Abstract
Some characteristic properties of two ruled surfaces whose principal normal vectors are parallel along their striction curves in are examined by assuming that the distance between two central planes at suitable points is constant, . In case of which two ruled surfaces are close, the relationship between the integral invariants of this ruled surfaces is computed.
1. Introduction
The basis notions about ruled surfaces in are given in [1]. Parallel -equidistant ruled surfaces are described, and some of their characteristic properties are given in Valeontis’s article entitled “Parallel -Äquidistante Regelflächen” [2]. Integral invariants, shape operators and spherical indicators of Parallel -equidistant ruled surfaces were computed by Masal and Kuruoğlu in the articles [3–6]. Mannheim curves were described in Liu and Wang’s article entitled “Mannheim partner curve in 3-Space” [7]. Some characteristic properties of Mannheim curves and Mannheim offsets of ruled surfaces were studied in [8, 9].
2. Preliminaries
Let be a differentiable curve with arc-length parameter , and, be the Frenet frame of at the point , where The Frenet formulas of are If is a curve and is a generator vector, then the ruled surface has the following parameter representation: Namely, a ruled surface is a surface generated by the motion of a straight line along . Furthermore, if is a closed curve, then this surfaces is called closed ruled surface. Moreover, the drall , the striction , the apex angle , and the pitch of the closed ruled surface are defined by The angle of the pitches, pitches, and dralls of the closed ruled surface generated by the Frenet vectors , , and are see [1].
Definition 1. Let the equation of ruled surface be . The corresponding planes of the spaces , , and along the striction line of this ruled surfaces are called asymptotic, polar, and central planes, respectively, [9].
Definition 2. If the unit tangent vectors of the anchor curves, and , are and which are so-called generator vectors, respectively, then parametric equations of the two ruled surfaces are and in . If the generator vectors are parallel, and the distance between polar planes in suitable points are constant, then this couple ruled surface are called -equidistant ruled surfaces [9].
Theorem 3. The relationships between the angles of the pitch, pitches, and dralls of parallel -equidistant are given as follows: where see [3].
Definition 4. Consider two space curves and , where is a real interval that has at least four continuous derivatives. If there exists a corresponding relationship between the space curves and such that the principal normal lines of coincide with the binormal lines of at the corresponding points of the curves, then is called as a Mannheim curve, and is called as a Mannheim partner curve of . The pair of is said to be a Mannheim pair [7].
Definition 5. Let be the unit tangent vector of the curve, . If makes a constant angle with a fixed line, then the curve is called the helix curve [1].
Theorem 6. The distance between corresponding points of the Mannheim partner curves in is constant [8].
Theorem 7. For a curve in , there is a curve , so that is a Mannheim pair [8].
Theorem 8. Let curvature and torsion of be and , respectively, and is Mannheim curve , . See [7].
3. Parallel -Equidistant Ruled Surfaces
Definition 9. Let and be two curves, and let and be the Frenet frames of and at the points and , respectively, in . If the unit principal normal vectors, and , are generator vectors and and are anchor curve, then parametric equations of the two ruled surfaces are and . For this surfaces, if principal normal vectors and are parallel, and the distance between central planes in suitable points are constant, then this couple ruled surface are called parallel -equidistant ruled surfaces [9].
Let striction curve and curvatures of ruled surface be , and , respectively. Let striction curve and curvatures of ruled surface be , , and , respectively. In this situation, parametric equations of striction curves are
If the Frenet formulas are written in the last equation, then we have
If vector is written related to frame , we get
where , , and . Here, , , and are distance between polar, central, and asymptotic planes, respectively.
Theorem 10. Let striction curves of and parallel -equidistant ruled surfaces be and . Then, the relation between striction curves are given as follows:
Proof. By substituting (10) into (11), we get Since vectors and are parallel vectors, then we can write By differentiating the last equation, we have From the last equation, we have Since vectors and are parallel vectors, then we can write Substituting (14) and (16) into the last equation, then we obtain By (11), we have the following results.
Corollary 11. The distance between central planes of and parallel -equidistant ruled surfaces is
Corollary 12. If and pairs are Mannheim pairs, the distance between central planes of and parallel -equidistant ruled surfaces is
Corollary 13. Let the unit tangent vectors of the striction curves of surfaces and be and , respectively. Let and be the angles between vectors , , and their projection vectors on central plane, respectively. In this case, the distance between central planes of parallel -equidistant ruled surfaces and can be obtained by the following equation (Figure 1):
Proof. By some algebraic manipulations, and can be calculated as follows: By differentiating (14), we have From the last equation, we can write Substituting (14) and (23) into equation , we get From (11), we have
Corollary 14. If and pairs are Mannheim pairs, then the distance between central planes of surfaces and is
Theorem 15. Let and be parallel -equidistant ruled surfaces. Then, the relation between Frenet frame of and of is given as follows: where is the angle between the vector and the vector .
Proof. Let be the angle between the vector and the vector . In this case, we can write Since the vector is parallel to the vector , we have This completes the proof of the theorem.
Theorem 16. Let and be parallel -equidistant ruled surfaces. Let and be arc parameters of anchor curves of and , respectively. If , and , are curvatures of anchor curves of and , respectively, there are following equations between these curvatures:
Proof. Since and parallel -equidistant ruled surfaces, . Differentiating this equation related to , we have Multiplying the last equation with and , we have
Theorem 17. The relations between apex angles of closed parallel -equidistant ruled surfaces and (1), (2)(3),
Proof. The apex angle of closed ruled surface which is generated by the unit tangent vector is . Substituting (14) into the last equation, we get Substituting (9) into the last equation, we get Impending, Substituting (33) into the last equation, we have From (5), we can write From (5), the apex angles of closed ruled surfaces which are generated with vector and are The apex angle of closed ruled surface which is generated with vector is . Substituting (14) into the last equation, we get Substituting (9) into the last equation, we have Impending, Substituting (32) into the last equation, we have
Corollary 18. If and pairs are Mannheim pairs, then the relations between apex angles of closed parallel -equidistant ruled surfaces and are given as follows:
(1), (2),(3), .
Theorem 19. If we specially take helix curve instead of anchor curve of closed parallel -equidistant ruled surface, there is relation between and as follows:
Proof. From (5), the pitch of closed ruled surface which is generated vector is . Substituting (14) into the last equation, we get Substituting (9) into the last equation, we get Impending, Substituting (32) into this last equation, we have Since anchor curve is helix curve, then curvatures are constant. In this situation, we have Substituting (5) into the last equation, we obtain
Theorem 20. Let and be Frenet frame of anchor curves of and closed parallel -equidistant ruled surfaces. Then, the relations between dralls of ruled surfaces and are given as follows:
Proof. From (5), the drall of closed ruled surface which is generated vector is From (5), the drall of closed ruled surface which is generated vector is Substituting (32) and (33) into the last equation, we have From (5), The drall of closed ruled surface which is generated vector is , Substituting (5) and (33) into this last equation, we have