Abstract

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 F1 and F2 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 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 earth’s surface, and so forth. Focal surfaces are also used as a surface interrogation tool to analyse the “quality” 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 and of the surface in are associated with surfaces at a constant distance from the edge of regression on that formed along directions of lying in planes and , respectively.

2. Surfaces at a Constant Distance from the Edge of Regression on a Surface

Definition 1. Let and be two surfaces in Euclidean space and let be a unit normal vector and let be tangent space at point of surface and let be orthonormal bases of Take a unit vector , where are constant and If there is a function defined bywhere , then the surface is called the surface at a constant distance from the edge of regression on the surface

Here, if , then and so and are parallel surfaces. Now, we represent parametrization of surfaces at a constant distance from the edge of regression on . Let be a parametrization of , so we can write thatIn case is a basis of , then we can write that , where are, respectively, partial derivatives of according to and . Since , a parametric representation of isThus, it is obtained thatand if we get , then we have

Calculation of and gives us thatHere, are calculated as in [1]. We choose curvature lines instead of parameter curves of and let and be arc length of these curvature lines. Thus, the following equations are obtained:From (6) and (7), we findand is a basis of . If we denote by unit normal vector of , then iswhere are principal curvatures of the surface Ifwe can writeHere, in case of and since and are not linearly independent, is not a regular surface. We will not consider this case [1].

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].

We represent surfaces parametrically as vector-valued functions . Given a set of unit vectors , a line congruence is defined:where is called the signed distance between and [8]. Let be unit normal vector of the surface. If , then is a normal congruence. A focal surface is a special normal congruence. The parametric representation of the focal surfaces of is given bywhere 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, and [11].

The generalization of this classical concept leads to the generalized focal surfaces:where the scalar function depends on the principal curvatures and of the surface The real number is used as a scale factor. If the curvatures are very small you need a very large number to distinguish the two surfaces and 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].

4. The Relationship between Focal Surfaces and Surfaces at a Constant Distance from the Edge of Regression on a Surface

Theorem 2. Let surface be given by parametrical One considers all surfaces at a constant distance from the edge of regression on that formed along directions of lying in plane Normals of these surfaces at points corresponding to point generate a spatial family of line of which top is center of first principal curvature at

Proof. Surfaces at a constant distance from the edge of regression on that formed along directions of lying in plane are defined by These surfaces and their unit normal vectors are, respectively, denoted by and We will demonstrate that intersection point of lines which pass from the point and are in direction is
The normal vector of the surface at the point isHere, it is clear that is in plane Suppose that line passing from the point and being in direction is and a representative point of is ; then, the equation of isBesides, suppose that line passing from the point and being in direction is and a representative point of is ; then, equation of isWe find intersection point of these lines. Since it is studied in plane of vectors , the point can be taken as beginning point. If we arrange the lines and , then we findFrom here, it is clear that intersection point of and is So, intersection point of the lines and is the point in plane

Corollary 3. Directions of normals of all surfaces at a constant distance from the edge of regression on that formed along directions of lying in plane intersect at a single point. This point which is referred in Theorem 2 is on the focal surface

We know thatfrom definition of focal surfaces. Moreover, we can see easily the following equations from Figure 1:orThese equations show us that the focal surface of the surface can be stated by surfaces at a constant distance from the edge of regression on that formed along directions of lying in plane If or , then the focal surfaces of surfaces , and will be the same. This case has been expressed in following theorem.

Theorem 4. Focal surfaces of the surface and surfaces at a constant distance from the edge of regression on that formed along directions of lying in plane are the same if and only if first principal curvature of the surface is constant.

Proof. Suppose that focal surfaces of surfaces and formed along directions of lying in plane intersect; then, mentioned in (21) must beFirst principal curvature of formed along directions of lying in plane , that is, for , is calculated by Tarakci as [1]Besides, from Figure 1, since is distance between points of and lying in plane , we can writeIf we substitute (24) and (25) in (23) and make necessary arrangements, we obtainThus, we have The converse statement is trivial. Hence, our theorem is proved.

Theorem 5. Let surface be given by parametrical We consider all surfaces at a constant distance from the edge of regression on that formed along directions of lying in plane Normals of these surfaces at points corresponding to point generate a spatial family of line of which top is center of second principal curvature at

Proof. Surfaces at a constant distance from the edge of regression on that formed along directions of lying in plane are defined by These surfaces and their unit normal vectors are, respectively, denoted by and We will demonstrate that intersection point of lines which pass from the point and are in direction is
The normal vector of the surface at the point isHere, it is clear that is in plane Suppose that line passing from the point and being in direction is and a representative point of is ; then, equation of isBesides, suppose that line passing from the point of the surface and being in direction is and a representative point of is ; then, equation of isWe find intersection point of these two lines. Since it is studied in plane of vectors , the point can be taken as beginning point. If we arrange the lines and , then we findFrom here, it is clear that intersection point of and is So, intersection point of the lines and is the point in plane

Corollary 6. The point which is referred in Theorem 5 is on the focal surface

Similar to Figure 1, we can write equationsorThese equations show us that the focal surface of the surface can be stated by surfaces at a constant distance from the edge of regression on that formed along directions of lying in plane If or , then the focal surfaces of surfaces , and will be the same. This case has been expressed in following theorem.

Theorem 7. Focal surfaces of the surface and surfaces at a constant distance from the edge of regression on that formed along directions of lying in plane are the same if and only if second principal curvature of the surface is constant.

Proof. Suppose that focal surfaces of surfaces and formed along directions of lying in plane intersect; then, mentioned in (32) must beSecond principal curvature of formed along directions of lying in plane , that is, for , is calculated by Tarakci as [1]Besides, similar to Figure 1, since is the distance between points of and lying in plane , we can writeIf we substitute (35) and (36) in (34) and make necessary arrangements, we obtainThus, we have The converse statement is trivial. Hence, our theorem is proved.

Points on the surface 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.