl. Math. Inf. Sci. 9 No / 8 X T X T Ω 0 X T X T Ω X T Ω T. Now suppose that X is fixe in K let us calculate its velocity X with respect to K. Then we obtain that X X X X T Ω ΩR. If we efine a new ual vector whose components in the relative system are Ψ i Ω i Ω i where i Ψ Ψ Ψ Ψ Ψ R Ψ R Ψ R then we get 4 X X 5 where Ψ is the Pfaffian vector corresponing to the ual spherical motion K/K. To calculate the acceleration J f X of X we have J f X X X Ψ X Ψ Ψ Ψ X Ψ X Ψ Ψ Ψ X X Ψ X Ψ X Ψ X Ψ X. By using the matrix form the relations 4 yiel where X MX J f X M ṀX M 0 Ψ Ψ Ψ 0 Ψ Ψ Ψ 0 Ṁ M 0 ψ Ψ Ψ 0 Ψ. Ψ Ψ 0 Hence we obtain M Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ. Ψ Ψ Ψ Ψ Ψ Ψ If Ψ Ψ R then Ψ Ψ 0. Thus M 0 Ψ 0 Ψ hence the velocity vector of X K is given by X MX Ψ X Ψ X 0 Ψ X R X R X T X Ψ X R X R Ψ X X X R X R. X By using the same arguments it is easy to see that B R. Therefore N B T R X R X R X X X This completes the proof. X R R X R R X R X R. From Theorem the following corollary can be obtaine. Corollary. On the one-parameter motion K/K the unit Darboux vector is D 0 τ T κ B τ κ τ X R X R κ R τ κ X where κ k εk τ k εk are the first secon ual curvatures respectively X X X X. Theorem. Consier a point T on the unit moving ual sphere K of the ual-spherical motion K/K corresponing to one-parameter spatial motion H/H

4 8 N. Abazari et. al. : The Pitch the Angle of Pitch the Distribution... such that uring the motion H/H the line T raws a rule surface T on the fixe space H. Then the istribution parameter of this rule surface is T p ψ ψ. Proof. The eclaration of the variation of a point T accoring to the canonical coorinate frame of the T Ψ T Ψ T Ψ X R X R. X ϕ ϕ T ϕ ψ ψ ψ ψ p ψ where Φ ϕ εϕ enotes the ual arc element of the rule surfacet. Theorem. Consier a point B on the unit moving ual sphere K such that uring the motion H/H the line B raws a rule surface B on the fixe space H. Then the istribution parameter of this rule surface is unefine. Proof. The eclaration of the variation of the point B accoring to the canonical coorinate frame on the B Ψ B Ψ B 0 since B R. So on the motion K/K the point B is fixe the istribution parameter of the rule surface is unefine. Theorem 4. Consier a point N on the unit moving ual sphere K such that uring the motion H/H the line N raws a rule surface N on the fixe space H. Then the istribution parameter of this rule surface is N ψ ψ p. Proof. The eclaration of the variation of the point N accoring to the canonical coorinate frame on the N Ψ N ψ ψ N X R X R. X ψ p ψ N where Φ ϕεϕ is the ual arc element of the rule surfacen p is the pitch of the motion. Theorem 5. Consier a point D 0 on the unit moving ual sphere K such that uring the motion H/H the line D 0 raws a rule surface D 0 on the fixe space H. Then the istribution parameter of this rule surface is kk k k D 0 k k k k p where p is the pitch of the motion k k are the first secon ual curvatures respectively. Proof. The eclaration of the variation of the point D 0 accoring to the canonical coorinate frame on the D0 D 0 τψ X R X R. τ k X kk k k D 0 k k p k k where Φ ϕεϕ is the ual arc element of the rule surfaced 0. 4 The Pitch the Angle of Pitch There are four cases of the pitch angle of pitch of the close rule surfaces T N B D 0 on the canonical coorinate frame. Theorem 6. The pitch the angle of pitch the ual angle of pitch of the close rule surface T on the canonical coorinate frame are vanishing. Proof. We know that T X R X R X is vali where T t ε t X i x i ε x i R i ri ε ri for i t x r x r x t x λ r λ r x r x r where λ x x x x x λ x x x x x. The ual vector D ε Ψ for Ψ Ψ R Ψ ψ εψ is the Steiner vector where ψ r

5 Appl. Math. Inf. Sci. 9 No / 8 ψ r ψ r. Consequently the angle of pitch is vanishing i.e. λ T t ψ r x r x r 0 x the pitch is efine as l T t From the equations t ψ r x x x t. λ r λ r x r x r ψ x r r ψ x r r t ψ r r ψ x r x r x x x ψ x r r ψ x r r since we have r r r r the pitch is The ual angle of pitch is r r r r l T 0. Λ T l T ελ T 0. HenceT is an extenable rule surface. Theorem 7. The pitch the angle of pitch the ual angle of pitch of the close rule surface N on the canonical coorinate frame are vanishing. Proof. We know that N X X R X R is vali where N n ε n X i x i ε x i R i ri ε ri i n x r x r x n x x x x r x x x x x x x r x r x r. Since the Steiner vector is efine as D ε Ψ for Ψ Ψ R Ψ ψ εψ where ψ r ψ r ψ r the angle of pitch is vanishing i.e. λ N n ψ r x r x r 0 x the pitch is efine as l N n From the equations n ψ r n. x x x x r x x x x x x x x x x x r x r x r ψ r r r ψ r n ψ r r ψ x r x r x x x ψ r r

6 84 N. Abazari et. al. : The Pitch the Angle of Pitch the Distribution... x r ψ x r since we have r r r r the pitch is The ual angle of pitch is r r r r l N 0. Λ N l N ελ N 0. SoN is an extenable rule surface. Theorem 8. The pitch the angle of pitch the ual angle of pitch of the close rule surface B on the canonical coorinate frame are vanishing. Proof. This is clear since we have B R so this rule surface is extenable. Theorem 9. The pitch the angle of pitch the ual angle of pitch of the close rule surface D 0 on the canonical coorinate frame are given by l D0 Λ D0 respectively. k k λ D0 k k Proof. We know that [ k ψ k ψ k k k ψ ] [ ] k ψ k ψ k ε k k ψ D 0 τ T κ B τ κ τ X R X R κ R τ κ X is vali where D 0 ε κ k εk τ k εk X i x i ε x i R i r i ε ri for i. We have 0 k k k x x r x r k r 0 x k r k x r k x r k k x r k r k r k k k k k x x k k k x k x r x r. From the efinition of the Steiner vector the angle of pitch is λ D0 0 ψ r k x r x r k r k k x k ψ k k the pitch ofd 0 is given by From 0 k x x l D k k 0 ψ k k k k k x x r ψ r r ψ r k ψ r k x k x x ψ r x r x r k r ψ r r k r x ψ r k ψ x since we have r r r r r r r r

7 Appl. Math. Inf. Sci. 9 No / 85 the pitch is l D0 [ ] k k ψ k ψ. k The ual angle of pitch ofd 0 is Λ D0 l D0 ελ D0 k k k [ k ε k k This completes the proof. References ] ψ k ψ ψ. [] Wilhelm Blaschke. Zur Bewegungsgeometrie auf er Kugel. S.-B. Heielberger Aka. Wiss. Math.-Nat. Kl. : [] Wilhelm Blaschke Hans Robert Müller. Ebene Kinematik. Verlag von R. Olenbourg München 956. [] Hasan Hilmi Hacısalihoğlu. On the pitch of a close rule surface. Mech. Mach. Theory 7: [4] Hasan Hilmi Hacısalihoğlu. On close spherical motions. Quart. Appl. Math. 9: [5] Hasan Hilmi Hacısalihoğlu. Geometry of motions the theory of quaternions. Ankara University 98. In Turkish. [6] Ömer Köse. Contributions to the theory of integral invariants of a close rule surface. Mech. Mach. Math. : [7] Ömer Köse. Kinematic ifferential geometry of a rigi boy in spatial motion using ual vector calculus. I. Appl. Math. Comput. 8: [8] Ömer Köse Celal Cem Sarıoğlu B. Karabey İlhan Karakılıc. Kinematic ifferential geometry of a rigi boy in spatial motion using ual vector calculus. II. Appl. Math. Comput. 8: [9] Mehmet Öner H. Hüseyin Uğurlu Ali Çalışkan. The Euler-Savary analogue equations of a point trajectory in Lorentzian spatial motion. Proc. Nat. Aca. Sci. Inia Sect. A 8: [0] Maxim V. Shamolin. Spatial motion of a rigi boy in a resisting meium. Internat. Appl. Mech. 467: [] Hellmuth Stachel. Instantaneous spatial kinematics invariants of the axoes. In Proceeings Ball 000 Symposium number Cambrige 000. [] Euar Stuy. Geometrie er Dynamen. B. G. Teubner Leipzig 90. [] An Tzu Yang. Application of quaternion algebra ual numbers to the analysis of spatial mechanisms. PhD thesis Columbia University 96. Nemat Abazari was born in Arabil Iran in 97. He receive the B. S. egree from the University of Tabriz Iran in 994 M. S. egree from the Valiasr University of Rafsanjan Rafsanjan Iran 00 Ph. D. egree in Geometry Department of Mathematics Ankara University Ankara Turkey in 0. He is currently an Associate Professor in University of Mohaghegh Arabili Arabil Iran since 0. Ilgin Sağer receive the M. S. egree in ifferential geometry from Ankara University Turkey in 005. She was working as a lecturer in Izmir University of Economics Turkey between Currently she is a octoral stuent in Ankara University Turkey. Her research interests are symplectic geometry the theory of elastic curves in Lorentz Minkowski spaces. H. Hilmi Hacisalihoğlu receive the Ph. D. egree in ifferential geometry from Ankara University Turkey in 966. He is the author of books in the area of ifferential geometry written in his native language more than 80 national international publications Eitor-in-Chief of many national journals. Professor Hacisalihoğlu is the Vice Presient of the Balkan Mathematicians Union a member of the organization commitee of the Balkan Mathematical Olympia. He is a member of Turkish European Mathematicians Associations. He has been on uty in aministrative acaemic positions in many universities in Turkey he is the epartment hea of Bilecik Seyh Eebali University Turkey since 007.