Special Space Curves Characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$
By using the facts that the condition$\det(α^{(1)}, α^{(2)}, α^{(3)}) = 0$ characterizes a plane curve and the condition $\det(α^{(2)}, α^{(3)}, α^{(4)}) = 0$ characterizes a curve of constant slope, we present special space curves characterized by the condition $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$...
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2014
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508100565401600 |
|---|---|
| author | Saracoglu, S. Yayli, Y. Сарацоґлу, С. Яйлі, І. |
| author_facet | Saracoglu, S. Yayli, Y. Сарацоґлу, С. Яйлі, І. |
| author_sort | Saracoglu, S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T10:25:15Z |
| description | By using the facts that the condition$\det(α^{(1)}, α^{(2)}, α^{(3)}) = 0$ characterizes a plane curve and the condition $\det(α^{(2)}, α^{(3)}, α^{(4)}) = 0$ characterizes a curve of constant slope, we present special space curves characterized by the condition $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$, in different approaches. It is shown that the space curve is Salkowski if and only if $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$. The approach used in our investigation can be useful in understanding the role of the curves characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ in differential geometry. |
| first_indexed | 2026-03-24T02:19:50Z |
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| fulltext |
UDC 517.9
S. Saracoglu (Bartın Univ., Turkey),
Y. Yayli (Ankara Univ., Turkey)
SPECIAL SPACE CURVES CHARACTERIZED BY det(α(3), α(4), α(5)) = 0
СПЕЦIАЛЬНI ПРОСТОРОВI КРИВI,
ЩО ХАРАКТЕРИЗУЮТЬСЯ УМОВОЮ det(α(3), α(4), α(5)) = 0
In this study, by using the facts that det(α(1), α(2), α(3)) = 0 characterizes а plane curve and det(α(2), α(3), α(4)) = 0
characterizes a curve of constant slope, we present the special space curves characterized by det(α(3), α(4), α(5)) = 0, in
different approaches. We show that the space curve is Salkowski if and only if det(α(3), α(4), α(5)) = 0. The approach
used in this paper is useful in understanding the role of the curves characterized by det(α(3), α(4), α(5)) = 0 in differential
geometry.
За допомогою тих фактiв, що умова det(α(1), α(2), α(3)) = 0 характеризує плоску криву, а умова det(α(2), α(3),
α(4)) = 0 — криву зi сталим нахилом, наведено спецiальнi просторовi кривi, що характеризуються умовами
det(α(3), α(4), α(5)) = 0, в рiзних пiдходах. Показано, що просторова крива є кривою Салковського тодi i тiль-
ки тодi, коли det(α(3), α(4), α(5)) = 0. Пiдхiд, що використовується в роботi, є корисним для розумiння ролi кривих,
що характеризуються умовою det(α(3), α(4), α(5)) = 0 в диференцiальнiй геометрiї.
1. Introduction. In the classical differential geometry general helices, Salkowski curves and spher-
ical helices are well-known curves. These curves are defined by the property that the tangent makes
a constant angle with a fixed straight line (the axis of the general helix) [2 – 5]. Among all the space
curves with constant curvature, Salkowski curves are those for which the normal vector maintains a
constant angle with a fixed direction in the space [3]. The study of these curves is given by Salkowski
in [4], Monterde in [3], Kula, Ekmekçi, Yaylı, İlarslan in [1] and Takenaka in [6]. Throughout this
paper, by using some characterizations from [6], we are going to present the characterizations of the
condition that det(α(3), α(4), α(5)) = 0 for special space curves in different approaches.
We first introduce the space curve α as
α(s) =
(
x(s), y(s), z(s)
)
,
where s is the arclength parameter and denote two geometrical quantities, curvature and torsion, by κ
and τ. These structures play essential role in the theory of space curve. Such as, circles and circular
helices are curves with constant curvature and torsion [3].
Furthermore, as it is seen from the known facts in [2, 5, 6],
(1) the condition that det(α(0), α(1), α(2)) = 0 characterizes a great circle, where α is a spherical
curve,
(2) the condition that det(α(1), α(2), α(3)) = 0 characterizes a plane curve,
(3) the condition that det(α(2), α(3), α(4)) = 0 characterizes a curve of constant slope.
In [6], Takenaka represents the diffuculty in solving det(α(3), α(4), α(5)) = 0. Therefore, he has
given the following form to put the complicated terms away:
det(α(3), α(4), α(5)) = κ4 det
ϕ1 ϕ2 ϕ3
κ κ′ κ′′
τ τ ′ τ ′′
, (1)
c© S. SARACOGLU, Y. YAYLI, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4 571
572 S. SARACOGLU, Y. YAYLI
where
ϕ1 = −
(
1
κ
)′
, (2)
ϕ2 = −
(
1
κ
)′′
− 1
κ
(κ2 + τ2),
ϕ3 = −
(
1
κ
)′′′
−
{
1
κ
(κ2 + τ2)
}′
− 1
2κ
(
κ2 + τ2
)′
.
Moreover, by taking a curve with constant curvature, he shows that the following conditions are
equivalent:
(i) det(α(2), α(3), α(4)) = 0,
(ii) τ(s) = ∓ a3(bs+ c)
{1− a4(bs+ c)2}1/2
, where b, c ∈ R and
−1
b
(1 + c) ≤ s ≤ 1
b
(1− c).
Then for the constant curvature κ = a, he has found the torsion as:
τ(s) = ± a3(bs+ c)[
1− a4(bs+ c)2
]1/2 . (3)
On the other hand, in [3], Monterde shows that normal vectors of the curve α parametrized by
arc-length with κ = 1 make a constant angle with a fixed line in space if and only if
τ(s) = ± s
[tan2 φ− s2]1/2
. (4)
As it is seen Salkowski curves are slant helices. In this study, we characterize Salkowski curves
by the help of determinants. Accordingly, we present new special characterization with slant helices.
Analogously, in this paper, considering the relationship between the space curves, we have found
important results. Moreover, we obtain that with constant curvature κ = 1, the space curve is
Salkowski if and only if
det(α(3), α(4), α(5)) = 0 (5)
and also we have showed the following three conditions are equivalent:
(1) the space curve α is Salkowski curve,
(2) det(α(3), α(4), α(5)) = 0,
(3) τ(s) = ± s
[tan2 φ− s2]1/2
.
If we look previous studies in that field, we only meet the studies on the calculations of curvatures.
In this study, we try to give some calculations on the family of determinants. and the special space
curves that are characterized by det(α(3), α(4), α(5)) = 0, in different approaches. We hope that this
study will gain different interpretation to the other studies in this field.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
SPECIAL SPACE CURVES CHARACTERIZED BY det(α(3), α(4), α(5)) = 0 573
2. Spherical indicatrice curves. In this section, we give the spherical indicatrice curves that are
characterized by det(α(3), α(4), α(5)) = 0 and discuss the main properties.
Proposition 1. Let
α : I → E3, (6)
s 7→ α(s)
be a space curve that is parametrized arc-length with κ ≡ 1. The tangent indicatrice of the space
curve (T ) is spherical helix if and only if
det(α(3), α(4), α(5)) = 0. (7)
Proof. Let (T ) be tangent curve as:
α′(s) = (T ) (8)
and then
T ′ = κN = N (9)
for κ ≡ 1. Here, ‖T ′(s)‖ = 1 and the arc-parameter of the curve (T ) is s and also the parameters of
the curves α and (T ) are the same. According to all of these, if (T ) is helix, then [6]
det(T (2), T (3), T (4)) = 0. (10)
Thus
T = α′(s) (11)
and then
det(α(3), α(4), α(5)) = 0. (12)
On the contrary, it can be easily proved as above smiliarly.
Theorem 1. Let
α : I → E3,
s 7→ α(s)
be a space curve that is parametrized arc-length with κ ≡ 1. The tangent indicatrice (T ) of α is
spherical helix if and only if
2τ ′′(1 + τ2)− 3τ(1 + τ2)′ = 0. (13)
Proof. The tangent indicatrix of the space curve α is spherical helix if and only if
det(α(3), α(4), α(5)) = 0
and also from [6]
det(α(3), α(4), α(5)) = κ4 det
ϕ1 ϕ2 ϕ3
κ κ′ κ′′
τ τ ′ τ ′′
. (14)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
574 S. SARACOGLU, Y. YAYLI
Here, for κ ≡ 1,
det(α(3), α(4), α(5)) = det
0 −(1 + τ2) −(1 + τ2)′ − 1
2
(1 + τ2)′
1 0 0
τ τ ′ τ ′′
= 0
then we obtain
2τ ′′(1 + τ2)− 3τ(1 + τ2)′ = 0. (15)
Here, τ is torsion of the curve α(s).
Corollary 1. Let
α : I → E3,
s 7→ α(s)
be a space curve that is parametrized arc-length with κ ≡ 1. The tangent indicatrice (T ) of the space
curve α is spherical helix if and only if
τ(s) = ± bs+ c
[1− (bs+ c)2]1/2
, (16)
where b 6= 0 and b, c ∈ R,
−1
b
(1 + c) ≤ s ≤ 1
b
(1− c).
Proof. Under the condition κ ≡ 1, we have ϕ1 = 0, ϕ2 = −(1 + τ2) and ϕ3 = −(1 + τ2)′ −
− 1
2
(1 + τ2)′ from the proof of the theorem above. By giving similliar calculations from [6], the
theorem can be easily proved.
Result. By making similiar calculations we can get that
τ = ± bs+ c
[1− (bs+ c)2]1/2
(17)
where b =
1
tanφ
, is the solution of differential equation as given
det(α(3), α(4), α(5)) = 2τ ′′(1 + τ2)− 3τ(1 + τ2)′ = 0. (18)
On the other hand, the following figure can be given as an example for showing the curve whose
κ and τ satisfy the condition above for κ = 1:
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Salkowski curves with κ ≡ 1 [3].
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
SPECIAL SPACE CURVES CHARACTERIZED BY det(α(3), α(4), α(5)) = 0 575
Now, we will introduce the condition that det(α(3), α(4), α(5)) = 0 for Salkowski curves in
different approaches.
Corollary 2. Let
α : I → E3,
s 7→ α(s)
be a space curve that is parametrized arc-length with κ ≡ 1. The space curve α is Salkowski curve
if and only if
det(α(3), α(4), α(5)) = 0.
Proof. If the space curve α is Salkowski and slant helix, then
〈N, d〉 = cos θ = constant, (19)
for a fixed line d in a space. Besides, it can be easily given that for the normal vector N of the space
curve α:
β(s) = T and β′(s) = N for κ ≡ 1. (20)
Here, 〈
β′(s), d
〉
= 〈N, d〉 = constant. (21)
In this case, the curve β(s) is spherical helix. From the Theorem 1, for β = α′,
det(β(2), β(3), β(4)) = 0 (22)
and then
det(α(3), α(4), α(5)) = 0.
In contrary,
det(α(3), α(4), α(5)) = 0
then for β = α(1), we obtain
det(β(2), β(3), β(4)) = 0.
Thus, β(s) is a spherical helix. In that case, the curve α(s) is slant helix. Then, we can easily get
that α(s) is Salkowski for κ ≡ 1 from the calculations above.
Corollary 3. Let
α : I → E3,
s 7→ α(s)
(23)
be a space curve that is parametrized arc-length with κ ≡ 1. The following three conditions are
equivalent:
1) the space curve α is Salkowski curve,
2) det(α(3), α(4), α(5)) = 0,
3) τ(s) = ± s
[tan2 φ− s2]1/2
.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
576 S. SARACOGLU, Y. YAYLI
Proof. In the first step, it should be shown that the curve α is Salkowski if and only if [3]
τ(s) = ± s
[tan2 φ− s2]1/2
.
In the second step, we should prove that det(α(3), α(4), α(5)) = 0 if and only if
τ(s) = ± s
[tan2 φ− s2]1/2
.
Here, by taking c = 0 and b =
1
tanφ
in Corollary 1, we get
τ(s) = ± s
[tan2 φ− s2]1/2
.
Hence, the theorem is proved.
3. Conclusions. The starting point of this study is to give the special space curves that are
characterized by det(α(3), α(4), α(5)) = 0, in different approach. We have developed this approach
with discussing main properties of spherical curves, slant helices, Salkowski curves and relationship
between these curves. At this time, different approaches that we give here have showed us the space
curve is Salkowski if and only if det(α(3), α(4), α(5)) = 0. Additionally, it is obtained that the tangent
indicatrice of the space curve (T ) is spherical helix if and only if det(α(3), α(4), α(5)) = 0.
1. Kula L., Ekmekçi N., Yaylı Y., İlarslan K. Characterizations of slant helices in Euclidean 3-space // Turk. J. Math. –
2010. – 34. – P. 261 – 273.
2. Kühnel W. Differential geometry curves-surfaces-manifolds. – Second ed. – Amer. Math. Soc., 2006.
3. Monterde J. Salkowski curves revisited: A family of curves with constant curvature and non-constat torsion // Comput.
Aided Geom. Des. – 2009. – 26. – P. 271 – 278.
4. Salkowski E. Zur Transformation von Raumkurven // Math. Ann. – 1909. – 66, № 4. – P. 517 – 557.
5. Struik D. J. Lectures on classical differential geometry. – Second ed. – New York: Dover Publ., 1988.
6. Takenaka Y. A space curve C characterized by det(C(3), C(4), C(5)) = 0 // Int. J. Contemp. Math. Sci. – 2011. – 6,
№ 20. – P. 971 – 984.
Received 05.06.12,
after revision — 04.10.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
|
| id | umjimathkievua-article-2160 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:19:50Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/83/3c163fb2a10acfef8863a0015041df83.pdf |
| spelling | umjimathkievua-article-21602019-12-05T10:25:15Z Special Space Curves Characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ Спеціальні просторові кривi, що характеризуються умовою $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ Saracoglu, S. Yayli, Y. Сарацоґлу, С. Яйлі, І. By using the facts that the condition$\det(α^{(1)}, α^{(2)}, α^{(3)}) = 0$ characterizes a plane curve and the condition $\det(α^{(2)}, α^{(3)}, α^{(4)}) = 0$ characterizes a curve of constant slope, we present special space curves characterized by the condition $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$, in different approaches. It is shown that the space curve is Salkowski if and only if $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$. The approach used in our investigation can be useful in understanding the role of the curves characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ in differential geometry. За допомогою тих фактів, що умова $\det(α^{(1)}, α^{(2)}, α^{(3)}) = 0$ характеризує плоску криву, а умова $\det(α^{(2)}, α^{(3)}, α^{(4)}) = 0$ — криву зі сталим нахилом, наведено спеціальні просторові криві, що характеризуються умовами $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$, в різних підходах. Показано, що просторова крива є кривою Салковського тоді i тільки тоді, коли $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$. Підхід, що використовується в роботі, є корисним для розуміння ролі кривих, що характеризуються умовою $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ в диференціальній геометрії. Institute of Mathematics, NAS of Ukraine 2014-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2160 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 4 (2014); 571-576 Український математичний журнал; Том 66 № 4 (2014); 571-576 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2160/1322 https://umj.imath.kiev.ua/index.php/umj/article/view/2160/1323 Copyright (c) 2014 Saracoglu S.; Yayli Y. |
| spellingShingle | Saracoglu, S. Yayli, Y. Сарацоґлу, С. Яйлі, І. Special Space Curves Characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ |
| title | Special Space Curves Characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ |
| title_alt | Спеціальні просторові кривi, що характеризуються умовою $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ |
| title_full | Special Space Curves Characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ |
| title_fullStr | Special Space Curves Characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ |
| title_full_unstemmed | Special Space Curves Characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ |
| title_short | Special Space Curves Characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ |
| title_sort | special space curves characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2160 |
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