# Some Characteristic Properties of Parallelz-Equidistant Ruled Surfaces

Authors: Engin As; Süleyman Şenyurt
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Some characteristic properties of two ruled surfaces whose principal normal vectors are parallel along their striction curves inE3 are examined by assuming that the distance between two central planes at suitable points is constant, E3. In case of which two ruled surfaces are close, the relationship between the integral invariants of this ruled surfaces is computed.


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## Body

## 1. Introduction

The basis notions about ruled surfaces inE3 are given in [1]. Parallel p-equidistant ruled surfaces are described, and some of their characteristic properties are given in Valeontis’s article entitled “Parallel p-Äquidistante Regelflächen” [2]. Integral invariants, shape operators and spherical indicators of Parallel p-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

Letα:I→E3 be a differentiable curve with arc-length parameter s, and, {u1,u2,u3} be the Frenet frame of α at the point α(s), where
(1)u1(s)=α′(s),u2(s)=α′′(s)∥α′′(s)∥,u3(s)=u1(s)∧u2(s).
The Frenet formulas of α are
(2)[u1′u2′u3′]=[0k10-k10k20-k20][u1u2u3].
If α is a curve and x is a generator vector, then the ruled surface X(s,v) has the following parameter representation:
(3)X(s,v)=α(s)+vx(s).
Namely, a ruled surface is a surface generated by the motion of a straight line x along α. Furthermore, if α is a closed curve, then this surfaces is called closed ruled surface. Moreover, the drall Px, the striction γ, the apex angle λx, and the pitch lx of the closed ruled surface are defined by
(4)Px=det(α′,x,x′)∥x′∥2,γ(s)=α(s)-〈x′(s),α′(s)〉∥x′(s)∥2x(s),λx=〈d,x〉,lx=〈V,x〉.
The angle of the pitches, pitches, and dralls of the closed ruled surface generated by the Frenet vectors u1, u2, and u3 are
(5)λu1=∮(α)k2ds,λu2=0,λu3=∮(α)k1ds,lu1=∮(α)ds,lu2=0,lu3=0,pu1=0,pu2=k2k12+k22,pu3=1k2,
see [1].Definition 1.
Let the equation of ruled surface beX(s,v)=α(s)+vu1(s). The corresponding planes of the spaces sp{u1,u2}, sp{u2,u3}, and sp{u3,u1} 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,α:I→E3 and α¯:I→E3, are u1(s) and v1(s) which are so-called generator vectors, respectively, then parametric equations of the two ruled surfaces are X(s,v)=α(s)+vu1(s) and X¯(s,v)=α¯(s)+vv1(s) in E3. If the generator vectors are parallel, and the distance p between polar planes in suitable points are constant, then this couple ruled surface are called p-equidistant ruled surfaces [9].Theorem 3.
The relationships between the angles of the pitch, pitches, and dralls of parallelp-equidistant are given as follows:
(6)λv1=λu1+a1,λv2=λu2=0,λv3=λu3+a2,Pv1=Pu1=0,Pv2=Pu2ds¯ds,Pv3=Pu3ds¯ds,lv1=k1k¯1(lu1+a3),lv2=lu2=lv3=lu3=0,
where
(7)a1=∮(pu1+zu2+qu3)k2ds,a2=∮(pu1+zu2+qu3)k1ds,a3=∮(pu1+zu2+qu3)ds,
see [3].Definition 4.
Consider two space curvesα:I→E3 and β:I→E3, where I is a real interval that has at least four continuous derivatives. If there exists a corresponding relationship between the space curves α and β such that the principal normal lines of α coincide with the binormal lines of β at the corresponding points of the curves, then α is called as a Mannheim curve, and β is called as a Mannheim partner curve of α. The pair of {α,β} is said to be a Mannheim pair [7].Definition 5.
LetT be the unit tangent vector of the curve, α:I→E3. If T makes a constant angle with a fixed line, then the curve α is called the helix curve [1].Theorem 6.
The distance between corresponding points of the Mannheim partner curves inE3 is constant [8].Theorem 7.
For a curveα in E3, there is a curve β, so that {α,β} is a Mannheim pair [8].Theorem 8.
Let curvature and torsion ofα be k1 and k2, respectively, and α is Mannheim curve ⇔λ=k1/(k12+k22), (λ≠0=const). See [7].


## 3. Parallelz-Equidistant Ruled Surfaces

Definition 9.
Letα and α¯ be two curves, and let {u1,u2,u3} and {v1,v2,v3} be the Frenet frames of α and α¯ at the points α(s) and α¯(s), respectively, in E3. If the unit principal normal vectors, u2(s) and v2(s), are generator vectors and α and α¯ are anchor curve, then parametric equations of the two ruled surfaces are S(s,v)=α(s)+vu2(s) and S¯(s,v)=α¯(s)+vv2(s). For this surfaces, if principal normal vectors u2 and v2 are parallel, and the distance z between central planes in suitable points are constant, then this couple ruled surface are called parallel z-equidistant ruled surfaces [9].
Let striction curve and curvatures ofS ruled surface be γ(s),k1(s), and k2(s), respectively. Let striction curve and curvatures of S¯ ruled surface be γ¯(s), k¯1(s), and k¯2(s), respectively. In this situation, parametric equations of striction curves are
(8)γ(s)=α(s)-〈u2′(s),α′(s)〉∥u2′(s)∥2u2(s),u2′(s)≠0,γ¯(s)=α¯(s)-〈v2′(s),α¯′(s)〉∥v2′(s)∥2v2(s),v2′(s)≠0.
If the Frenet formulas are written in the last equation, then we have
(9)γ(s)=α(s)-k1(s)k12(s)+k22(s)u2(s),(10)γ¯(s)=α¯(s)-k1¯(s)k¯12(s)+k¯22(s)v2(s).
If vector γγ¯ is written related to frame {u1,u2,u3}, we get
(11)γ¯-γ=pu1+zu2+qu3,|z|=z,|q|=q,|p|=p,
where p=〈γγ¯,u1〉, z=〈γγ¯,u2〉, and q=〈γγ¯,u3〉. Here, |p|, |z|, and |q| are distance between polar, central, and asymptotic planes, respectively.Theorem 10.
Let striction curves ofS and S¯ parallel z-equidistant ruled surfaces be γ and γ¯. Then, the relation between striction curves are given as follows:
(12)γ¯=γ+pu1+qu3+(((1+k12k12+k22)k1((1+k12k12+k22)+p′)(1+k12k12+k22)(k1((1+k12k12+k22)+p′)(k1((1+k12k12+k22)+p′)-k2(q′-(k1k2k12+k22)))×(k12+k22)-1(k1((1+k12k12+k22)+p′)(k1((1+k12k12+k22)+p′))u2.Proof.
By substituting (10) into (11), we get
(13)α¯-k1¯k¯12+k¯22v2=γ+pu1+zu2+qu3.
Since vectors v2 and u2 are parallel vectors, then we can write
(14)α¯=γ+pu1+(z+k¯1k¯12+k¯22)u2+qu3.
By differentiating the last equation, we have
(15)α¯′=[(1+k12k12+k22)+p′-(z+k¯1k¯12+k¯22)k1]u1+[-(k1k12+k22)′+pk1-qk2+(z+k¯1k¯12+k¯22)′]×u2+[-(k1k2k12+k22)+q′+(z+k¯1k¯12+k¯22)k2]×u3.
From the last equation, we have
(16)〈α¯′,u2′〉=-k1[(1+k12k12+k22)+p′]+k2[-(k1k2k12+k22)+q′]+(z+k¯1k¯12+k¯22)(k12+k22).
Since vectors v2 and u2 are parallel vectors, then we can write
(17)γ¯=α¯-〈u2′,α¯′〉∥u2′∥2u2.
Substituting (14) and (16) into the last equation, then we obtain
(18)γ¯=γ+pu1+qu3+((k1((1+k12k12+k22)+p′)-k2(q′-(k1k2k12+k22)))×(k12+k22)-1k12k12+k22)u2.
By (11), we have the following results.Corollary 11.
The distance between central planes ofS and S¯ parallel z-equidistant ruled surfaces is
(19)z=(k1((1+(k12k12+k22))+p′)-k2(q′-(k1k2k12+k22)))×(k12+k22)-1.Corollary 12.
If{α,γ} and {α¯,γ¯} pairs are Mannheim pairs, the distance between central planes of S and S¯ parallel z-equidistant ruled surfaces is
(20)z=λ(2+p′-q′k2k1).Corollary 13.
Let the unit tangent vectors of the striction curves of surfacesS and S¯ be T and T¯, respectively. Let δ and δ¯ be the angles between vectors T, T¯, and their projection vectors on central plane, respectively. In this case, the distance between central planes of parallel z-equidistant ruled surfaces S and S¯ can be obtained by the following equation (Figure 1):
(21)z=(k1(cosδsinβdsγds+p′)-k2(cosδcosβdsγds+q′))×(k12+k22)-1dsγds.Figure 1Proof.
By some algebraic manipulations,T and T¯ can be calculated as follows:
(22)T=(cosδsinβu1+sinδu2+cosδcosβu3)dsdsγ,(23)T¯=(cosδ¯sinβ¯v1+sinδ¯v2+cosδ¯cosβ¯v3)ds¯dsγ¯.
By differentiating (14), we have
(24)α¯′=[cosδsinβdsγds+p′-(z+(k¯1k¯12+k¯22))k1]u1+[sinγdsγds+pk1-qk2+(z+k¯1k¯12+k¯22)′]×u2+[(z+k¯1k¯12+k¯22)cosδcosβdsγds+q′+(z+k¯1k¯12+k¯22)k2]u3.
From the last equation, we can write
(25)〈α¯′,u2′〉=(-k1cosδsinβ-k1p′+k2cosδcosβ)×dsγds+k2q′+(z+k¯1k¯12+k¯22)×(k12+k22).
Substituting (14) and (23) into equation γ¯=α¯-(〈u2′,α¯′〉/∥u2′∥2)u2, we get
(26)γ¯=γ+pu1+qu3+[(k1(cosδsinβdsγds+p′)-k2(cosδcosβdsγds+q′))×(k12+k22)-1cosδsinβdsγds]u2.
From (11), we have
(27)z=((k12+k22)-1k1(cosδsinβdsγds+p′)-k2(cosδcosβdsγds+q′))×(k12+k22)-1.Corollary 14.
If{α,γ} and {α¯,γ¯} pairs are Mannheim pairs, then the distance between central planes of surfaces S and S¯ is
(28)z=λ[(cosδsinβ-k2k1cosδcosβ)dsγds+p′-k2k1q′].Theorem 15.
LetS and S¯ be parallel z-equidistant ruled surfaces. Then, the relation between Frenet frame of α,{u1,u2,u3} and of α-,{v1,v2,v3} is given as follows:
(29)v1=cosφu1-sinφu3,v2=u2,v3=sinφu1+cosφu3,
where φ is the angle between the vector v1 and the vector u1.Proof.
Letφ be the angle between the vector v1 and the vector u1. In this case, we can write
(30)v1=cosφu1+sinφu3.
Since the vector v2 is parallel to the vector u2, we have
(31)v2=u2,v3=v1∧v2,v3=-sinφu1+cosφu3.
This completes the proof of the theorem.Theorem 16.
LetS and S¯ be parallel z-equidistant ruled surfaces. Let s and s¯ be arc parameters of anchor curves of S and S¯, respectively. If k1, k2 and k¯1, k¯2 are curvatures of anchor curves of S and S¯, respectively, there are following equations between these curvatures:
(32)k¯1=(cosφk1-sinφk2)dsds¯,(33)k¯2=(sinφk1+cosφk2)dsds¯.Proof.
SinceS and S¯ parallel z-equidistant ruled surfaces, u2(s)=v2(s¯). Differentiating this equation related to s, we have
(34)-k1u1+k2u3=(-k¯1v1+k¯2v3)ds¯ds.
Multiplying the last equation with v1 and v3, we have
(35)k¯1=(cosφk1-sinφk2)dsds¯,k¯2=(sinφk1+cosφk2)dsds¯.Theorem 17.
The relations between apex angles of closed parallelz-equidistant ruled surfaces S and S¯(1)
λv1=cosφλu1+sinφλu3+b1, b1=∮(pu1+(z+(k¯1/(k-12+k-22)))u2+qu3)k¯2ds¯+∮((-k1/(k12+k22))·u2)k¯2ds¯(2)
λv2=λu2=0(3)
λv3=-sinφλu1+cosφλu3+b2, b2=∮(pu1+(z+(k¯1/(k-12+k-22)))u2+qu3)k¯1ds-+∮((-k1/(k12+k22))·u2)k¯1ds¯.Proof.
The apex angle of closed ruled surface which is generated by the unit tangent vectorv1 is λv1=∮(α¯)k¯2ds¯. Substituting (14) into the last equation, we get
(36)λv1=∮(γ+pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)k¯2ds¯,λv1=∮(γ)k¯2ds¯+∮(pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)k¯2ds¯.
Substituting (9) into the last equation, we get
(37)λv1=∮(α-(k1/(k12+k22))·u2)k¯2ds¯+∮(pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)k¯2ds¯,λv1=∮(α)k¯2ds¯+∮((-k1/(k12+k22))·u2)k¯2ds¯+∮(pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)k¯2ds¯.
Impending,
(38)b1=∮(pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)k¯2ds¯+∮((-k1/(k12+k22))·u2)k¯2ds¯,λv1=∮(α)k¯2ds¯+b1.
Substituting (33) into the last equation, we have
(39)λv1=sinφ∮(α)k1ds+cosφ∮(α)k2ds+b1.
From (5), we can write
(40)λv1=cosφλu1+sinφλu3+b1.
From (5), the apex angles of closed ruled surfaces which are generated with vector v2 and u2 are
(41)λv2=λu2=0.
The apex angle of closed ruled surface which is generated with vector v3 is λv3=∮(α¯)k¯1ds¯. Substituting (14) into the last equation, we get
(42)λv3=∮(γ+pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)k¯1ds¯,λv3=∮(γ)k¯1ds¯+∮(pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)k¯1ds¯.
Substituting (9) into the last equation, we have
(43)λv3=∮(α-(k1/(k12+k22))·u2)k¯1ds¯+∮(pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)k¯1ds¯,λv3=∮(α)k¯1ds¯+∮((-k1/(k12+k22))·u2)k¯1ds¯+∮(pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)k¯1ds¯.
Impending,
(44)b2=∮(pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)k¯1ds¯+∮((-k1/(k12+k22))·u2)k¯1ds¯,λv3=∮(α)k¯1ds¯+b2.
Substituting (32) into the last equation, we have
(45)λv3=-sinφλu1+cosφλu3+b2.Corollary 18.
If{α,γ} and {α¯,γ¯} pairs are Mannheim pairs, then the relations between apex angles of closed parallel z-equidistant ruled surfaces S and S¯ are given as follows:
(1)
λ
v
1
=
cos
φ
λ
u
1
+
sin
φ
λ
u
3
+
b
1, b1=∮(pu1+(z+λ¯)u2+qu3)k¯2ds¯+∮(-λ·u2)k¯2ds¯,
(2)
λ
v
2
=
λ
u
2
=
0,
(3)
λ
v
3
=
-
sin
φ
λ
u
1
+
cos
φ
λ
u
3
+
b
2, b2=∮(pu1+(z+λ¯)u2+qu3)k¯1ds¯+∮(-λ·u2)k¯1ds¯.Theorem 19.
If we specially take helix curve instead of anchor curve of closed parallelz-equidistant ruled surface, there is relation between lu1 and lv1 as follows:
(46)lv1=(cosφk1k¯1-sinφk2k¯1)lu1+b3,b3=∮(pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)ds¯+∮((-k1/(k12+k22))·u2)ds¯.Proof.
From (5), the pitch of closed ruled surface which is generated vector v1 is lv1=∮(α¯)ds¯. Substituting (14) into the last equation, we get
(47)lv1=∮(γ+pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)ds¯,lv1=∮(γ)ds¯+∮(pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)ds¯.
Substituting (9) into the last equation, we get
(48)lv1=∮(α-(k1/(k12+k22))·u2)ds¯+∮(pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)ds¯,lv1=∮(α)ds-+∮((-k1/(k12+k22))·u2)ds¯+∮(pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)ds¯.
Impending,
(49)b3=∮(pu1+(z+(k¯1/(k¯12+k¯22)))u2+qu3)ds¯+∮((-k1/(k12+k22))·u2)ds¯,lv1=∮(α)ds-+b3.
Substituting (32) into this last equation, we have
(50)lv1=∮(α)(cosφk1k¯1-sinφk2k¯1)ds+b3.
Since anchor curve is helix curve, then curvatures are constant. In this situation, we have
(51)lv1=(cosφk1k¯1-sinφk2k¯1)∮(α)ds+b3.
Substituting (5) into the last equation, we obtain
(52)lv1=(cosφk1k¯1-sinφk2k¯1)lu1+b3.Theorem 20.
Let{u1,u2,u3} and {v1,v2,v3} be Frenet frame of anchor curves of S and S¯ closed parallel z-equidistant ruled surfaces. Then, the relations between dralls of ruled surfaces S and S¯ are given as follows:
(53)Pv1=Pu1=0,Pv2=Pu2(cosφ+sinφk1k2)ds¯ds,Pv3=Pu3k2(sinφk1+cosφk2)ds¯ds.Proof.
From (5), the drall of closed ruled surface which is generated vector v1 is
(54)Pv1=0.
From (5), the drall of closed ruled surface which is generated vector v2 is
(55)Pv2=k¯2k-12+k-22.
Substituting (32) and (33) into the last equation, we have
(56)Pv2=((sinφk1+cosφk2)dsds¯)×([(cosφk1-sinφk2)dsds¯]2+[(sinφk1+cosφk2)dsds¯]2)-1,Pv2=cosφk2(k12+k22)ds¯ds+sinφk2(k12+k22)·k1k2ds¯ds.
From (5),
(57)Pv2=Pu2(cosφ+sinφk1k2)ds¯ds·
The drall of closed ruled surface which is generated vector v3 is Pv3=1/k¯2,
(58)Pv3=1k2k2k¯2.
Substituting (5) and (33) into this last equation, we have
(59)Pv3=Pu3k2(sinφk1+cosφk2)ds¯ds.



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