Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2015, Article ID 397126, 6 pages http://dx.doi.org/10.1155/2015/397126
Research Article The Relationship between Focal Surfaces and Surfaces at a Constant Distance from the Edge of Regression on a Surface Semra Yurttancikmaz and Omer Tarakci Department of Mathematics, Faculty of Science, Ataturk University, 25240 Erzurum, Turkey Correspondence should be addressed to Semra Yurttancikmaz;
[email protected] Received 7 July 2014; Accepted 8 September 2014 Academic Editor: John D. Clayton Copyright Β© 2015 S. Yurttancikmaz and O. Tarakci. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the relationship between focal surfaces and surfaces at a constant distance from the edge of regression on a surface. We show that focal surfaces F 1 and F 2 of the surface M can be obtained by means of some special surfaces at a constant distance from the edge of regression on the surface M.
1. Introduction Surfaces at a constant distance from the edge of regression on a surface were firstly defined by Tarakci in 2002 [1]. These surfaces were obtained by taking a surface instead of a curve in the study suggested by Hans Vogler in 1963. In the mentioned study, Hans Vogler asserted notion of curve at a constant distance from the edge of regression on a curve. Also, Tarakci and Hacisalihoglu calculated some properties and theorems which known for parallel surfaces for surfaces at a constant distance from the edge of regression on a surface [2]. Later, various authors became interested in surfaces at a constant distance from the edge of regression on a surface and investigated Euler theorem and Dupin indicatrix, conjugate tangent vectors, and asymptotic directions for this surface [3] and examined surfaces at a constant distance from the edge of regression on a surface in πΈ13 Minkowski space [4]. Another issue that we will use in this paper is the focal surface. Focal surfaces are known in the field of line congruence. Line congruence has been introduced in the field of visualization by Hagen et al. in 1991 [5]. They can be used to visualize the pressure and heat distribution on an airplane, temperature, rainfall, ozone over the earthβs surface, and so forth. Focal surfaces are also used as a surface interrogation tool to analyse the βqualityβ of the surface before further processing of the surface, for example, in a NC-milling operation [6]. Generalized focal surfaces are related to hedgehog
diagrams. Instead of drawing surface normals proportional to a surface value, only the point on the surface normal proportional to the function is drawing. The loci of all these points are the generalized focal surface. This method was introduced by Hagen and Hahmann [6, 7] and is based on the concept of focal surface which is known from line geometry. The focal surfaces are the loci of all focal points of special congruence, the normal congruence. In later years, focal surfaces have been studied by various authors in different fields. In this paper, we have discovered a new method to constitute focal surfaces by means of surfaces at a constant distance from the edge of regression on a surface. Focal surfaces πΉ1 and πΉ2 of the surface π in πΈ3 are associated with surfaces at a constant distance from the edge of regression on π that formed along directions of ππ lying in planes ππ{ππ’ , π} and ππ{πV , π}, respectively.
2. Surfaces at a Constant Distance from the Edge of Regression on a Surface Definition 1. Let π and ππ be two surfaces in πΈ3 Euclidean space and let ππ be a unit normal vector and let ππ π be tangent space at point π of surface π and let {ππ , ππ } be orthonormal bases of ππ π. Take a unit vector ππ = π1 ππ +
2
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π2 ππ + π3 ππ , where π1 , π2 , π3 β R are constant and π12 + π22 + π32 = 1. If there is a function π defined by π : π σ³¨β ππ ,
π (π) = π + πππ ,
Here, if π1 = π2 = 0, then ππ = ππ and so π and ππ are parallel surfaces. Now, we represent parametrization of surfaces at a constant distance from the edge of regression on π. Let (π, π) be a parametrization of π, so we can write that 2
(π’, V)
π (π’, V) .
and {ππ’ , πV } is a basis of π(ππ ). If we denote by ππ unit normal vector of ππ , then ππ is [π , π ] ππ = σ΅©σ΅© π’ V σ΅©σ΅© σ΅©σ΅©[ππ’ , πV ]σ΅©σ΅© = (π 1 π
1 (1 + π 3 π
2 ) ππ’ + π 2 π
2 (1 + π 3 π
1 ) πV
(3)
2
Γ (π21 π
12 (1 + π 3 π
2 ) + π22 π
22 (1 + π 3 π
1 ) 2 β1/2
2
+ (1 + π 3 π
1 ) (1 + π 3 π
2 ) )
,
where π
1 , π
2 are principal curvatures of the surface π. If 2
2
π΄ = (π21 π
12 (1 + π 3 π
2 ) + π22 π
22 (1 + π 3 π
1 ) 2
ππ = {π (π’, V) : π (π’, V)
2 1/2
(10)
+(1 + π 3 π
1 ) (1 + π 3 π
2 ) )
= π (π’, V) + π (π1 ππ’ (π’, V)
we can write (4)
+ π2 πV (π’, V) and if we get ππ1 = π 1 , ππ2 = π 2 , ππ3 = π 3 , then we have
+ π 2 πV (π’, V) + π 3 π (π’, V) ,
(5)
π21 + π22 + π23 = π2 } . Calculation of ππ’ and πV gives us that ππ’ = ππ’ + π 1 ππ’π’ + π 2 πVπ’ + π 3 ππ’ , πV = πV + π 1 ππ’V + π 2 πVV + π 3 πV .
(6)
Here, ππ’π’ , πVπ’ , ππ’V , πVV , ππ’ , πV are calculated as in [1]. We choose curvature lines instead of parameter curves of π and let π’ and V be arc length of these curvature lines. Thus, the following equations are obtained: ππ’π’ = β π
1 π, πVV = β π
2 π, ππ’V = πVπ’ = 0,
π 1 π
1 (1 + π 3 π
2 ) π π
(1 + π 3 π
1 ) ππ’ + 2 2 πV π΄ π΄
(11)
Here, in case of π
1 = π
2 and π 3 = β1/π
1 = β1/π
2 since ππ’ and πV are not linearly independent, ππ is not a regular surface. We will not consider this case [1].
ππ = {π (π’, V) : π (π’, V) = π (π’, V) + π 1 ππ’ (π’, V)
ππ =
(1 + π 3 π
1 ) (1 + π 3 π
2 ) + π. π΄
+ π3 π (π’, V))}
πV = π
2 πV .
(9)
+ (1 + π 3 π
1 ) (1 + π 3 π
2 ) π) 2
Thus, it is obtained that
ππ’ = π
1 ππ’ ,
(8)
πV = (1 + π 3 π
2 ) πV β π 2 π
2 π
(2)
In case {ππ’ , πV } is a basis of ππ π, then we can write that ππ = π1 ππ’ +π2 πV +π3 ππ , where ππ’ , πV are, respectively, partial derivatives of π according to π’ and V. Since ππ = {π(π) : π(π) = π + πππ }, a parametric representation of ππ is π (π’, V) = π (π’, V) + ππ (π’, V) .
ππ’ = (1 + π 3 π
1 ) ππ’ β π 1 π
1 π,
(1)
where π β R, then the surface ππ is called the surface at a constant distance from the edge of regression on the surface π.
π : π β πΈ σ³¨β π
From (6) and (7), we find
(7)
3. Focal Surfaces The differential geometry of smooth three-dimensional surfaces can be interpreted from one of two perspectives: in terms of oriented frames located on the surface or in terms of a pair of associated focal surfaces. These focal surfaces are swept by the loci of the principal curvatures radii. Considering fundamental facts from differential geometry, it is obvious that the centers of curvature of the normal section curves at a particular point on the surface fill out a certain segment of the normal vector at this point. The extremities of these segments are the centers of curvature of two principal directions. These two points are called the focal points of this particular normal [8]. This terminology is justified by the fact that a line congruence can be considered as the set of lines touching two surfaces, the focal surfaces of the line congruence. The points of contact between a line of the congruence and the two focal surfaces are the focal points of this line. It turns out that the focal points of a normal congruence are the centers of curvature of the two principal directions [9, 10].
Advances in Mathematical Physics
3 The normal vector of the surface πππ at the point ππ (π) is
We represent surfaces parametrically as vector-valued functions π(π’, V). Given a set of unit vectors π(π’, V), a line congruence is defined: πΆ (π’, V) = π (π’, V) + π· (π’, V) π (π’, V) ,
(12)
where π·(π’, V) is called the signed distance between π(π’, V) and π(π’, V) [8]. Let π(π’, V) be unit normal vector of the surface. If π(π’, V) = π(π’, V), then πΆ = πΆπ is a normal congruence. A focal surface is a special normal congruence. The parametric representation of the focal surfaces of πΆπ is given by 1 π (π’, V) ; πΉπ (π’, V) = π (π’, V) β π
π (π’, V)
π = 1, 2,
(13)
where π
1 , π
2 are the principal curvatures. Except for parabolic points and planar points where one or both principal curvatures are zero, each point on the base surface is associated with two focal points. Thus, generally, a smooth base surface has two focal surface sheets, πΉ1 (π’, V) and πΉ2 (π’, V) [11]. The generalization of this classical concept leads to the generalized focal surfaces: πΉ (π’, V) = π (π’, V) + ππ (π
1 , π
2 ) π (π’, V)
πππ = π 1π π
1 (π) ππ’ (π) + (1 + π 3π π
1 (π)) ππ .
Here, it is clear that πππ is in plane ππ{ππ’ , π}. Suppose that π line passing from the point ππ (π) and being in direction πππ(π) π is ππ and a representative point of ππ is π = (π₯, π¦) = π₯ππ’ (π) + π¦ππ ; then, the equation of ππ is σ³¨σ³¨β σ³¨σ³¨σ³¨σ³¨σ³¨β π ππ β
β
β
ππ = πππ (π) + π1 πππ(π) . π
π
being in direction πππ(π) is ππ and a representative point of ππ π is π
= (π₯, π¦); then, equation of ππ is σ³¨β σ³¨σ³¨σ³¨σ³¨σ³¨β π ππ β
β
β
ππ
= πππ (π) + π2 πππ(π) , π
Theorem 2. Let surface π be given by parametrical π(π’, V). One considers all surfaces at a constant distance from the edge of regression on π that formed along directions of ππ lying in plane ππ{ππ’ , π}. Normals of these surfaces at points π(π) corresponding to point π β π generate a spatial family of line of which top is center of first principal curvature πΆ1 = πβ (1/π
1 (π))ππ at π. Proof. Surfaces at a constant distance from the edge of regression on π that formed along directions of ππ lying in plane ππ{ππ’ , π} are defined by
ππ β
β
β
(π₯, π¦) = (π 1π , π 3π ) + π1 (π 1π π
1 , 1 + π 3π π
1 ) , ππ β
β
β
π¦ =
1 + π 3π π
1 π 1π π
1
π₯β
1 , π
1
ππ β
β
β
(π₯, π¦) = (π 1π , π 3π ) + π2 (π 1π π
1 , 1 + π 3π π
1 ) , ππ β
β
β
π¦ =
1 + π 3π π
1 π 1π π
1
π₯β
(19)
1 . π
1
Corollary 3. Directions of normals of all surfaces at a constant distance from the edge of regression on π that formed along directions of ππ lying in plane ππ{ππ’ , π} intersect at a single point. This point πΆ1 = π β (1/π
1 (π))ππ which is referred in Theorem 2 is on the focal surface πΉ1 . We know that πΉ1 (π) = π β
1 π π
1 π
(20)
from definition of focal surfaces. Moreover, we can see easily the following equations from Figure 1: π
πΉ1 (π) = ππ (π) β ππ πππ(π)
(15)
These surfaces and their unit normal vectors are, respectively, denoted by πππ and πππ . We will demonstrate that intersection point of lines which pass from the point ππ (π) and are in π direction πππ(π) is πΆ1 = π β (1/π
1 (π))ππ . π
(18)
From here, it is clear that intersection point of ππ and ππ is (π₯, π¦) = (0, β1/π
1 ). So, intersection point of the lines ππ and ππ is the point πΆ1 = πβ(1/π
1 (π))ππ in plane ππ{ππ’ (π), ππ }.
4. The Relationship between Focal Surfaces and Surfaces at a Constant Distance from the Edge of Regression on a Surface
ππ (π) = π + π 1π ππ’ (π) + π 3π ππ .
π = 1, 2, . . . .
We find intersection point of these lines. Since it is studied in plane of vectors {ππ’ (π), ππ }, the point π can be taken as beginning point. If we arrange the lines ππ and ππ , then we find
with π β R, (14)
π = 1, 2, . . . ,
(17)
Besides, suppose that line passing from the point ππ (π) and
where the scalar function π depends on the principal curvatures π
1 = π
1 (π’, V) and π
2 = π
2 (π’, V) of the surface π. The real number π is used as a scale factor. If the curvatures are very small you need a very large number π to distinguish the two surfaces π(π’, V) and πΉ(π’, V) on the screen. Variation of this factor can also improve the visibility of several properties of the focal surface; for example, one can get intersections clearer [6].
ππ : π σ³¨β πππ ,
(16)
π
(21)
or π
πΉ1 (π) = ππ (π) β ππ πππ(π) . π
(22)
These equations show us that the focal surface πΉ1 of the surface π can be stated by surfaces at a constant distance from
4
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the edge of regression on π that formed along directions of π π ππ lying in plane ππ{ππ’ , π}. If ππ = 1/π
1 π or ππ = 1/π
1 π , then the focal surfaces πΉ1 of surfaces π, πππ , and πππ will be the same. This case has been expressed in following theorem.
Nfπ
fi (P)
Theorem 4. Focal surfaces πΉ1 of the surface π and surfaces at a constant distance from the edge of regression on π that formed along directions of ππ lying in plane ππ{ππ’ , π} are the same if and only if first principal curvature π
1 of the surface π is constant. Proof. Suppose that focal surfaces πΉ1 of surfaces π and ππ formed along directions of ππ lying in plane ππ{ππ’ , π} intersect; then, ππ mentioned in (21) must be ππ =
1 π
π
1 π
.
NP
Mfπ
ZPπ
fj (P) ZPπ
P
Nfπ
Mfπ
πu
M 1 π
1
(23)
π
First principal curvature π
1 of ππ formed along directions of ππ lying in plane ππ{ππ’ , π}, that is, for π 2 = 0, is calculated by Tarakci as [1]
C1 = F1 (P)
F1 dj di
π
π
1 =
1 2
βπ21 π
12 + (1 + π 3 π
1 )
(
π 1 (ππ
1 /ππ’) π21 π
12
2
+ (1 + π 3 π
1 )
+ π
1 ) . (24)
σ³¨σ³¨σ³¨σ³¨σ³¨σ³¨β Besides, from Figure 1, since ππ = |πΆ1 ππ (π)| is distance between points of πΆ1 = (0, β1/π
1 ) and ππ (π) = (π 1 , π 3 ) lying in plane ππ{ππ’ , π}, we can write σ΅¨σ΅¨σ³¨σ³¨σ³¨σ³¨σ³¨σ³¨βσ΅¨σ΅¨ 1 2 ππ = σ΅¨σ΅¨σ΅¨σ΅¨πΆ1 ππ (π)σ΅¨σ΅¨σ΅¨σ΅¨ = β π21 + (π 3 + ) . π
1 σ΅¨ σ΅¨
(25)
If we substitute (24) and (25) in (23) and make necessary arrangements, we obtain ππ
1 = 0. ππ’
(26)
Thus, we have π
1 = const. The converse statement is trivial. Hence, our theorem is proved. Theorem 5. Let surface π be given by parametrical π(π’, V). We consider all surfaces at a constant distance from the edge of regression on π that formed along directions of ππ lying in plane ππ{πV , π}. Normals of these surfaces at points π(π) corresponding to point π β π generate a spatial family of line of which top is center of second principal curvature πΆ2 = πβ (1/π
2 (π))ππ at π. Proof. Surfaces at a constant distance from the edge of regression on π that formed along directions of ππ lying in plane ππ{πV , π} are defined by ππ : π σ³¨β πππ ,
π = 1, 2, . . . ,
ππ (π) = π + π 2π πV (π) + π 3π ππ .
(27)
Figure 1: Directions of normals of all surfaces at a constant distance from the edge of regression on π that formed along directions of ππ lying in plane ππ{ππ’ , π} and their intersection point (focal point).
These surfaces and their unit normal vectors are, respectively, denoted by πππ and πππ . We will demonstrate that intersection point of lines which pass from the point ππ (π) and are in π direction πππ(π) is πΆ2 = π β (1/π
2 (π))ππ . π
The normal vector of the surface πππ at the point ππ (π) is πππ = π 2π π
2 (π) πV (π) + (1 + π 3π π
2 (π)) ππ .
(28)
Here, it is clear that πππ is in plane ππ{πV , π}. Suppose that π line passing from the point ππ (π) and being in direction πππ(π) π is ππ and a representative point of ππ is π = (π₯, π¦) = π₯πV (π) + π¦ππ ; then, equation of ππ is σ³¨σ³¨β σ³¨σ³¨σ³¨σ³¨σ³¨β π ππ β
β
β
ππ = πππ (π) + π1 πππ(π) . π
(29)
Besides, suppose that line passing from the point ππ (π) of the π
surface πππ and being in direction πππ(π) is ππ and a represenπ tative point of ππ is π
= (π₯, π¦); then, equation of ππ is σ³¨β σ³¨σ³¨σ³¨σ³¨σ³¨β π ππ β
β
β
ππ
= πππ (π) + π2 πππ(π) , π
π = 1, 2, . . . .
(30)
We find intersection point of these two lines. Since it is studied in plane of vectors {πV (π), ππ }, the point π can be taken
Advances in Mathematical Physics
5
as beginning point. If we arrange the lines ππ and ππ , then we find ππ β
β
β
(π₯, π¦) = (π 2π , π 3π ) + π1 (π 2π π
1 , 1 + π 3π π
2 ) , ππ β
β
β
π¦ =
1 + π 3π π
2 π 2π π
2
σ΅¨σ΅¨σ³¨σ³¨σ³¨σ³¨σ³¨σ³¨βσ΅¨σ΅¨ 1 2 ππ = σ΅¨σ΅¨σ΅¨σ΅¨πΆ2 ππ (π)σ΅¨σ΅¨σ΅¨σ΅¨ = β π22 + (π 3 + ) . π
2 σ΅¨ σ΅¨
1 π₯β , π
2
ππ β
β
β
(π₯, π¦) = (π 2π , π 3π ) + π2 (π 2π π
2 , 1 + π 3π π
2 ) , ππ β
β
β
π¦ =
1 + π 3π π
2
π₯β
π 2π π
2
σ³¨σ³¨σ³¨σ³¨σ³¨σ³¨β Besides, similar to Figure 1, since ππ = |πΆ2 ππ (π)| is the distance between points of πΆ2 = (0, β1/π
2 ) and ππ (π) = (π 2 , π 3 ) lying in plane ππ{πV , π}, we can write
(31)
If we substitute (35) and (36) in (34) and make necessary arrangements, we obtain ππ
2 = 0. πV
1 . π
2
From here, it is clear that intersection point of ππ and ππ is (π₯, π¦) = (0, β1/π
2 ). So, intersection point of the lines ππ and ππ is the point πΆ2 = π β (1/π
2 (π))ππ in plane ππ{πV (π), ππ }. Corollary 6. The point πΆ2 = πβ(1/π
2 (π))ππ which is referred in Theorem 5 is on the focal surface πΉ2 . Similar to Figure 1, we can write equations π
πΉ2 (π) = ππ (π) β ππ πππ(π)
(32)
π
(36)
(37)
Thus, we have π
2 = const. The converse statement is trivial. Hence, our theorem is proved. Points on the surface π can have the same curvature in all directions. These points correspond to the umbilics, around which local surface is sphere-like. Since normal rays of umbilic points pass through a single point, the focal mesh formed by vertices around an umbilic point can shrink into a point [11].
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
or π
πΉ2 (π) = ππ (π) β ππ πππ(π) .
(33)
π
These equations show us that the focal surface πΉ2 of the surface π can be stated by surfaces at a constant distance from the edge of regression on π that formed along directions of π π ππ lying in plane ππ{πV , π}. If ππ = 1/π
2 π or ππ = 1/π
2 π , then the focal surfaces πΉ2 of surfaces π, πππ , and πππ will be the same. This case has been expressed in following theorem. Theorem 7. Focal surfaces πΉ2 of the surface π and surfaces at a constant distance from the edge of regression on π that formed along directions of ππ lying in plane ππ{πV , π} are the same if and only if second principal curvature π
2 of the surface π is constant. Proof. Suppose that focal surfaces πΉ2 of surfaces π and ππ formed along directions of ππ lying in plane ππ{πV , π} intersect; then, ππ mentioned in (32) must be ππ =
1 π
π
2 π
.
(34)
π
Second principal curvature π
2 of ππ formed along directions of ππ lying in plane ππ{πV , π}, that is, for π 1 = 0, is calculated by Tarakci as [1] π
π
2 =
1 2
βπ22 π
22 + (1 + π 3 π
2 )
(
π 2 (ππ
2 /πV) π22 π
22
2
+ (1 + π 3 π
2 )
+ π
2 ) . (35)
References Β¨ Tarakci, Surfaces at a constant distance from the edge of reg[1] O. ression on a surface [Ph.D. thesis], Ankara University Institute of Science, Ankara, Turkey, 2002. Β¨ Tarakci and H. H. HacisalihoΛglu, βSurfaces at a constant [2] O. distance from the edge of regression on a surface,β Applied Mathematics and Computation, vol. 155, no. 1, pp. 81β93, 2004. Β¨ usaglam, and C. Ekici, βConjugate [3] N. Aktan, A. GΒ¨orgΒ¨ulΒ¨u, E. OzΒ¨ tangent vectors and asymptotic directions for surfaces at a constant distance from edge of regression on a surface,β International Journal of Pure and Applied Mathematics, vol. 33, no. 1, pp. 127β133, 2006. Β¨ Kalkan, βSurfaces at a constant distance from [4] D. SaΛglam and O. the edge of regression on a surface in πΈ13 ,β Differential GeometryβDynamical Systems, vol. 12, pp. 187β200, 2010. [5] H. Hagen, H. Pottmann, and A. Divivier, βVisualization functions on a surface,β Journal of Visualization and Animation, vol. 2, pp. 52β58, 1991. [6] H. Hagen and S. Hahmann, βGeneralized focal surfaces: a new method for surface interrogation,β in Proceedings of the IEEE Conference on Visualization (Visualization β92), pp. 70β76, Boston, Mass, USA, October 1992. [7] H. Hagen and S. Hahmann, βVisualization of curvature behaviour of free-form curves and surfaces,β Computer-Aided Design, vol. 27, no. 7, pp. 545β552, 1995. [8] H. Hagen, S. Hahmann, T. Schreiber, Y. Nakajima, B. Wordenweber, and P. Hollemann-Grundstedt, βSurface interrogation algorithms,β IEEE Computer Graphics and Applications, vol. 12, no. 5, pp. 53β60, 1992.
6 [9] J. Hoschek, Linien-Geometrie, BI, Wissensehaffs, Zurich, Switzerland, 1971. [10] K. Strubecker, Differentialgeometrie III, De Gruyter, Berlin, Germany, 1959. [11] J. Yu, X. Yin, X. Gu, L. McMillan, and S. Gortler, βFocal Surfaces of discrete geometry,β in Eurographics Symposium on Geometry Processing, 2007.
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