Abstract: This paper mainly studies the Singularities of smooth mapping. The singularities of the families of Gauss maps corresponding to the family of mappings are studied and the shape of these families and their singularities using mathematica program are illustrated and plotted.  By changing the control parameters we find some singular points for the family which can be classified according to the famous theorems in singularity theory. Using the Hessian matrix we can obtain the singular points and singular set. Geometrically these singularities can be plotted but the classification of them can&apos;t be a valuable for all points. Using the terminology of level set which tell us the type of singular points like folds (level sets is start line), cusp (level set is semicubical parabola). In general there is no existance of some famous types.  where ∆ is the discriminant set and it is a plan. Remark 4 The family of contours is given from z = k (constant), and the family of zero level set corresponding to k = 0 are given through the figurer

[ M A Soliman

E Dahi

M A Soliman

Nassar H Abdel-All

Soad A Hassan

CiteSeerX

Life Science Journal, 2011;8(3)         http://www.lifesciencesite.com 
  
482 
 
Families of maps Singularities and its Gauss maps 
 
M. A. Soliman1, Nassar. H. Abdel-All1, Soad. A. Hassan1 and E. Dahi2 
 
1. Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt 
2. Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt 
saodali@ymail.com 
 
Abstract: This paper mainly studies the Singularities of smooth mapping. The singularities of the families of Gauss 
maps corresponding to the family of mappings are studied and the shape of these families and their singularities 
using mathematica program are illustrated and plotted. 
[M. A. Soliman, Nassar. H. Abdel-All, Soad. A. Hassan and E. Dahi. Families of maps Singularities and its Gauss 
maps. Life Science Journal. 2011;8(3):482-487] (ISSN: 1097-8135). http://www.lifesciencesite.com. 
 
Keywords: extrinsic differential geometry, Families of Maps, Gauss map, theory of singularities. 
 
1. Families of Maps (Scaler Function) 
Let 
),...,,,...,( 11 aauu rnfz =  
be the family of r parameter of hypersurface in Rn. 
Where RaRu rrn aU ⊂=⊂∈ + ,1α , 
the vector a is a vector of control parameter. 
The discriminate set of these families can be 
calculate by the solution of these (n) equations 
0=αp such that 
α
α
u
zp
∂
∂
=  and finding 
αu ’s as 
function in ar say )( ααα auu = and substitution in 
the relation f = 0 we obtain the discriminate set for 
the considered formal as in the following form 
0),...,( 1 ==∆ raaf  
By changing the control parameters we find some 
singular points for the family which can be classified 
according to the famous theorems in singularity 
theory. Using the Hessian matrix we can obtain the 
singular points and singular set. Geometrically these 
singularities can be plotted but the classification of 
them can’t be a valuable for all points. Using the 
terminology of level set which tell us the type of 
singular points like folds (level sets is start line), cusp 
(level set is semicubical parabola). In general there is 
no existance of some famous types. 
 
Remark 1 For the simple ideas for finding and 
calculated singular points and singular sets see [1, 2, 
3]. 
Definition 1 [1] The level set attached to the 
hypersurface M is defined as the following: let un = 
Z(u1, u2, …, un-1) = c, c is constant, if c = 0 that given 
the level set V0 = {(u1, u2, …, un-1): un = 0} and the 
other level sets are 
Vc={(u1, u2, …, un-1): un = 0},    0≠c  
Another version of the definition of level sets is 
contours as given in the following 
Definition 3 We say the point p on a surface M with 
a parametric representation is a contour point if and 
only if                  N.pc=0 
where N is the normal vector field on the surface M 
and c is the view point. The contour line or contour, 
for short, of a surface is the set of all its contour 
points. 
The determination of the contour line of a surface in 
the general case involves a numerical method to find 
the zeros of a real-valued function of n real variables 
in a domain (u1, u2, …, un) ∈  U. An algorithm and its 
implementation can be found in [6]. 
2. Application 
Let us consider a 3-parameter family of surfaces 
defined by mong’s form as the following: 
Z=f (u, v; a, b, c) =au4+bu2v+u2+cv2,   a, b, c ∈R  (1) 
This family is denoted byσ , where a, b, c are the 
parameters of the familyσ . The bifurcation diagram 
of the zeroes of this family (of functions of u; v 
depending on the parameters (a, b, c) is given from: 
0==∆ c                    (2) 
where ∆  is the discriminant set and it is a plan. 
Remark 4 The family of contours is given from z = k 
(constant), and the family of zero level set 
corresponding to k = 0 are given through the figurers 
[1, 2, 3]. 
The families (1) are classified into serval subclasses 
as the following: 
1) a = b = c = 0, in this case the family f has A1 
singularity (fold). 
2) b = c = 0, a < 0 then f has a singularity (cusp). 
3) b = c = 0, a > 0 in this case the family f has A1 
singularity (fold). 
4) a = c = 0, b < 0 or b > 0 in this case the family f 
has a fold. 
5) a = b = 0, c < 0 or c > 0 in this case the family f 
has ± B2 singularity (the normal form of this set is 
Bk = xk + y2; k≥ 2). 
6)  a; b; c > 0, a; b; c < 0.           7) a = 0, b < 0; c < 0. 
Life Science Journal, 2011;8(3)         http://www.lifesciencesite.com 
  
483 
 
8)  a = 0, b > 0; c > 0.                 9) a = 0, b < 0; c > 0. 
10) a = 0, b > 0; c < 0.              11) b = 0, a < 0; c < 0. 
12) b = 0, a > 0; c > 0.             13) b = 0, a < 0; c > 0. 
14) b = 0, a > 0; c < 0.             15) c = 0, a < 0; b < 0. 
16) c = 0, a > 0; b > 0.             17) c = 0, a < 0; b > 0. 
18) c = 0, a > 0; b < 0. 
The previous families are plotted by iσ , i = 
1, 2,…,18 which corresponding to the conditions ( 1), 
2),..., 18)) and their geometric interpretations are 
given through the figures [4]- [16]. 
we denote the previous families by iσ   i = 
1, 2,…,18 which corresponding to the codomain ( 1), 
2),..., 18)). 
The normal vector field to the family of surfaces (1), 
is given by: 
N(u, v; a, b, c)={-2(u+2au3+buv),-(bu2+2cv),1}  (3) 
For the subfamily iσ , it is easy to see that the 
normal vector fields has a planer swallowtail when a, 
b, c > 0 or a, b, c < 0 as shown in figure (17). 
Since the family (3) of function lies in the 
plane z =1, so we can make a modification to the 
families (N(u, v; a, b, c)) as in the following: 
Nmod(u, v; a, b, c)={-2(u+2au3+buv),-(bu2+2cv)}  (4) 
The singularities of this family can be deferral using 
the rank of its Jacobian matrix which is given by: 
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−
−++−
cbu
bubvau
22
2)61(2 2
      (5) 
so the discriminant set(singular set) is given as: 
S :=4c-4b2u2+24acu2+4bcv = 0     (6) 
From which we have : 
S:=u2= ⎟
⎠
⎞
⎜
⎝
⎛ +
− b
v
acb
bc 1
62
,   0,62 ≠≠ bacb   (7) 
This relation represents a parabola in the 
plan u, v with  ⎟
⎠
⎞
⎜
⎝
⎛ −
b
1,0 , i. e, is non defined for all 
points lies on the hyperbolic paraboloid. Thus the 
image of singular set under the modified normal 
vector field N is 
Nmod(S)= 
⎭
⎬
⎫
⎩
⎨
⎧ −
−
−
− 2
2
3
2 )4(32,)4(2 u
b
acb
b
cu
c
acb (8) 
It is easy to see that the normal vector field of the 
family has a cusp point for all points except for the 
points lies on the hyperbolic paraboloid 
042 =− acb  with conditions b; c ≠  0 see figures 
(18). The shapes of discriminate set, zero level sets, 
for the family subclasses iσ of the given family and 
the normal vector field for the subclasses iσ  and 
singular point are shown in fig [1] to [18]. 
 
 
3. Gauss and Mean Curvature 
The notion of curvature of a surface is a 
great deal complicated than the notion of curvature of 
a curve. Let α  be a curve in R3. and let p be a point 
in on the trace ofα .The curvature of α  at p 
measures the rate at which α  leaves the tangent line 
to α  at p. By analogy. the curvature of a surface 
M⊂  R3 at p ∈  M should measure the rate at which 
M leaves the tangent plane to M at p. But for surface 
a difficulty arises that was not present for curves: 
although a curve can separate from one of its tangent 
lines in only two direction. a surface separates from 
one of its tangent planes in infinitely many directions. 
In general the rate of departure of a surface from one 
of its tangent planes depends on the direction. 
It is usually possible to glance at almost any 
surface and recognize which points are elliptic, 
hyperbolic, parabolic or planer. at the planer points 
we find the Gauss curvature achieves to maximum 
value at thats points and mean curvature is planer. I. e 
Gauss curvature and mean curvature have a relation 
with elliptic, hyperbolic, parabolic and planer points. 
Now we calculate the Gauss curvature and 
mean curvature of that family of functions. The 
shapes of Gauss curvature and mean curvature and 
their contour are plotted to show the singularities of 
these families. 
The first fundamental coefficients of the 
family under consideration are given as: 
G11=1+4(u+2au3+buv)2, G22=1+(bu2+2cv)2 and 
G12=2(bu2+2cv) (u+2au3+buv) 
and its discriminant (metric) is given by: 
 
G= 
 
 
The second fundamental coefficients Lij of this family 
are: 
L11=2(1+6au2+bv)Q,   L22=2cQ,   L12=2buQ          (9) 
Where Q= 2322 )2(4)2(1 buvauucvbu +++++  
and the discriminant L = det(Lij) is given by 
L=4(c-b2u2+6acu2+bcv)Q               (10) 
From above equations and by simple 
calculation we obtain the Gauss and mean curvatures 
as the following: 
 
K= 
(11) 
 
H= 
Life Science Journal, 2011;8(3)         http://www.lifesciencesite.com 
  
484 
 
  
/  
respectively. 
 
From (11) its easy to see that K = 0 when u; v lie on 
the parabola equation: 
u2= ⎟
⎠
⎞
⎜
⎝
⎛ +
− b
v
acb
bc 1
62
 
i. e., K = 0 when u; v lie on the singular set (6). 
Finally the shapes of Gauss Curvature and Mean 
Curvature and its contours corresponding some 
iσ are plotted in figure [19] - [29]. 
 
Figures 
 
 
 
 
 
 
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References 
1.  V. I. Arnol’d, S .M. Gusein-Zade and A. N. Varchenko, 
Singularities of Differentiable Maps vol. I. Birkhauser 
(1986). 
2. V. I. Arnol’d: The Theory of Singularities and Its 
Applications ; Cambridge Univ. Press, Cambridge, (1991). 
3.  T. Banchoff,” Cusps of Gauss mappings” Publisher: 
Longman Scicence Technology, May, (1986). 
4. M. P. Do Carmo, Differential Geometry of Curves and 
Surfaces, Prentice-Hall, (1976). 
5.  M. P. Do Carmo, Rimannian Geometry, Birkhuser Basel, 
(1992). 
6.  M. Failing, E. Malkowsky, Ein effizienter 
Nullstellenalgorithmus zur computergrafis chen Darstellung 
spezieller Kurven und Flachen, Mitt. Math. Sem. Giessen, 
229, (1996), 11-25. 
7.  M. Golubitsky, V. Guillemin, Stable mappings and thier 
singularities; Springer Verlag, New York, Inc, (1973). 
8.  A. Gry, Modern differential Geometry of Curves and 
Surfaces, studies in advanced mathmatices. CRC press,Inc, 
(1993). 
9.  L. u. Yung-Chen: Singularity Theory and an Introduction to 
Catastrophe Theory Springer-Verlag, New York, Inc (1976). 
 
8/21/2011
 


Families of maps Singularities and its Gauss maps

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Families of maps Singularities and its Gauss maps

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