Properties of Generalized Offset Curves and Surfaces

This paper proposes a definition of generalized offsets for curves and surfaces, which have the variable offset distance and direction, by using the local coordinate system. Based on this definition, some analytic properties and theorems of generalized offsets are put forward. The regularity and the topological property of generalized offsets are simply given by representing the generalized offset as the standard offset. Some examples are provided as well to show the applications of generalized offsets. The conclusions in this paper can be taken as the foundation for further study on extending the standard offset.


Introduction
Offset curves/surfaces, also called parallel curves/surfaces, are defined as locus of the points which are at constant distance along the normal vector from the generator curves/surfaces.In the field of computer aided geometric design (CAGD), offset curves and surfaces have got considerable attention since they are widely used in various practical applications such as tolerance analysis, geometric optics, and robot path-planning [1,2].The study on the offset of curve and surface has been one of the hotpots in CAGD [3].
In some of the engineering applications, we need to extend the concept of standard offset, which has constant distance along the normal vector from the generator such as geodesic offset where constant distance is replaced by geodesic distance (distance measured from a curve on a surface along the geodesic curve drawn orthogonally to the curve) and generalized offset where offset direction is not necessarily along the normal direction.Generalized offset surfaces were first introduced by Brechner [3] and have been extended further, from the differential geometric as well as algebraic points of view, by Pottmann [4].Arrondo et al. [5] presented a formula for computing the genus of irreducible generalized offset curves to projective irreducible plane curves with only affine ordinary singularities over an algebraically closed field.Lin and Rokne [6] defined the variable-radius generalized offset parametric curves and surfaces.The envelopes of these variable offset parametric curves and surfaces are computed explicitly.J. R. Sendra and J. Sendra [7] presented a complete algebraic analysis of degeneration and the existence of simple and special components of generalized offsets to irreducible hypersurfaces over algebraically closed fields of characteristic zero.A notion of a similarity surface offset was introduced by Georgiev [8] and applied to different constructions of rational generalized offsets.There are also some literatures on generalized offsets which primarily focus on solving some concrete problems [4,9,10].But the general definition, properties, and complete analytic conclusions for generalized offsets have not yet been presented.
Some algebraic properties on standard offsets are known to classical geometers.The study of algebraic and geometric properties on offsets has been an active research area since it arises in practical applications.Farouki and Neff [11,12] analyzed the basic geometric and topological properties of plane offset curves and provided algorithm to compute the implicit equation.We expect that generalized offsets would have more interesting properties and practical applications.In this paper, a strict definition of generalized offsets, which have the variable offset distance and direction, is given.The 2 Journal of Applied Mathematics offset distance and direction are determined by the local coordinate systems.Though using the local coordinate systems to define a curve is not new [13], the definition of offset curves and surfaces by the local coordinate systems has never been presented before.According to this definition, similar to the standard offsets, we are concerned with the enumeration of certain fundamental geometric and algebraic characteristics for generalized offsets.The relationships between generalized and standard offsets are discussed.
This paper studies the generalized offsets of curves and surfaces in two primary segments.In each segment, we firstly give the definition and regularity of generalized offsets which can be explicitly expressed by the local coordinate systems, secondly we analyze the relationship between generalized and standard offsets, then we discuss some major properties of generalized offsets, and finally some examples are given to illustrate the applications of generalized offsets.The results in this paper will be the foundation for further study on extending the standard offset.Most analytic and topological properties of the generalized offset are addressed in this paper, which provide a series of fundamental conclusions for further study in the related field of generalized offsets.

The Definition and Regularity of Generalized Offset Curves.
For a planar parametric curve r = r(), the well-known Frenet [14,15] equations are given as follows: where  is the curvature,  is the arc length such that / = |r  |, and e and n are, respectively, the unit tangent vector and the normal vector at each point of the curve r().For the convenience of representation, we define a unit vector z such that So we have Based on the local coordinate system (e, n) in the plane, a generalized curve offset with the variable offset distance and offset direction can be defined.
Definition 1.For a planar smooth parametric curve r = r() with the regular parameter  ∈ [0, 1], its generalized offset curve r  () with the variable offset distance and offset direction is defined by where  1 () and  2 () are the functions of .
Thus the offset direction depends on  2 (t)n and  1 ()e, and the offset distance is √ 2 1 () +  , Regarding the regularity of generalized offset curves, we have the following theorem.Theorem 2. If there exists  0 ∈ [0, 1] to satisfy the equation and for any arbitrary small positive number , then r  ( 0 ) is a nonregular point of the generalized offset curve r  .

Relationship between Generalized and Standard Offset
Curves.We will prove that the generalized offset curve can be represented as the standard offset curve.
In order to establish the above relationship, the following two conditions must be satisfied.
(i) The inner product of vectors n 1 and r  1 must be zero.That is, Our goal is to get the values of  and  by solving the above differential equations.From ( 18) and ( 19), we get Note that and we have Analyzing the following four cases, we can get the values of  and .
Therefore, in any case there exist two functions  and  to guarantee that the generalized offset r  = r +  1 e +  2 n can be expressed as the standard offset r  = r 1 + n 1 , where ) n, That is, we can find r 1 () so that r  () becomes the standard offset of r 1 ().So far we have proved that the generalized offset can be represented as the standard offset.Based on the current results of standard offsets, we can continue the research on the regularity and integral properties of generalized offset curves.This theorem also helps us to obtain the simpler and conciser expressions.The following paragraph explains the details.

Properties of Generalized Offset Curves
From the expression of standard offsets r  = r 1 + n 1 , where = const, and  1 ,  2 , ,  are the functions of , we have Moreover where   and  1 are the curvatures of r  and r 1 at each point, respectively.In the above case, there is another expression for the nonregular point of r  .Since r( 0 ) is a nonregular point if  1 = −1/, and ,  are not both zero.
Therefore we can study the properties of the generalized offsets by using the similar approaches as what Farouki and Neff [12] had done for the standard offsets.
Proof.Based upon the following relationships we can easily prove Theorem 4.

(iii) Length and area
We can calculate the lengths  1 and   of the curves r 1 and r  , respectively.Since  1 = |r  1 |, then The area between r() and its generalized offset r  () is denoted by  (Figure 1). 1 and  1 are the area elements.See Figure 4.
At first, we compute Since it follows that (iv) Topological property The distance between a regular curve r 1 = r 1 (),  ∈ [0, 1] and a point  in the same plane is defined as follows: For the standard offset r  = r 1 + n 1 , we have the following theorm.
Theorem 5.The distance (r  (), r) between the point r  () of the generalized offset and the curve r = r(),  ∈ [0, 1] satisfies one of the following conditions: Each of the open intervals (  ,  +1 ) and  = 0, . . ., . is delineated by the self-intersections.
Then one of the following propositions holds For the generalized offset r 0 (), we have Therefore we prove Theorem 5, which is shown in Figure 2.
According to Theorem 5, each of the segments {r  (),  ∈ (  ,  +1 )} of the offset curve among its self-intersections should either be retained or rejected in its entirety when forming the trimmed offset.

Remark.
The curves in three-dimensional space can also be discussed analogously.As we know, a curve is not planar if and only if the torsion of the curve is not zero.Therefore, different from a planar parametric curve, the Frenet equations for a spatial parametric curve r() are where  is the torsion, and we have Based on the local coordinate system (e, n, z) in the space, a generalized curve offset with the variable offset distance and offset direction can be defined.The properties of generalized offset curves can be given similarly.Since a torsion item is added in the Frenet equations, the calculations may become more complicated and the conclusions may not be expressed simply.

The Definition and Regularity of Generalized Offset
Surfaces.Note that the symbols used in Section 3 are all redefined.
For a regular parameter surface r(, V) = ((, V), (, V), (, V)), its two unit tangent vectors in the directions of  and V and its unit normal vector are given by [17] where r  (, V) and r V (, V) are the corresponding partial derivatives of r(, V) about parameters  and V. (e 1 , e 2 , n) forms a right-handed system.Based on the local natural coordinate system (e 1 , e 2 , n) of surface r(, V), a generalized surface offset r  (, V) with the variable offset direction and the variable offset distance can be defined.
Definition 6.For a regular smooth parametric surface r(, V), (, V) ∈ [0, 1] × [0, 1], the generalized offset surface r  (, V) with the variable offset distance and offset direction is defined by where  1 (, V),  2 (, V), and  3 (, V) are the functions of the variables  and V.The offset direction and the offset distance are determined by  1 e 1 ,  2 e 2 , and  3 n.
For a regular smooth parametric surface r(, V), the wellknown first and second fundamental quantities and the Gauss curvature [14] are given as follows: Let e 1 × n = e  1 , e 2 × n = e  2 , and the angle between e 1 and e 2 is ; then Thus the related parametric partial derivatives of generalized offset surface r  (, V) can be obtained by Thus the unit tangent vectors and unit normal vector of surface offsets r  (, V) are given as follows: where   ,   ,   ,   ,    ,    , and   are, respectively, the basic quantities and Gauss curvature of surface offset r  (, V).Moreover, we can get the tangent plane and normal line at one particular point of surface r  (, V).
Let r  ( 0 , V 0 ) be a nonregular point of r  (, V); then Note that Regarding the regularity of generalized offset surfaces, we have the following theorem.
From the above explanation, we can easily prove Theorem 7.
In most cases the local natural coordinate system (e 1 , e 2 , n) at each point of a regular parameter surface r(, V) is not the orthonormal coordinate system.In order to discuss the relationship between generalized and standard offset surfaces, we need to do some parameter transformation of surface r(, V) firstly.According to the theorem [14], for every point at the regular parameter surface r(, V), we can find a neighbourhood and a new parameter system (ũ, Ṽ) to make the new local natural coordinate system (e ũ, e ṽ, n) be the orthonormal coordinate system.This theorem guarantees the existence of orthonormal parameter curve net on the regular parameter surface.Therefore for any regular parameter surface, we can make the local natural coordinate system be orthonormal by this means.In the following two paragraphs we suppose that the local natural coordinate system (e 1 , e 2 , n) of a regular surface r(, V) is the orthonormal coordinate system.

Relationship between Generalized and Standard Offset
Surfaces.We will prove that the generalized offset surface can be represented as the standard offset surface.
where , , and  are the functions of  and V, The parametric partial derivatives of surface r 1 (, V) are where   ,  V ,   ,  V ,   ,  V are the corresponding partial derivatives.
In order to establish the above relationship, the following two conditions must be satisfied: (i) n 1 must be the unit normal vector of r 1 .That is, (ii)  is constant.That is, It follows that Our goal is to get the values of , , and  by solving the above differential equations.From (66) and (69), we get From ( 67) and (70), we get Let From ( 71) and (72), we get By solving (69), (70), and (74), we get where  is an arbitrary constant, , , and  can not all be zero.Thus When  =  =  = 0, , , and  only need to satisfy (69) and (70).Therefore, in any case there are three functions , , and  to guarantee the generalized offset r  = r+ 1 e 1 + 2 e 2 + 3 n can be expressed as the standard offset r  (, V) = r 1 (, V) + n 1 (, V),  = const.That is, we can find r 1 (, V) so that r  (, V) becomes the standard offset of r 1 (, V).
So far we have proved that the generalized offset can be transformed to the standard offset.Based on the current results of standard offsets, we can continue the research on the properties of generalized offset surfaces.This theorem also helps us to obtain the simpler and conciser expressions.The following paragraph explains the details.

Properties of Generalized Offset Surfaces.
To study the properties of the generalized offset surfaces, we can use the similar approaches which have been introduced in offset curves.Here we only give the integral and topological properties of generalized offset surfaces.

(i) The area of an offset surface
The area of generalized offset surface is denoted by .We consider the area element , which is shown in Figure 3. Consider the following: then For the generalized offset we have (93) Therefore we prove the above theorem, which is illustrated in Figure 5.

Applications
Offset for curves and surfaces plays an important role in CAGD.It can be widely used in varieties of applications [18].Generalized offsets are the extending of standard offsets, which have more flexible properties.By using programming language VC++, some simple examples are given in the following to show how to use generalized offset technique.
In computer aided plane flower design, let a circle parameter curve be the original curve.We choose the union normal direction or deviate a certain angle from the normal direction as the offset direction, and some trigonometric functions as the offset distance.For example, the original circle parameter curve is which is shown in Figure 6.By this means, we can get the plane flowers, which have all kinds of beautiful shapes.Pasting them on cloth after machining, we obtain the following pattern shown in Figure 7.
The generalized offset surfaces can be widely used in 3D modeling, and more complicated 3D shapes could be defined by using dynamic offset direction and distance.Let a sphere be the original surface.Similar to the plane flower design, we choose the union normal direction or the direction deviating a fixed angle from the normal direction at each point of the sphere as the offset direction, respectively, and some trigonometric functions as the offset distance.We can get the following two models shown in Figures 8 and 9.
We have introduced some applications of generalized offset in 2D graphic design and 3D modeling.Generalized offsets are more flexible since they provide the variable direction and distance.The generalized offset technique is useful especially when the shape design is derived from an  existing graph or a 3D object.Moreover the mathematical expressions of offset curves and surfaces can be simplified by using the concepts and theorems given in this paper.

Conclusions
In this paper a strict definition of generalized offsets for curves and surfaces is given.By proving that the generalized offset can be represented as the standard offset, we get a series of conclusions on the properties of generalized offsets.The conclusions given in this paper cover most of the fundamental properties of generalized offsets and can be taken as the foundation for further study on generalized offsets and their application.

Figure 2 :
Figure 2: Topological property of the curves.

Figure 5 :
Figure 5: Topological property of the surfaces.

Figure 8 :
Figure 8: Generalized offset surface with the fixed offset direction.

Figure 9 :
Figure 9: Generalized offset surface with the variable offset direction.
22 ().From the above definition, the related parametric derivatives of r  can be obtained by r   = r  ⋅ e +  ⋅ e  +   ⋅ n +  ⋅ n

Theorem 3 .
The generalized offset r  = r +  1 e +  2 n can be represented as a standard offset: r  = r 1 + n 1 , where r 1 is a new planar smooth parametric curve,  is constant, and n 1 is the unit normal vector of r 1 .