ON THE SECOND ORDER INVOLUTE CURVES IN E3
Article electronically published on April 10, 2017
Com mun.Fac.Sci.Univ.Ank.Series A1
Volum e 66, Numb er 2, Pages 332–339 (2017)
DOI: 10.1501/Com mua1_ 0000000823
ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series=A1
ON THE SECOND ORDER INVOLUTE CURVES IN E3
¼
ŞEYDA KILIÇO GLU
AND SÜLEYMAN ŞENYURT
Abstract. In this study we worked on the involute of involute curve of curve
. We called them the second order involute of curve
in E3 . All Frenet
apparatus of the second order involute of curve
are examined in terms of
Frenet apparatus of the curve . Further we show that; Frenet vector …elds
of the second order involute curve 2 can be written based on the principal
normal vector …eld of curve . Besides, we illustrate examples of our results.
The involute of the curve is well known by the mathematicians especially the di¤erential geometry scientists. There are many important consequences and properties
of curves. Involute curves have been studied by some authors [1, 2, 3, 5]. Let
: I ! E3 be the C 2 class di¤erentiable unit speed curve denote by fT; N; Bg
the moving Frenet frame. For an arbitrary curve 2 E3 , with …rst and second
curvature, and respectively, the Frenet formulae are given by [3]
8
0
>
<T = N
N0 =
T+ B
>
: 0
B =
N:
(0.1)
The tangent lines to a curve generate a surface called the tangent surface of .
A curve 1 which lies on the tangent surface of and intersects the tangent lines
orthogonally is called an involute of . The equation of the involutes is,
1
(s) = (s) + (s)T (s);
(s) = c
s; c 2 R;
(0.2)
where c is constant, [3]. The relationship are between Frenet apparatus of this
curves as follows, [5].
Received by the editors: April 27, 2016; Accepted: March 05, 2017.
2010 Mathematics Subject Classi…cation. 53A04 - 53A05.
Key words and phrases. Involute curve, second order involute curve, Frenet apparatus.
c 2 0 1 7 A n ka ra U n ive rsity
C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rsité d ’A n ka ra . S é rie s A 1 . M a th e m a tic s a n d S ta tistic s.
332
ON THE SECOND O RDER INVOLUTE CURVES IN E3
8
T1 = N
>
>
>
<
N1 = p
>
>
>
:B1 = p
and
1
=
p
2
+
2
T+p
2
+
2
2
+
2
T+p
2
+
2
+ 2
;
(c s)
B
1
=
(c
(0.3)
B;
0
2
2
s) (
333
2
+
2)
:
(0.4)
: I ! E3 , the vector W is called Darboux vector which
For any unit speed curve
is de…ned by [2]
W = T + B:
(0.5)
If we consider the normalization of the Darboux C =
1
kW k W ,
we have Figure 1
Figure 1. Darboux vector
sin ' = p
2
+
2
=
kW k
; cos ' = p
2
+
2
=
kW k
(0.6)
and
C = sin 'T + cos 'B
(0.7)
where \(W; B) = ', [4]. Substituting the equation (0.6) into equation (0.3) and
(0.4), we can write [1],
and
8
>
<T1 = N
N1 = cos 'T + sin 'B
>
:
B1 = sin 'T + cos 'B;
1
=
sec '
;
1
=
'0
:
(0.8)
(0.9)
334
¼
ŞEYDA KILIÇO GLU
AND SÜLEYM AN ŞENYURT
1. Second Order Involute Curve
: I ! E3 and 2 : I ! E3 are the arclengthed curves with the arcparameters
1
s1 and s2 , respectively. The quantities fT1 ; N1 ; B1 ; 1 ; 1 g and fT2 ; N2 ; B2 ; 2 ; 2 g
are collectively Frenet-Serret apparatus of the curve 1 and the involute 2 , respectively. 1 has the parametrization with arclength s as the involute curve of (s).
Also 2 has the parametrization with arclength s as the involute curve of 1 (s),
hence we can give the following de…nitions in terms of the parameter s. Let 2 (s2 )
be the involute of the curve 1 (s) then we have the following equation
2
(s) =
1
(s) +
1 T1
(s) :
(1.1)
Figure 2. Involute of involute of the curve
Theorem 1. The distance between corresponding points of the involute curve 1
and its involute 2 curve is
Z
=
c
ds; c1 = constant; 8s 2 I:
(1.2)
1
1
Proof. Di¤erentiating (1.1), we can write
ds2
0
=
N+ 1 B
1 T +
1 +
ds
where T1 = N and hT1 ; T2 i = 0 is. If we multiply internal both sides of the equation
with T1 we have,
0
=0
1 +
) T2
)
1
)
1
where c1 2 R and c1 is constant.
0
=
= c1
Z
ds
ON THE SECOND O RDER INVOLUTE CURVES IN E3
335
Substituting the equation (0.2) and (0.3) into equation (1.1), this give as following
de…nition:
De…nition 1. : I ! E3 be an unit speed curve. If 1 is an involute of and 2
is an involute of 1 , then the curve 2 is called second order involute curve of .
2
(s) =
(s) + (s)T (s) +
1 (s)N
is the expression of the second order involute curve
(s)
(1.3)
.
Theorem 2. The Frenet vector …elds of the second order involute
Frenet apparatus of the curve are
2
, based in the
8
>
T2 =
T+
B
>
>
kW k
kW k
>
>
>
1
>
3
<N =
q
nT + kW k4 N + 2 nB
2
2
6
2
kW k kW k + ( n)
>
>
>
1
>
2
>
>
kW k2 T
nN + kW k2 B
B =q
>
: 2
2
kW k6 + ( 2 n)
(1.4)
Proof. It is easy to say that Frenet vectors of the second order involute
on the Frenet apparatus of the curve 1 are
8
>
T2 = N1
>
>
>
<
N2 = p
>
>
>
>
:B 2 = p
1
2
1
+
2
1
1
2
1
+
2
1
T1 + p
T1 + p
1
2
1
2
+
2
; based
B1
(1.5)
1
1
1
2
2
+
B1 :
1
Substituting (0.3) and (0.4) into equation (1.5), we have
T2 = N1 = p
N2 =
and
B2 =
1 T1 +
p
2+
1
1
B1
2
1
=
T+ B
=
2+ 2
1
q
2
kW k kW k6 + ( 2 n)
T+ B
;
kW k
3
nT + kW k4 N +
T + 1 B1
1
p1
=q
kW k2 T
2+ 2
2
6
2
1
1
kW k + ( n)
1
2
2
nB
nN + kW k2 B :
¼
ŞEYDA KILIÇO GLU
AND SÜLEYM AN ŞENYURT
336
where kW k =
p
2
+
2
6
3
6
T2
6
1
6
4 N2 5 =
kW k 6
6
B2
4
2
2
p
p
0
and
= n 6= 0, which has the following matrix form
3
0
72
3
7 T
2
kW k4
n
7
p
p
7
2 n)2
kW k6 +( 2 n)2
kW k6 +( 2 n)2 7 4 N 5
7 B
5
2
2
kW
k
n
p
p
2
2
2
3
n
kW k6 +(
kW k2
kW k6 +(
2 n)
kW k6 +(
2 n)
kW k6 +(
2 n)
(1.6)
Theorem 3. The …rst and the second curvatures of the second order involute
based on the Frenet apparatus of the curve are respectively.
2
=
s
4 2
n
2
kW k6 + ( 2 n)
;
2
6
1 kW k
=
2
2
n
kW k3
6
1 kW k (kW k
+
0
2
2
=
k
=
s
2
0
k
^
2
2
00
03
k
2
=
detf
k
2
2
;
0
2
^
4 2
n
2
=
2
2
Also it is easy to say that, the torsion of second order involute
00
;
2
2
000
002
g
2
n
kW k3
1 kW k (kW k
6
+
;
0
4 n2 )
:
2
, we di¤eren-
(1.8)
;
kW k6 + ( 2 n)
2
6
1 kW k
0
2
is
2
(1.7)
4 n2 )
Proof. In order to calculate the curvature and torsion of the curve
tiate
8
0
=
>
1T +
1 B;
2
>
>
>
>
>
>
>
< 00 =
0
2
T kW k2 1 N + 0 1 +
B;
1
2
>
>
>
>
00
02
3
> 000 =
>
kW k2 1 T
+ kW k2 +
1
2
>
>
:
00
2
0
+
B:
1 + kW k 1 +
The curvature of second order involute
2
is
N
ON THE SECOND O RDER INVOLUTE CURVES IN E3
Theorem 4. Let unit Darboux vector …eld of involute
expressed in terms of Frenet apparatus of the curve
1
C1 = q
'0 2 + ( sec ')2
1
337
be C1 . This vector is
tan 'T + '0 N + B
(1.9)
Proof. The vector C1 is the direction of the Darboux vector W1 of the involute
curve 1 we can write
C1 = sin '1 T1 + cos '1 B1 ;
(1.10)
where
cos '1 = p
1
2
1
+
2
1
; sin '1 = p
1
2
1
+
2
:
(1.11)
1
Substituting the equation (0.9) into equation (1.11), we can write
cos '1 = q
'0
sec '
; sin '1 = q
:
'0 2 + ( sec ')2
'0 2 + ( sec ')2
(1.12)
Substituting the equation (...truncated)
