On singularities of the Galilean spherical darboux ruled surface of a space curve in $G_3$
We study the singularities of Galilean height functions intrinsically related to the Frenet frame along a curve embedded into the Galilean space. We establish the relationships between the singularities of the discriminant and the sets of bifurcations of the function and geometric invariants of curv...
Saved in:
| Date: | 2010 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2010
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2962 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508968944664576 |
|---|---|
| author | Şahin, T. Yilmaz, M. Шахін, Т. Йилмаз, М. |
| author_facet | Şahin, T. Yilmaz, M. Шахін, Т. Йилмаз, М. |
| author_sort | Şahin, T. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:41:19Z |
| description | We study the singularities of Galilean height functions intrinsically related to the Frenet frame along a curve embedded into the Galilean space. We establish the relationships between the singularities of the discriminant and the sets of bifurcations of the function and geometric invariants of curves in the Galilean space. |
| first_indexed | 2026-03-24T02:33:39Z |
| format | Article |
| fulltext |
UDC 517.9
T. Şahin, M. Yilmaz (Ondokuz Mayıs Univ., Turkey)
ON SINGULARITIES OF THE GALILEAN SPHERICAL
DARBOUX RULED SURFACE OF A SPACE CURVE IN G3
ПРО ОСОБЛИВОСТI СФЕРИЧНО-ГАЛIЛЕЄВОЇ ЛIНIЙЧАТОЇ
ПОВЕРХНI ДАРБУ ПРОСТОРОВОЇ КРИВОЇ В G3
We study the singularities of Galilean height functions intrinsically related to Frenet frame along a curve em-
bedded into Galilean space. We establish the relationships between singularities of discriminant and bifurcation
sets of the function and geometric invariants of curves in Galilean space.
Дослiджено особливостi галiлеївських функцiй висоти, що внутрiшньо пов’язанi iз рамкою Френе
вздовж кривої, вкладеної у галiлеївський простiр. Встановлено спiввiдношення мiж особливостями
множини дискримiнантiв та множини бiфуркацiй функцiї i геометричними iнварiантами кривих у галi-
леївському просторi.
1. Introduction. Singularity theory, being a direct descendant of differential calculus,
is certain to have a great deal of interest to say about geometry and therefore about all
the branches of mathematics, physics and other disciplines where the geometrical spirit
is a guiding light.
The crucial idea of a versal unfolding is contributed by R. Thom in 1975 which
was also emerging in algebraic geometry at the same time. Most of the deeper and more
interesting results in [1] hinged on Thom’s versal unfolding idea, and it became a central
tool in almost all applications of singularity theory inside and outside mathematics.
Several geometers were interested in studying the singularities and generic differen-
tial geometry in Euclidean space [1 – 6]. The main point of studying singularity is defin-
ing real-valued functions such as squared-distance function and height function defined
on a curve or on a surface. The classical invariants of extrinsic differential geometry
can be treated as singularities of these two functions. Also, some good approximations
to singularity theory in affine geometry can be found in [7 – 10]. Related to the theory,
some geometrical applications can be found in [11, 12].
Besides Euclidean geometry, a range of new types of geometries have been invented
and developed in the last two centuries. They can be introduced in a variety of manners.
One possible way is through projective manner, where one can express metric properties
through projective relations. For this purpose a fixed conic (called absolute) in infinity
is taken and all metric relations may be considered as projective relations with respect
to the absolute. This approach is due to A. Cayley and F. Klein. F. Klein noticed that
due to the nature of the absolute, various geometries are possible [13]. Among these
geometries, there is also Galilean geometry which is our matter in this paper.
In this paper we will introduce the notion of Galilean height function on space
curves in G3, Galilean space. This function is quite useful for the study of singularities
of Galilean spherical Darboux ruled surface of space curves in G3. We also introduce
the notion of the line of striction of the Galilean spherical Darboux ruled surface and
Galilean spherical Darboux images of space curves in G3.
As a consequence, we apply ordinary techniques of singularity theory for the func-
tion and describe the relationships between the singularities of the above three subjects
and differential geometric invariants of space curves in G3. We also explain by an ex-
ample that Galilean spherical Darboux ruled surface of space curves in G3 is a planed
c© T. ŞAHİN, M. YILMAZ, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10 1377
1378 T. ŞAHİN, M. YILMAZ
surface while Euclidean spherical Darboux ruled surface of space curves in E3 is a non-
planed surface (see Fig. 2).
The techniques used in this paper depend heavily on those in the book of Bruce and
Giblin [1].
2. Preliminaries on Galilean geometry. “All geometry is projective geometry”
(A. Cayley). From A. Cayley point of view, G3 is a real 3-dimensional projective space
P 3 (R), is the set of equivalence classes of ∼ on R4−{0} by equivalence relation x ∼ y
iff x = λy for some λ ∈ R\ {0}. Thus, P 3 (R) obtained as a factor space on R4\ {0}
by ∼, i.e., P 3 (R) =̃
(
R4 − {0}
)
/ ∼ [14]. We can think of P 3 (R) more geometrically
as set of lines through the origin in R4. G3 is a real Cayley – Klein space equipped with
the projective metric of signature (0, 0,+,+), as showed in [15]. The absolute of the
Galilean geometry is an ordered triple {w, f, I}, where w is the ideal (absolute) plane,
f is the line (absolute line) in w and I is the fixed elliptic involution of points of f .
The points, the lines and the planes of P 3 (R) are the one-dimensional, two-dimensional
and three-dimensional subspaces of R4, respectively [16]. Therefore, G3 contains R3 as
a proper subset and the complement in G3 to w is diffeomorphic to R3.
Let P be any point of R3 with affine coordinates (x, y, z). Write (x, y, z) as(
X1
X0
,
X2
X0
,
X3
X0
)
, where X0 is some common deminator. Call (X0, X1, X2, X3) the ho-
mogeneous coordinates of P . Thus, the homogeneous coordinates (X0 : X1 : X2 : X3)
and ρ (X0 : X1 : X2 : X3) refer to the same point, for all ρ ∈ R−{0} [16]. We now can
introduce homogeneous coordinates in G3 in such a way that the absolute plane w is
given by X0 = 0, the absolute line f by X0 = X1 = 0 and the elliptic involution I by
(0 : 0 : X2 : X3)→ (0 : 0 : X3 : −X2) .
In affine coordinates, the distance between the points Pi = (xi, yi, zi) for i = 1, 2,
is defined by
d (P1, P2) =
|x2 − x1| , if x1 6= x2,√
(y2 − y1)2 + (z2 − z1)2 if x1 = x2.
(1)
In the nonhomogeneous coordinates the isometries group B6 has the form
x = a+ x,
y = b+ cx+ y cosϕ+ z sinϕ, (2)
z = d+ ex− y sinϕ+ z cosϕ,
where a, b, c, d, e and ϕ are real numbers. The group of motions of G3 is a six-parameter
group [17].
A vector A (x, y, z) is said to be non-isotropic if x 6= 0. All unit non-isotropic
vectors are of the form (1, y, z). For isotropic vectors x = 0 holds.
For a curve γ : I → G3 , I ⊂ R parametrized by the invariant parameter s = x,
given in the coordinate form
γ (x) = (x, y (x) , z (x)) ,
the curvature κ (x) and the torsion τ (x) are defined by
κ (x) =
√
y′′ (x)
2
+ z′′ (x)
2
,
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
ON SINGULARITIES OF THE GALILEAN SPHERICAL DARBOUX RULED SURFACE . . . 1379
τ (x) =
det (γ′ (x) , γ′′ (x) , γ′′′ (x))
κ2 (x)
(3)
and the associated moving trihedron is given by
t (x) = γ′ (x) = (1, y′ (x) , z′ (x)) ,
n (x) =
1
κ (x)
(0, y′′ (x) , z′′ (x)) , (4)
b (x) =
1
κ (x)
(0,−z′′ (x) , y′′ (x)) .
The vectors t (x) , n (x) and b (x) are called the vectors of the tangent, principal normal
and the binormal line, respectively [17]. Therefore, the Frenet – Serret formulas can be
written in matrix notation as tn
b
′ =
0 κ 0
0 0 τ
0 −τ 0
tn
b
. (5)
From the equations in (4) and (5) one gets an important relation
γ′′′ (x) = κ′ (x)n (x) + κ (x) τ (x) b (x) .
For any unit special curve γ : I → G3, we call D (x) = τ (x) t (x) + κ (x) b (x)
a Darboux vector of γ [18]. By using the Darboux vector, Frenet – Serret formulas can
be rewritten as follows:
t (x) = D (x)×G t (x) ,
n (x) = D (x)×G n (x) , (6)
b (x) = D (x)×G b (x) ,
where the Galilean cross product ×Gis defined by
a×G b =
∣∣∣∣∣∣
0 e2 e3
a1 a2 a3
b1 b2 b3
∣∣∣∣∣∣ (7)
for a =(a1, a2, a3) and b =(b1, b2, b3) [19, 20].
According to the absolute figure, there are two types of (ideal) lines in the Galilean
space-isotropic lines which intersect the absolute line f and non-isotropic lines which
do not. A plane is called Euclidean if it contains f , otherwise it is called isotropic. In the
given affine coordinates, isotropic vectors are of the form (0, y, z), whereas Euclidean
planes are of the form x = k, k ∈ R.
A ruled surface in the Galilean G3 is a surface that admits a parametrization
ϕ (u, v) = β (u) + va (u) ,
where β is an admissible curve (the directrix), a is a nowhere vanishing vector field
(field of generators) along the curve β and u, v are parameters, u ∈ I ⊂ R, v ∈ R.
According to the absolute figure of G3, we distinguish the following three types of ruled
surfaces in G3 :
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1380 T. ŞAHİN, M. YILMAZ
Type A. Nonconodial or conodial ruled surfaces whose striction line does not lie in
a Euclidean plane.
Type B. Ruled surfaces with the striction line in a Euclidean plane.
Type C. Conodial ruled surfaces with the absolute line as the directional line in
infinity [18].
The Galilean sphere S2
G is defined by S2
G = {(x, y, z) ∈ G3 | |x− x0| = r} .
For more on Galilean geometry, one can refer to [18, 20] and references there in.
3. Singularities of some functions in Galilean geometry. We define a spherical
curve d : I → S2
G by d (x) =
D (x)
‖D (x)‖G
and surface
dR (γ) = {d (x) + un (x) | u ∈ R, x ∈ I} , (8)
β (x) =
{
d (x)− 1
τ (x)
(κ
τ
)′
(x)n (x) | x ∈ I
}
. (9)
We call the image of d the Galilean spherical Darboux image, the surface dR (γ) the
Galilean spherical Darboux ruled surface of γ and the curve β (x) the line of striction
of the Darboux ruled surface.
Theorem 1. Let γ : I → G3 be a unit speed curve. Then we have the following:
(1) The line of stiriction of the Galilean spherical Darboux ruled surface image is
locally diffeomorphic to the ordinary cusp C at β (x0) if and if only(κ
τ
)′′
(x) =
τ ′ (x)
τ (x)
(κ
τ
)′
(x) and
(κ
τ
)′′′
(x) 6= τ ′′ (x)
τ (x)
(κ
τ
)′
(x) .
(2) (a) The Galilean spherical Darboux ruled surface is locally diffeomorphic to the
cuspidal edge C × R at d (x0) + u0n (x0) if and if only
u0 = − 1
τ (x0)
(κ
τ
)′
(x0) and
(κ
τ
)′′
(x) 6= τ ′ (x)
τ (x)
(κ
τ
)′
(x) .
(b) The Galilean spherical Darboux ruled surface is locally diffeomorphic to the
swallowtail SW at d (x0) + u0n (x0) if and if only
u0 = − 1
τ (x0)
(κ
τ
)′
(x0) ,
(κ
τ
)′′
(x) =
τ ′ (x)
τ (x)
(κ
τ
)′
(x) ,
and (κ
τ
)′′′
(x) 6= τ ′′ (x)
τ (x)
(κ
τ
)′
(x) .
Here, C =
{
(x1, x2) : x
2
1 = x32
}
is ordinary cusp and
SW =
{
(x1, x2, x3) : x1 = 3u4 + u2v, x2 = 4u3 + 2uv, x3 = v
}
is the swallowtail (see Fig. 1).
The main aim of this paper is proving the preceding theorem, Theorem 1. For this
issue, we will study the singularities of height function in Galilean space in Section 3.1.
Also, since we need the unfoldings of functions in G3, we describe the content of them
in Section 3.2.
3.1. Families of smooth functions on a space curve in Galilean geometry. In this
section families of function on a space curve and surface will be defined which are useful
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
ON SINGULARITIES OF THE GALILEAN SPHERICAL DARBOUX RULED SURFACE . . . 1381
Fig. 1. The cusp curve, the cuspidal edge, the swallowtail surface.
for the study of singularities. Let γ : I → G3 be a unit speed curve with κ (x) 6= 0. We
will assume that τ (x) 6= 0 throughout this paper.
3.1.1. Height function in Galilean space. We now define a two-parameter family
of smooth functions on I:
Fh : I × S2
G → R
by Fh (x,v) = |t (x) b (x) v|. Here, |a b c| denotes the determinant of the matrix
(a b c) . We call Fh a Galilean height function (or a normal directed height function)
on γ. We denote that fhv (x) = Fh (x,v) for any v ∈ S2
G. Then, we have the following
proposition.
Proposition 1. Let γ : I → G3 be a unit speed curve with κ (x) 6= 0 and τ (x) 6=
6= 0. Then,
(1) f ′hv (x) = 0 if and only if there exist real numbers µ ∈ R, such that
v = ±t (x) + µn (x)±
(κ
τ
)
(x) b (x) ,
(2) f ′hv (x) = f ′′hv (x) = 0 if and only if
v = ±
(
t (x)− 1
τ (x)
(κ
τ
)′
(x)n (x) +
(κ
τ
)
(x) b (x)
)
,
(3) f ′hv (x) = f ′′hv (x) = f ′′′hv (x) = 0 if and only if
v = ±
(
t (x)− 1
τ (x)
(κ
τ
)′
(x)n (x) +
(κ
τ
)
(x) b (x)
)
,
(κ
τ
)′′
(x) =
τ ′ (x)
τ (x)
(κ
τ
)′
(x) ,
(4) f ′hv (x) = f ′′hv (x) = f ′′′hv (x) = f
(4)
hv (x) = 0 if and only if
v = ±
(
t (x)− 1
τ (x)
(κ
τ
)′
(x)n (x) +
(κ
τ
)
(x) b (x)
)
,
(κ
τ
)′′′
(x) =
τ ′′ (x)
τ (x)
(κ
τ
)′
(x) .
Proof. By the Frenet – Serret formula, we have the following calculations:
(i) f ′hv (x) = κ (x) |n (x) b (x)v| − τ (x) |t (x)n (x)v| ,
(ii) f ′′hv (x) = κ′ (x) |n (x) b (x)v| − τ ′ (x) |t (x)n (x)v| − τ2 (x) |t (x) b (x)v| ,
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1382 T. ŞAHİN, M. YILMAZ
(iii) f ′′′hv (x)=
(
κ′′ (x)−κ (x) τ2 (x)
)
|n (x)b (x)v|+
(
τ3 (x)−τ ′′ (x)
)
|t (x)n (x)v| −
−3τ (x) τ ′ (x) |t (x) b (x)v| ,
(iv) f (4)hv (x)=
(
κ′′′(x)−κ′(x)τ2 (x)−5κ(x)τ(x)τ ′(x)
)
|n(x)b(x)v|+
(
6τ2(x)τ ′(x)−
−τ ′′′ (x)) |t (x)n (x)v|+
(
τ4 (x)− 3τ ′2 (x)− 4τ (x) τ ′′ (x)
)
|t (x) b (x)v| .
(1) The assertion is trivial by the formula (i) from the above calculations. By the
assumption that v ∈S2
G, we have v = ±t (x) + µn (x) + λb(x). It follows (i) that
f ′hv (x) = ±κ (x)−λτ (x) . Since τ (x) 6= 0, f ′hv (x) = 0 if and only if λ = ±
(κ
τ
)
(x).
Therefore we have
v = ±t (x) + µn (x)±
(κ
τ
)
(x) b (x) .
(2) By (1), we have v = ±t (x) + µn (x) ±
(κ
τ
)
(x) b (x). It follows from (ii) that
f ′′hv (x) = ±κ′ (x) ± τ ′ (x)
(κ
τ
)
(x) + µτ2 (x) . Since τ (x) 6= 0, f ′′hv (x) = 0 if and
only if µ = ∓ 1
τ (x)
(κ
τ
)
(x). Therefore we have
v = ±
(
t (x)− 1
τ (x)
(κ
τ
)′
(x)n (x) +
(κ
τ
)
(x) b (x)
)
.
(3) If we substitute the formula
v = ±
(
t (x)− 1
τ (x)
(κ
τ
)′
(x)n (x) +
(κ
τ
)
(x) b (x)
)
into (iii), then we have
κ′′ (x) τ2 (x)− κ (x) τ (x) τ ′′ (x)− 3κ′ (x) τ (x) τ ′ (x) + 3κ (x) τ ′2 (x) = 0.
Therefore, we have
(κ
τ
)′′
(x) =
τ ′ (x)
τ (x)
(κ
τ
)′
(x) the assertion (3) follows.
(4) We also substitute the formula (3) into (iv), then we have
κ′′′ (x) τ3 (x)− κ (x) τ2 (x) τ ′′′ (x) + 3κ (x) τ ′3 (x) + 4κ (x) τ (x) τ ′ (x) τ ′′ (x) =
= +3κ′ (x) τ (x) τ ′2 (x) + 4κ′ (x) τ2 (x) τ ′′ (x) . (10)
If
(κ
τ
)′′
(x) =
τ ′ (x)
τ (x)
(κ
τ
)′
(x) then we can show that
(κ
τ
)′′′
(x) =
τ ′′ (x)
τ (x)
(κ
τ
)′
(x) .
We have assertion (4)
3κ (x) τ ′3 (x) + 4κ (x) τ (x) τ ′ (x) τ ′′ (x) = 0.
Proposition 1 is proved.
We now study the geometric properties of the spherical Darboux ruled surface of
space curves in G3. By the propositions in the last section, we can recognize that the
function
(κ
τ
)′
(x) and the modified Darboux vector
( τ
κ
)
(x) t (x) + b (x) are impor-
tant subjects. If
(κ
τ
)
(x) ≡ c (constant) then the curve γ (x) in G3 has been classically
known as a helix in Galilean space [18]. Galilean cycle is the only curves of constant cur-
vature in plane [20]. For a unit speed regular curve γ (x) has tangent curve σ : I → S2
G,
σ (x) = t (x) is called the Galilean spherical tangential image of γ (x) .
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
ON SINGULARITIES OF THE GALILEAN SPHERICAL DARBOUX RULED SURFACE . . . 1383
Proposition 2. Let γ : I → G3 be a unit speed regular curve. Then γ(x) is a helix
if and only if the modified Darboux vector d (x) is a constant vector. In this case we
have the following assertions:
(1) The Galilean spherical tangential image σ (x) of γ (x) is a cycle on the unit
Galilean sphere S2
G.
(2) The Galilean spherical Darboux ruled surface of γ (x) is a plane given by
e+ un (x) . Where e = d (x) .
Proof. By the Frenet – Serret formulas, we can show that D̃′ (x) =
( τ
κ
)′
(x) t (x).
Therefore, γ(x) is a helix if and only if D̃′ (x) ≡ 0. This condition is equivalent to the
condition that D̃ (x) is a constant vector. In this case we have
σ (x) = t (x) ,
σ′ (x) = κ (x)n (x) ,
σ′′ (x) = κ′ (x)n (x) + κ (x) τ (x) b (x) .
The curvature of σ (x) is κσ (x) =
(κ
τ
)
(x) = constant. This means that the Galilean
spherical tangential image σ (x) is a cycle on the unit Galilean sphere S2
G [20]. The
assertion (2) is clear by definition.
Proposition 2 is proved.
The singularities of the Galilean spherical Darboux image describe how the shape
of the curve γ is similar to a helix.
3.2. Unfoldings of functions by one-variable. In this section, we will use some
general results on singularity theory for families of function germs. Let
F : (I × Rr, (x0, w0))→ R
be a function germ. We call F an r-parameter unfolding of f , where f (x) = Fw0
(x,w0) .
We say that f has Ak-singularity at x0 if f (p) (x0) = 0 for all 1 ≤ p ≤ k and
f (k+1) (x0) 6= 0. We also say that f has A≥k-singularity at x0 if f (p) (x0) = 0 for all
1 ≤ p ≤ k. Let F be an unfolding of f and f (x) has Ak-singularity (k ≥ 1) at x0. We
denote the (k − 1)-jet of the partial derivative
∂F
∂wi
at x0 by Jk−1
(
∂F
∂wi
(x,w0)
)
(x0) =
=
∑k−1
j=1
αijx
j for i = 1, ..., r. Then F is called a (p)-versal unfolding if the
(
(k − 1)×
×r
)
-matrix of coefficients (αij) has rank k − 1, k − 1 ≤ r. Under the same conditions
as the above, then F is called a versal unfolding if the (k × r)-matrix of coefficients
(α0i, αij) has rank k, k ≤ r, where α0i =
∂F
∂wi
(x0, w0) .
We now introduce important sets concerning the unfoldings relative to the above
notions. The bifurcation set BF of F is the set
BF =
{
w ∈ Rr| ∂F
∂w
=
∂2F
∂w2
= 0 at (x,w)
}
.
The discriminant set of F is the set
DF =
{
w ∈ Rr| ∂F
∂w
= 0 at (x,w)
}
.
Then we have the following well-known result [1].
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1384 T. ŞAHİN, M. YILMAZ
Theorem 2. Let F : (I × Rr, (x0, w0)) → R be an r-parameter unfolding of
f (x) which has Ak-singularity at x0.
(1) Suppose that F is a (p)-versal unfolding:
(a) if k = 2, then BF is locally diffeomorphic to {0} × Rr−1;
(b) if k = 3, then BF is locally diffeomorphic to C × Rr−2;
(c) if k = 4, then BF is locally diffeomorphic to SW × Rr−3.
(2) Suppose that F is a versal unfolding:
(a) if k = 1, then DF is locally diffeomorphic to {0} × Rr−1;
(b) if k = 2, then DF is locally diffeomorphic to C × Rr−2;
(c) if k = 3, then DF is locally diffeomorphic to SW × Rr−3.
Here, C =
{
(x1, x2) : x
2
1 = x32
}
is ordinary cusp and
SW =
{
(x1, x2, x3) : x1 = 3u4 + u2v, x2 = 4u3 + 2uv, x3 = v
}
is the swallowtail (see Fig. 1).
For the proof of Theorem 1, we have the following key propositions.
Proposition 3. Let Fh : I × S2
G → R be the Galilean height function on a unit
speed curve γ (x) . If fhv0 has Ak-singularity (k = 2, 3) at x0, then Fh is a (p)-versal
unfolding of fhv0 .
Proof. We denote by γ (x) = (x, y (x) , z (x)) and v = (1, v2, v3) . By definition,
we have
Fh (x,v) = |t (x) b (x) v| =
=
1
κ (x)
[−z′′ (x) v3 − y′′ (x) v2 + y′ (x) y′′ (x) + z′ (x) z′′ (x)] .
Let Jk−1
(
∂Fh
∂vi
(x, v0)
)
(x0) be the (k − 1)-jet of
∂Fh
∂vi
, i = 2, 3, at x0; then we have
J3
(
∂Fh
∂vi
(x, v0)
)
(x0) = −n′i (x0)x−
1
2
n′′i (x0)x
2 − 1
6
n′′′i (x0)x
3, i = 2, 3.
Here, n (x) = (0, n2, n3) =
1
κ (x)
(0, y′′ (x) , z′′ (x)) by the equation (5). We distin-
guish two cases.
Case (1). When fhv0 has the A2-singularity at x0, we can define (1× 2)-matrix A
as follows:
A =
[(
−y
′′ (x0)
κ (x0)
)′ (
−z
′′ (x0)
κ (x0)
)′]
.
We also have A(x) = −n′ (x) = −τ (x) b (x) 6= 0 by the equation (5). Therefore we
have RankA = 1.
Case (2). When fhv0 has the A3-singularity at x0, we define (2× 2)-matrix A as
follows:
B =
(
−y
′′ (x0)
κ (x0)
)′ (
−z
′′ (x0)
κ (x0)
)′
(
−y
′′ (x0)
κ (x0)
)′′ (
−z
′′ (x0)
κ (x0)
)′′
to be nonsingular. That is to say detB = |B| 6= 0. Here, we can show by direct
calculations but rather long calculation. We will use a method simpler than that. By the
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
ON SINGULARITIES OF THE GALILEAN SPHERICAL DARBOUX RULED SURFACE . . . 1385
Frenet – Serret formulas (5), we have the following calculation:
|B| = |t (x0) n′ (x0) n′′ (x0)| .
If the necessary derivatives of the Frenet – Serret formulas (5) is written, then we have
|B| = −τ3 (x0) . Since τ (x) 6= 0, the rank of B is 2.
Proposition 3 is proved.
Let’s define a function F̃h : I × S2
G × R → R by F̃h (x, v, w) = F (x, v) − w and
fhv,w (x) = F̃h (x, v, w) .
Proposition 4. If fhv0,w0
has Ak-singularity (k = 1, 2, 3) at x0, then Fh is a
versal unfolding of fhv0,w0
.
Proof. Using the same notations of Proposition 3, we have
F̃h (x, v, v1) =
1
κ (x)
[−z′′ (x) v3 − y′′ (x) v2 + y′ (x) y′′ (x) + z′ (x) z′′ (x)]− v1.
Let Jk−1
(
∂F̃h
∂vi
(x, v0)
)
(x0) be the (k − 1)-jet of
∂F̃h
∂vi
, i = 1, 2, 3, at x0; then we
have
∂F̃h
∂v1
(x0, v0) + J2
(
∂F̃h
∂v1
(x, v0)
)
(x0) = −1,
∂F̃h
∂vi
(x0, v0) + J2
(
∂F̃h
∂vi
(x, v0)
)
(x0) = −ni (x0)− n′i (x0)x− n′′i (x0)
x2
2
, i = 2, 3.
Now, we will distinguish three cases.
Case (1). When fhv0,w0 has the A1-singularity at x0, we define (1× 2)-matrix C as
follows:
C =
[
−1
(
−y
′′ (x0)
κ (x0)
) (
−z
′′ (x0)
κ (x0)
)]
.
The rank of C is clearly 1.
Case (2). When fhv0,w0 has the A2-singularity at x0, we require (2× 3)-matrix:
D =
−1
(
−y
′′ (x0)
κ (x0)
) (
−z
′′ (x0)
κ (x0)
)
0
(
−y
′′ (x0)
κ (x0)
)′ (
−z
′′ (x0)
κ (x0)
)′
to have the maximal rank. By the case 1 in Proposition 3, the second line of D does not
vanish. Thus the rank of D is 2.
Case (3). When fhv0,w0
has the A3-singularity at x0, we define (3× 3)-matrix:
E =
−1
(
−y
′′ (x0)
κ (x0)
) (
−z
′′ (x0)
κ (x0)
)
0
(
−y
′′ (x0)
κ (x0)
)′ (
−z
′′ (x0)
κ (x0)
)′
0
(
−y
′′ (x0)
κ (x0)
)′′ (
−z
′′ (x0)
κ (x0)
)′′
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1386 T. ŞAHİN, M. YILMAZ
Fig. 2. The Euclidean spherical Darboux ruled surface, the Galilean spherical Darboux ruled
surface, the line of striction of dR (γ).
to be nonsingular. By the case 2 in Proposition 4, determinant of E does not vanish. It
means that the rank of E is 3.
Proposition 4 is proved.
Proof of Theorem 1 follows from, Propositions 1, 3, 4 and Theorem 2.
Example 1. Consider the curve γ : I ⊂ R → E3, γ (x) =
(
x,
x2√
2
,
x3
3
)
. For
an arbitrary speed curve γ : I → E3 , I ⊂ R the associated moving trihedron is given
by
T (x) =
γ′ (x)
‖γ′ (x)‖
, B (x) =
γ′ (x)× γ′′ (x)
‖γ′ (x)× γ′′ (x)‖
, N (x) = B (x)× T (x)
and the curvature κ (x) and the torsion τ (x) are defined by
κ (x) =
‖γ′ (x)× γ′′ (x)‖
‖γ′ (x)‖3
, τ (x) =
det (γ′ (x) , γ′′ (x) , γ′′′ (x))
κ2 (x)
.
The vectors t (x) , n (x) and b (x) are called the vectors of the tangent, principal normal
and the binormal line, respectively [21].
We compute the Frenet apparatus of the curve γ (x) =
(
x,
x2√
2
,
x3
3
)
. If the neces-
sary derivatives of the Frenet – Serret formulas is written, then we have
T (x) =
1
1 + x2
(
1,
√
2x, x2
)
, B (x) =
1
1 + x2
(
x2,−
√
2x, 1
)
,
N (x) =
1
1 + x2
(
−
√
2x, 1− x2,
√
2x
)
, κ (x) = τ (x) =
√
2
(1 + x2)
2 .
Therefore we compute dR (γ) = {d (x) + uN (x) | u ∈ R, x ∈ I} surface, here a spher-
ical curve d : I → S2 by d (x) =
D (x)
‖D (x)‖
(see Fig. 2). Hence, we have
d (x) + uN (x) =
κ (x)√
κ2 (x) + τ2 (x)
(
τ (x)
κ (x)
T (x) +B (x)
)
+ uN (x) =
=
(
1 + x2√
2
− u
√
2x
1 + x2
, u
1− x2
1 + x2
,
1 + x2√
2
+ u
√
2x
1 + x2
)
. (11)
We also consider the curve γ : I ⊂ R → G3, γ (x) =
(
x,
x2√
2
,
x3
3
)
. If the necessary
derivatives of the Frenet – Serret formulas (5) is written, then we have
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
ON SINGULARITIES OF THE GALILEAN SPHERICAL DARBOUX RULED SURFACE . . . 1387
t (x) =
(
1,
√
2x, x2
)
, n (x) =
1√
2 + 4x2
(
0,
√
2, 2x
)
,
b (x) =
1√
2 + 4x2
(
0,−2x,
√
2
)
, κ (x) =
√
2 + 4x2, τ (x) =
√
2
2 + 4x2
.
Therefore we compute dR (γ) = {d (x) + un (x) | u ∈ R, x ∈ I} surface, here a spher-
ical curve d : I → S2
G by d (x) =
D (x)
‖D (x)‖G
. Hence, we have
d (x) + un (x) =
(
t (x) +
κ (x)
τ (x)
b (x)
)
+ un (x) =
=
(
1,−
√
2x− 4
√
2x3 +
u√
1 + 2x2
, 2 + 5x2 +
2ux√
2 + 4x2
)
.
Acknowledgement. The authors would like to thank the referee for the helpful
suggestions.
1. Bruce J. W., Giblin P. J. Curves and singularities. – Second ed. – Cambridge Univ. Press, 1992.
2. Arnol’d V. I., Gusein-Zade S. M., Varchenko A. N. Singularities of differentiable maps. – Birkhäuser,
1986. – Vol. 1.
3. Bruce J. W. On singularities, envelopes and elementary differential geometry // Math. Proc. Cambridge
Phil. Soc. – 1981. – 89. – P. 43 – 48.
4. Bruce J. W., Giblin P. J. Generic geometry // Amer. Math. Mon. – 1983. – 90. – P. 529 – 561.
5. Mond D. Singularities of the tangent developable surface of a space curve // Quart. J. Math. – 1989. –
40. – P. 79 – 91.
6. Pottmann H., Hofer M. Geometry of the squared-distance functions to curves and surfaces // Techn.
Rept. – 2002. – № 90.
7. Fidal D. L., Giblin P. J. Generic 1-parameter families of caustic by reflection in the plane // Math. Proc.
Cambridge Phil. Soc. – 1984. – 96. – P. 425 – 432.
8. Fidal D. L. The existance of sextactik point // Math. Proc. Cambridge Phil. Soc. – 1984. – 96. –
P. 433 – 436.
9. Izumıya S., Sano T. Generic affine differential geometry of space curves // Proc. Edinburgh Math. Soc.
– 1998. – 128A. – P. 301 – 314.
10. Sano T. Bifurcations of affine invariants for one parameter family of generic convex plane curves //
Hokkaido Univ. Math. J. – 1998.
11. Banchoff T., Gaffney T., McCrory C. Cusps of Gauss mappings // Res. Notes Math. – London: Pitman,
1982. – 55.
12. Wall C. T. C. Geometric properties of generic differentiable manifolds // Geometry and Topology III
(Lect. Notes Math.). – 1976. – 597. – P. 707 – 774.
13. Divjak B., Milin Šipuš Ž. Some special surfaces in the pseudo-Galilean space // Acta Math. hung. –
2008. – 118, № 3. – P. 209 – 226.
14. Cox D., Little J., O’shea D. Ideals, variets, and algorithms. – Second ed. – New York: Springer, 1997.
15. Molnar E. The projective interpretation of the eight 3-dimensional homogeneous geometries // Beitr.
Algebra und Geom. – 1997. – 38. – S. 261 – 288.
16. Casse R. Projective geometry an introduction. – Oxford Univ. Press, 2006. – P. 45 – 51.
17. Pavkovič B. J., Kamenarovič I. The equiform differential geometry of curves in the Galilean space G3
// Glas. Mat. – 1987. – 22(42). – P. 449 – 457.
18. Röschel O. Die geometrie des Galileischen raumes // Habilitationssch. – Inst. Math. und angew. Geom.
– 1984.
19. Milin Šipuš Ž. Ruled Weingarten surfaces in the Galilean space // Period. math. hung. – 2008. – 56,
№ 2. – P. 213 – 225.
20. Yaglom I. M. A simple non-Euclidean geometry and physical basis. – New York: Springer, 1979.
21. Do Carmo M. P. Differential geometry of curves and surfaces. – New Jersey: Prentice-Hall, 1976.
Received 19.03.10,
after revision — 13.07.10
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
|
| id | umjimathkievua-article-2962 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:33:39Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/53/fc199daa60c8715ea07a224579ff2653.pdf |
| spelling | umjimathkievua-article-29622020-03-18T19:41:19Z On singularities of the Galilean spherical darboux ruled surface of a space curve in $G_3$ Про особливості сферично-галілеєвої лінійчатої поверхні Дарбу просторової кривої в $G_3$ Şahin, T. Yilmaz, M. Шахін, Т. Йилмаз, М. We study the singularities of Galilean height functions intrinsically related to the Frenet frame along a curve embedded into the Galilean space. We establish the relationships between the singularities of the discriminant and the sets of bifurcations of the function and geometric invariants of curves in the Galilean space. Досліджено особливості галілеївських функцій висоти, що внутрішньо пов'язані із рамкою Френе вздовж кривої, вкладеної у галілеївський простір. Встановлено співвідношення між особливостями множини дискримінантів та множини біфуркацій функції і геометричними інваріантами кривих у галілеївському просторі. Institute of Mathematics, NAS of Ukraine 2010-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2962 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 10 (2010); 1377–1387 Український математичний журнал; Том 62 № 10 (2010); 1377–1387 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2962/2674 https://umj.imath.kiev.ua/index.php/umj/article/view/2962/2675 Copyright (c) 2010 Şahin T.; Yilmaz M. |
| spellingShingle | Şahin, T. Yilmaz, M. Шахін, Т. Йилмаз, М. On singularities of the Galilean spherical darboux ruled surface of a space curve in $G_3$ |
| title | On singularities of the Galilean spherical darboux ruled surface of a space curve in $G_3$ |
| title_alt | Про особливості сферично-галілеєвої лінійчатої поверхні Дарбу просторової кривої в $G_3$ |
| title_full | On singularities of the Galilean spherical darboux ruled surface of a space curve in $G_3$ |
| title_fullStr | On singularities of the Galilean spherical darboux ruled surface of a space curve in $G_3$ |
| title_full_unstemmed | On singularities of the Galilean spherical darboux ruled surface of a space curve in $G_3$ |
| title_short | On singularities of the Galilean spherical darboux ruled surface of a space curve in $G_3$ |
| title_sort | on singularities of the galilean spherical darboux ruled surface of a space curve in $g_3$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2962 |
| work_keys_str_mv | AT sahint onsingularitiesofthegalileansphericaldarbouxruledsurfaceofaspacecurveing3 AT yilmazm onsingularitiesofthegalileansphericaldarbouxruledsurfaceofaspacecurveing3 AT šahínt onsingularitiesofthegalileansphericaldarbouxruledsurfaceofaspacecurveing3 AT jilmazm onsingularitiesofthegalileansphericaldarbouxruledsurfaceofaspacecurveing3 AT sahint proosoblivostísferičnogalíleêvoílíníjčatoípoverhnídarbuprostorovoíkrivoívg3 AT yilmazm proosoblivostísferičnogalíleêvoílíníjčatoípoverhnídarbuprostorovoíkrivoívg3 AT šahínt proosoblivostísferičnogalíleêvoílíníjčatoípoverhnídarbuprostorovoíkrivoívg3 AT jilmazm proosoblivostísferičnogalíleêvoílíníjčatoípoverhnídarbuprostorovoíkrivoívg3 |