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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.

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 [

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 [

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 [

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.

For a planar parametric curve

For a planar smooth parametric curve

Thus the offset direction depends on

Regarding the regularity of generalized offset curves, we have the following theorem.

If there exists

Let

When

When

We will prove that the generalized offset curve can be represented as the standard offset curve.

The generalized offset

Let

The inner product of vectors

Our goal is to get the values of

From (

When

where

When

When

When

where

Therefore, in any case there exist two functions

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.

Let

Therefore we can study the properties of the generalized offsets by using the similar approaches as what Farouki and Neff [

Evolute

We construct

Turning point, inflection, and vertex

Let

If

the turning point, inflection, and vertex on

the turning point on

the inflection on

Based upon the following relationships

Length and area

We can calculate the lengths

The area between

The area element between

At first, we compute

Topological property

The distance between a regular curve

The distance

We have the following:

Then one of the following propositions holds

Topological property of the curves.

According to Theorem

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

Based on the local coordinate system

Note that the symbols used in Section

For a regular parameter surface

For a regular smooth parametric surface

For a regular smooth parametric surface

Let

If

From the above explanation, we can easily prove Theorem

In most cases the local natural coordinate system

We will prove that the generalized offset surface can be represented as the standard offset surface.

The generalized offset

Let

In order to establish the above relationship, the following two conditions must be satisfied:

Therefore, in any case there are three functions

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.

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.

The area of an offset surface

The area of generalized offset surface

The volume between

The area element of

The volume element between

The volume between

After several computations, we get

(ii) Topological property

For a regular parameter surface

The distance

We have the following:

Then one of the following propositions holds

Topological property of the surfaces.

According to Theorem

Offset for curves and surfaces plays an important role in CAGD. It can be widely used in varieties of applications [

Generalized offset curves.

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

Thus the generalized offset

Generalized offset curve.

Flower cloth.

Generalized offset surfaces.

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

Generalized offset surface with the fixed offset direction.

Generalized offset surface with the variable offset direction.

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.

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.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Fujian Provincial Natural Science Foundation of China (Grant no. 2012J01013) and the Panjinlong Discipline Construction Foundation of Jimei University, China (Grant no. ZC2012022). The authors would like to thank the anonymous referees for their comments.