Properties of Generalized Offset Curves and Surfaces

Abstract

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.

1. 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 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.

2. Generalized Offset Curves

2.1. The Definition and Regularity of Generalized Offset Curves

For a planar parametric curve 𝐫=𝐫(𝑡), the well-known Frenet [14, 15] equations are given as follows: 𝑑𝐞𝑑𝑠=−𝑘𝐧,𝑑𝐧𝑑𝑠=𝑘𝐞, (1) where 𝑘 is the curvature, 𝑠 is the arc length such that 𝑑𝑠/𝑑𝑡=|𝐫^′|, and 𝐞 and 𝐧 are, respectively, the unit tangent vector and the normal vector at each point of the curve 𝐫(𝑡). For the convenience of representation, we define a unit vector 𝐳 such that 𝐫𝐞=^′𝐫^′,𝐳=𝐧×𝐞,𝐞=𝐳×𝐧,𝐧=𝐞×𝐳. (2) So we have 𝐫𝑘=^′×𝐫^′′⋅𝐳𝐫^′3. (3) 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(𝑡) with the regular parameter 𝑡∈[0,1], its generalized offset curve 𝐫_𝑜(𝑡) with the variable offset distance and offset direction is defined by 𝐫_𝑜(𝑡)=𝐫(𝑡)+𝑑₁(𝑡)𝐞+𝑑₂(𝑡)𝐧, (4) where 𝑑₁(𝑡) and 𝑑₂(𝑡) are the functions of 𝑡.

Thus the offset direction depends on 𝑑₂(𝑡)𝐧 and 𝑑₁(𝑡)𝐞, and the offset distance is 𝑑²₁(𝑡)+𝑑²₂(𝑡). From the above definition, the related parametric derivatives of 𝐫_𝑜 can be obtained by 𝐫^′_𝑜=𝐫^′+𝑑^′₁⋅𝐞+𝑑₁⋅𝐞^′+𝑑^′₂⋅𝐧+𝑑₂⋅𝐧^′=𝐫^′+𝑑^′₁+𝑑₂⋅𝑘⋅𝐫^′𝐞+𝑑^′₂−𝑑₁⋅𝑘⋅𝐫^′𝐧. (5) Let 𝛼=|𝐫^′|+𝑑^′₁+𝑑₂⋅𝑘⋅|𝐫^′| and let 𝛽=𝑑^′₂−𝑑₁⋅𝑘⋅|𝐫^′|; we get 𝐫^′_𝑜𝐫=𝛼⋅𝐞+𝛽⋅𝐧,_𝑜′′=𝛼^′⋅𝐞+𝛼⋅𝐞^′+𝛽^′⋅𝐧+𝛽⋅𝐧^′=𝛼^′+𝛽⋅𝑘⋅𝐫^′𝐞+𝛽^′−𝛼⋅𝑘⋅𝐫^′𝐧. (6) Therefore we have 𝐞_𝑜=𝐫^′_𝑜𝐫^′_𝑜=𝛼𝐞+𝛽𝐧𝛼²+𝛽²,𝐧_𝑜=𝐞_𝑜𝛼()()×𝐳=𝐞×𝐳+𝛽𝐧×𝐳𝛼²+𝛽²=𝛼𝐧−𝛽𝐞𝛼²+𝛽²,𝑘_𝑜=𝐫^′_𝑜×𝐫_𝑜^′′⋅𝐳𝐫^′_𝑜³=𝛼^′𝛽−𝛼𝛽^′+𝛼²+𝛽²⋅𝑘⋅𝐫^′𝛼²+𝛽^23/2. (7)

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

Theorem 2. If there exists 𝑡₀∈[0,1] to satisfy the equation 𝑑^′₁𝑡₀𝑑₁𝑡₀+𝑑^′₂𝑡₀𝑑₂𝑡₀+𝑑₁𝑡₀⋅𝐫^′𝑡₀=0, (8) and for any arbitrary small positive number 𝛿, −1𝑘=𝑑₂𝑡₀𝑑1+^′1𝑡₀𝐫^′𝑡₀,𝑤ℎ𝑒𝑛𝑑₂1≥𝛿>0,𝑑₁𝑡₀𝑑^′₂𝑡₀𝐫^′𝑡₀,𝑤ℎ𝑒𝑛𝑑₁≥𝛿>0, (9) then 𝐫_𝑜(𝑡₀) is a nonregular point of the generalized offset curve 𝐫_𝑜.

Proof. Let 𝐫_𝑜(𝑡₀) be any non-regular point of the offset curve 𝐫_𝑜(𝑡); then |𝐫^′_𝑜(𝑡₀)|=𝛼²+𝛽²=0. We get 𝛼=𝛽=0. Thus 𝑑₂⋅𝐫^′⋅𝑘+𝑑^′₁+𝐫^′=0,−𝑑₁⋅𝐫^′⋅𝑘+𝑑^′₂=0. (10)(11) We discuss the following two cases. (i)When |𝑑₂|≥𝛿>0, from (10) it follows that 1𝑘=−𝑑₂𝑑1+^′1𝐫^′. (12)Substituting it into (11), we have 𝑑^′₁𝑑₁+𝑑^′₂𝑑₂+𝑑₁𝐫^′=0. (13)(ii)When |𝑑₁|≥𝛿>0, from (11) it follows that 1𝑘=𝑑₁⋅𝑑^′₂𝐫^′. (14)Substituting it into (10), we also have 𝑑^′₁𝑑₁+𝑑^′₂𝑑₂+𝑑₁𝐫^′=0. (15) Hence we prove Theorem 2.

2.2. Relationship between Generalized and Standard Offset Curves

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

Theorem 3. The generalized offset 𝐫_𝑜=𝐫+𝑑₁𝐞+𝑑₂𝐧 can be represented as a standard offset: 𝐫_𝑜=𝐫₁+𝑑𝐧₁, where 𝐫₁ is a new planar smooth parametric curve, 𝑑 is constant, and 𝐧₁ is the unit normal vector of 𝐫₁.

Proof. Let 𝐫_𝑜=𝐫+𝑑₁𝐞+𝑑₂𝐧=(𝐫+𝐴𝐞+𝐵𝐧)+𝑑₁−𝐴𝐞+𝑑₂−𝐵𝐧≜𝐫₁+𝑑𝐧₁, (16) where 𝐴 and 𝐵 are the functions of 𝑡, 𝑑=𝑑₁−𝐴²+𝑑₂−𝐵²,𝐧₁=𝑑₁−𝐴𝑑𝑑𝐞+²−𝐵𝑑𝐫𝐧,_′1=𝐫^′+𝐴^′𝐞+𝐴𝐞^′+𝐵^′𝐧+𝐵𝐧^′=𝐫^′+𝐴^′+𝐵⋅𝑘⋅𝐫^′𝐞+𝐵^′−𝐴⋅𝑘⋅𝐫^′𝐧. (17) In order to establish the above relationship, the following two conditions must be satisfied. (i)The inner product of vectors 𝑑𝐧₁ and 𝐫^′₁ must be zero. That is, 𝑑₁−𝐴𝐫^′+𝐴^′+𝐵⋅𝑘⋅𝐫^′+𝑑₂−𝐵𝐵^′−𝐴⋅𝑘⋅𝐫^′=0. (18)(ii)(𝑑₁−𝐴)²+(𝑑₂−𝐵)² must be constant. That is, 𝑑₁−𝐴𝑑^′₁−𝐴^′+𝑑₂−𝐵𝑑^′₂−𝐵^′=0. (19)
Our goal is to get the values of 𝐴 and 𝐵 by solving the above differential equations.
From (18) and (19), we get 𝐫^′+𝑑₂⋅𝑘⋅𝐫^′+𝑑^′₁𝐴=𝑑₁⋅𝑘⋅𝐫^′−𝑑^′₂𝐵+𝑑₁𝐫^′+𝑑^′₁𝑑₁+𝑑^′₂𝑑₂. (20) Note that 𝑑₁𝐫^′+𝑑^′₁𝑑₁+𝑑^′₂𝑑₂=𝑑₁𝐫^′+𝑑^′₁𝑑₁+𝑑₂𝛽+𝑑₁⋅𝑘⋅𝐫^′=𝑑₁𝛼+𝑑₂𝛽, (21) and we have 𝛼𝑑₁−𝐴+𝛽𝑑₂−𝐵=0. (22) Analyzing the following four cases, we can get the values of 𝐴 and 𝐵. (1)When 𝛼=𝛽=0, 𝐴 and 𝐵 only need to satisfy (19). That is, 𝑑₁−𝐴²+𝑑₂−𝐵²=𝐶₁, (23) where 𝐶₁>0 is an arbitrary constant. Thus one of 𝐴 and 𝐵 can be determined arbitrarily. For instance, if 𝐴=𝑔(𝑡) is given, then 𝐵=𝑑₂±𝐶₁−𝑑₁−𝑔(𝑡)². (24)(2)When 𝛼≠0 and 𝛽=0, from (22) it follows that 𝐴=𝑑₁ and 𝐴^′=𝑑^′₁. Substituting it into (19), we have (𝑑₂−𝐵)(𝑑^′₂−𝐵^′)=0. Since 𝑑₂≠𝐵, then 𝑑^′₂=𝐵^′. Therefore 𝐵=𝑑₂(𝑡)+𝐶₂, where 𝐶₂ is an arbitrary nonzero constant.(3)When 𝛼=0 and 𝛽≠0, from (22) it follows that 𝐵=𝑑₂ and 𝐵^′=𝑑^′₂. Substituting it into (19), we have (𝑑₁−𝐴)(𝑑^′₁−𝐴^′)=0. Since 𝑑₁≠𝐴, then 𝑑^′₁=𝐴^′. Therefore 𝐴=𝑑₁(𝑡)+𝐶₃, where 𝐶₃ is an arbitrary nonzero constant.(4)When 𝛼≠0 and 𝛽≠0, by solving (19) and (22), we get 𝐴=𝑑₁−𝛽𝐶𝛼²+𝛽²,𝐵=𝑑₂+𝛼𝐶𝛼²+𝛽², (25) where 𝐶 is an arbitrary constant.
Therefore, in any case there exist two functions 𝐴 and 𝐵 to guarantee that the generalized offset 𝐫_𝑜=𝐫+𝑑₁𝐞+𝑑₂𝐧 can be expressed as the standard offset 𝐫_𝑜=𝐫₁+𝑑𝐧₁, where 𝐫₁𝑑=𝐫+𝐴𝐞+𝐵𝐧=𝐫+₁−𝛽𝐶𝛼²+𝛽²𝐞+𝑑₂+𝛼𝐶𝛼²+𝛽²𝐧,𝑑=𝑑₁−𝐴²+𝑑₂−𝐵²=|𝐶|,𝐧₁=𝛽𝛼²+𝛽²𝛼𝐞−𝛼²+𝛽²𝐧. (26) That is, we can find 𝐫₁(𝑡) so that 𝐫_𝑜(𝑡) becomes the standard offset of 𝐫₁(𝑡).
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.

2.3. Properties of Generalized Offset Curves

Let 𝜆=|𝑟^′|+𝐴^′+𝐵⋅𝑘⋅|𝑟^′|  and 𝜔=𝐵^′−𝐴⋅𝑘⋅|𝑟^′|. From the expression of standard offsets 𝐫_𝑜=𝐫₁+𝑑𝐧₁, where 𝐫₁=𝐫+𝐴𝐞+𝐵𝐧,𝐧₁=𝑑₁−𝐴𝑑𝑑𝐞+²−𝐵𝑑𝐧, (27)𝑑=const, and 𝑑₁,𝑑₂,𝐴,𝐵 are the functions of 𝑡, we have 𝐫^′_𝟏𝑘=𝜆𝐞+𝜔𝐧,₁=𝜆^′𝜔−𝜆𝜔^′+𝜆²+𝜔²⋅𝑘⋅𝐫^′𝜆²+𝜔^23/2,𝐞₁=𝐫^′₁𝐫^′₁=𝜆𝐞+𝜔𝐧𝜆²+𝜔²,𝐧₁=𝜆𝐧−𝜔𝐞𝜆²+𝜔². (28) Moreover 𝐫^′_𝑜=1+𝑑𝑘₁⋅𝜆²+𝜔²⋅𝐞₁,𝑘_𝑜=11+𝑑𝑘₁𝑘₁,𝐞_𝑜=sgn1+𝑑𝑘₁𝐞₁,𝐧_𝑜=sgn1+𝑑𝑘₁𝐧₁, (29) where 𝑘_𝑜 and 𝑘₁ are the curvatures of 𝐫_𝑜 and 𝐫₁ at each point, respectively. In the above case, there is another expression for the nonregular point of 𝐫_𝑜. Since 𝐫^′_𝑜=1+𝑑𝑘₁⋅𝐫^′₁=1+𝑑𝑘₁⋅𝜆²+𝜔², (30)𝐫(𝑡₀) is a nonregular point if 𝑘₁=−1/𝑑, and 𝜆,𝜔 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.(i)Evolute

We construct 𝐫_𝜀(𝑡)=𝐫₁(𝑡)−𝜌₁(𝑡)⋅𝐧₁, (31) where 𝜌₁≡𝜌₁(𝑡)=1/𝑘₁(𝑘₁≠0). At the nonregular point 𝐫(𝑡₀), it follows that 𝜌₁=1/𝑘₁=−𝑑; hence we also have 𝐫_𝑜(𝜏)=𝐫₁(𝜏)+𝑑𝐧₁=𝐫₁(𝜏)−𝜌₁𝐧₁=𝐫_𝜀(𝜏). (32) On the other hand, we have 𝐫_𝜀𝑑=𝐫−¹−𝐴1+𝑑𝑘₁𝑑𝑘₁𝑑𝐞−²−𝐵1+𝑑𝑘₁𝑑𝑘₁𝐧. (33) Moreover, from 𝐫_𝑜−1𝑘_𝑜𝐧_𝑜=𝐫₁−1𝑘₁𝐧₁, (34) we can get the relations as follows: 𝑑₁+𝛽𝑘_𝑜𝛼²+𝛽²𝜔=𝐴+𝑘₁𝜆²+𝜔²,𝑑₂−𝛼𝑘_𝑜𝛼²+𝛽²𝜆=𝐵−𝑘₁𝜆²+𝜔². (35)(ii)Turning point, inflection, and vertex

Let 𝐫(𝑡)=(𝑥(𝑡),𝑦(𝑡))^𝑇; then 𝐫(𝑡₀) is called a turning point [16] if 𝑥^′(𝑡₀)=0 and 𝑦^′(𝑡₀)≠0, or 𝑥^′(𝑡₀)≠0 and 𝑦^′(𝑡₀)=0, and 𝐫(𝑡₀) is called an inflection if 𝑘(𝑡₀)=0 and 𝐫(𝑡₀) are called a vertex if 𝑑𝑘(𝑡₀)/𝑑𝑠=0.

Theorem 4. If 𝑘₁≠−1/𝑑, and 𝜆,𝜔 are not both zero, then (1)the turning point, inflection, and vertex on 𝐫_𝑜(𝑡) are, respectively, in one-to-one correspondence to those on 𝐫₁(𝑡);(2)the turning point on 𝐫_𝑜(𝑡) is in one-to-one correspondence to that on 𝐫(𝑡) as 𝜔=0;(3)the inflection on 𝐫_𝑜(𝑡) is in one-to-one correspondence to that on 𝐫(𝑡) as 𝜆^′𝜔=𝜆𝜔^′.

Proof. Based upon the following relationships 𝐞_𝑜=sgn1+𝑑𝑘₁⋅𝜆𝐞+𝜔𝐧𝜆²+𝜔²,𝑘_𝑜=11+𝑑𝑘₁⋅𝜆^′𝜔−𝜆𝜔^′+𝜆²+𝜔²⋅𝑘⋅𝑟^′𝜆²+𝜔^23/2,𝑑𝑘_𝑜𝑑𝑠_𝑜=1+𝑑𝑘₁⁻³⋅𝑑𝑘₁𝑑𝑠₁, (36) we can easily prove Theorem 4.

(iii)Length and area

We can calculate the lengths 𝑙₁ and 𝑙_𝑜 of the curves 𝐫₁ and 𝐫_𝑜, respectively. Since 𝑑𝑙₁=|𝐫^′_𝟏|𝑑𝑡, then 𝑙₁=¹₀𝐫^′₁𝑑𝑡=¹₀𝜆²+𝜔²𝑙𝑑𝑡,_𝑜=¹₀𝐫^′_𝑜𝑑𝑡=¹₀1+𝑑𝑘₁⋅𝐫^′₁𝑑𝑡. (37)

The area between 𝐫(𝑡) and its generalized offset 𝐫_𝑜(𝑡) is denoted by 𝑀 (Figure 1). 𝑑𝑆₁ and 𝑑𝑆₁ are the area elements. See Figure 4.

Figure 1 

The area element between 𝐫(𝑡) and 𝐫𝑜(𝑡).

At first, we compute 𝑑𝑆₁=𝐫_𝑜(𝑡)−𝐫(𝑡)×[𝐫(𝑡+Δ𝑡)−𝐫(𝑡)]2,𝑑𝑆₂=𝐫_𝑜(𝑡)−𝐫_𝑜(𝑡+Δ𝑡)×𝐫(𝑡+Δ𝑡)−𝐫_𝑜(𝑡+Δ𝑡)2. (38) Since 𝐫(𝑡+Δ𝑡)=𝐫(𝑡)+Δ𝑡⋅𝐫^′(𝑡)+𝑂Δ𝑡²,𝐫_𝑜(𝑡+Δ𝑡)=𝐫_𝑜(𝑡)+Δ𝑡⋅𝐫^′_𝑜(𝑡)+𝑂Δ𝑡², (39) it follows that 𝑑𝑆₁=𝑑₁𝐞+𝑑₂𝐧×Δ𝑡⋅𝐫^′(𝑡)2=12𝑑₂⋅𝐫^′𝑑𝑡,𝑑𝑆₂=Δ𝑡⋅𝐫^′_𝑜(𝑡)×𝐫(𝑡)−𝐫_𝑜(𝑡)+Δ𝑡𝐫^′(𝑡)−𝐫^′_𝑜(𝑡)2=12⋅Δ𝑡⋅𝑑^′₁𝑑₂−𝑑^′₂𝑑₁+𝑑₂⋅𝐫^′+𝑑²₁+𝑑²₂⋅𝑘⋅𝐫^′. (40) Therefore 𝑑𝑀=𝑑𝑆₁+𝑑𝑆₂, and 𝑀=¹₀1𝑑𝑀=2¹₀𝑑₂⋅𝐫^′+𝑑𝑡₁₀𝑑^′₁𝑑₂−𝑑^′₂𝑑₁+𝑑₂⋅𝐫^′+𝑑²₁+𝑑²₂⋅𝑘⋅𝐫^′𝑑𝑡. (41)(iv)Topological property

The distance between a regular curve 𝐫₁=𝐫₁(𝑡),𝑡∈[0,1] and a point 𝑄 in the same plane is defined as follows: 𝛿𝑄,𝐫₁=inf_𝑡∈[0,1]𝑄−𝐫₁(𝑡). (42) For the standard offset 𝐫_𝑜=𝐫₁+𝑑𝐧₁, we have the following theorm.

Theorem 5. The distance 𝛿(𝐫_𝑜(𝜏),𝐫) between the point 𝐫_𝑜(𝜏) of the generalized offset and the curve 𝐫=𝐫(𝑡),𝑡∈[0,1] satisfies one of the following conditions: 𝛿𝐫_𝑜(𝜏),𝐫=|𝑑|+𝐴²+𝐵²,𝜏∈𝑖_𝑘,𝑖_𝑘+1,𝛿𝐫_𝑜(𝜏),𝐫<|𝑑|+𝐴²+𝐵²,𝜏∈𝑖_𝑘,𝑖_𝑘+1,𝑘=0,…,𝑁,𝑁∈𝑍⁺. (43) Each of the open intervals (𝑖_𝑘,𝑖_𝑘+1) and 𝑘=0,…,𝑁. is delineated by the self-intersections.

Proof. We have the following:(1)𝛿(𝐫_𝑜(𝜏),𝐫₁)≤|𝑑|,𝜏∈[0,1];(2)𝑖₀=0, 𝑖_𝑁+1=1, 𝑖₁,…,𝑖_𝑁∈(0,1), 𝑁∈𝑍⁺ are the self-intersections of 𝐫_𝑜.
Then one of the following propositions holds 𝛿𝐫_𝑜(𝜏),𝐫₁≡|𝑑|,𝜏∈𝑖_𝑘,𝑖_𝑘+1,𝛿𝐫_𝑜(𝜏),𝐫₁<|𝑑|,𝜏∈𝑖_𝑘,𝑖_𝑘+1,𝑘=0,…,𝑁 (44) For the generalized offset 𝐫₀(𝑡), we have 𝐫₁(𝑡)−𝐫(𝑡)=𝐴²+𝐵²,𝑡∈[0,1]. (45) Considering 𝜏∈(𝑖_𝑘,𝑖_𝑘+1), let 𝑝∈𝐫₁ such that 𝐫_𝑜(𝜏)−𝑝=𝛿𝐫_𝑜(𝜏),𝐫₁, (46) and 𝑞∈𝐫=𝐫(𝑡), 𝑡∈[0,1] is the point with the same parameter 𝑡 of 𝑝 on 𝐫; then we have 𝑝−𝑞=𝐴²+𝐵². (47) Thus 𝛿𝐫_𝑜(𝜏),𝐫≤𝐫_𝑜(𝜏)−𝐫≤𝛿𝐫_𝑜(𝜏),𝐫₁+𝐴²+𝐵². (48) Therefore we prove Theorem 5, which is shown in Figure 2.

Figure 2 

Topological property of the curves.

According to Theorem 5, each of the segments {𝐫_𝑜(𝑡),𝑡∈(𝑖_𝑘,𝑖_𝑘+1)} of the offset curve among its self-intersections should either be retained or rejected in its entirety when forming the trimmed offset.

2.4. 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(𝑡) are 𝑑𝐞𝑑𝑠=−𝑘𝐧,𝑑𝐧𝑑𝑠=𝑘𝐞+𝜏𝐳,𝑑𝐳𝑑𝑠=−𝜏𝐧, (49) where 𝜏 is the torsion, and we have 𝐫𝜏=^′×𝐫^′′⋅𝐫^′′′𝐫^′×𝐫^′′2. (50)

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.

3. Generalized Offset Surfaces

3.1. 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(𝑢,𝑣)=(𝑥(𝑢,𝑣),𝑦(𝑢,𝑣),𝑧(𝑢,𝑣)), its two unit tangent vectors in the directions of 𝑢 and 𝑣 and its unit normal vector are given by [17] 𝐞₁=𝐫_𝑢(𝑢,𝑣)𝐫_𝑢(𝑢,𝑣),𝐞₂=𝐫_𝑣(𝑢,𝑣)𝐫_𝑣(,𝐫𝑢,𝑣)𝐧=^𝑢(𝑢,𝑣)×𝐫_𝑣(𝑢,𝑣)𝐫_𝑢(𝑢,𝑣)×𝐫_𝑣(,𝑢,𝑣) (51) where 𝐫_𝑢(𝑢,𝑣) and 𝐫_𝑣(𝑢,𝑣) are the corresponding partial derivatives of 𝐫(𝑢,𝑣) about parameters 𝑢 and 𝑣. (𝐞₁,𝐞₂,𝐧) forms a right-handed system. Based on the local natural coordinate system (𝐞₁,𝐞₂,𝐧) of surface 𝐫(𝑢,𝑣), a generalized surface offset 𝐫^𝑜(𝑢,𝑣) with the variable offset direction and the variable offset distance can be defined.

Definition 6. For a regular smooth parametric surface 𝐫(𝑢,𝑣),(𝑢,𝑣)∈[0,1]×[0,1], the generalized offset surface 𝐫^𝑜(𝑢,𝑣) with the variable offset distance and offset direction is defined by 𝐫^𝑜(𝑢,𝑣)=𝐫(𝑢,𝑣)+𝑑₁(𝑢,𝑣)𝐞₁+𝑑₂(𝑢,𝑣)𝐞₂+𝑑₃(𝑢,𝑣)𝐧, (52) where 𝑑₁(𝑢,𝑣), 𝑑₂(𝑢,𝑣), and 𝑑₃(𝑢,𝑣) are the functions of the variables 𝑢 and 𝑣. The offset direction and the offset distance are determined by 𝑑₁𝐞₁, 𝑑₂𝐞₂, and 𝑑₃𝐧.

For a regular smooth parametric surface 𝐫(𝑢,𝑣), the well-known first and second fundamental quantities and the Gauss curvature [14] are given as follows: 𝐸=𝐫²_𝑢=𝐴²₁,𝐺=𝐫²_𝑣=𝐴²₂,𝐹=𝐫_𝑢⋅𝐫_𝑣,𝐷=𝑛⋅𝐫_𝑢𝑢,𝐷^′=𝑛⋅𝐫_𝑢𝑣,𝐷^′′=𝑛⋅𝐫_𝑣𝑣,𝑘=𝐷𝐷^′′−𝐷^′2𝐸𝐺−𝐹². (53) Let 𝐞₁×𝐧=𝐞^′₁,𝐞₂×𝐧=𝐞^′₂, and the angle between 𝐞₁ and 𝐞₂ is 𝜃; then 𝐞₁×𝐞₂=𝐧sin𝜃,𝐞^′₁×𝐞^′₂𝐞=𝐧sin𝜃,_′1⋅𝐞₂=−sin𝜃,𝐞^′₂⋅𝐞₁=sin𝜃. (54) Thus the related parametric partial derivatives of generalized offset surface 𝐫^𝑜(𝑢,𝑣) can be obtained by 𝐫^𝑜_𝑢(𝑢,𝑣)=𝐫_𝑢+𝑑_1𝑢𝐞₁+𝑑₁𝐞_1𝑢+𝑑_2𝑢𝐞₂+𝑑₂𝐞_2𝑢+𝑑_3𝑢𝐧+𝑑₃𝐧_𝑢=𝐴₁+𝑑_1𝑢+𝑑₂𝐴_1𝑣𝐴₂−𝑑₃𝐴₁𝑅₁𝐞₁+𝑑_2𝑢−𝑑₁𝐴_1𝑣𝐴₂𝐞₂+𝑑_3𝑢+𝑑₁𝐴₁𝑅₁𝐧≜𝐵₁𝐞₁+𝐵₂𝐞₂+𝐵₃𝐧,𝐫^𝑜_𝑣(𝑢,𝑣)=𝐫_𝑣+𝑑_1𝑣𝐞₁+𝑑₁𝐞_1𝑣+𝑑_2𝑣𝐞₂+𝑑₂𝐞_2𝑣+𝑑_3𝑣𝐧+𝑑₃𝐧_𝑣=𝑑_1𝑣−𝑑₂𝐴_2𝑢𝐴₁𝐞₁+𝐴₂+𝑑₁𝐴_2𝑢𝐴₁+𝑑_2𝑣−𝑑₃𝐴₂𝑅₂𝐞₂+𝑑_3𝑣+𝑑₂𝐴₂𝑅₂𝐧≜𝐶₁𝐞₁+𝐶₂𝐞₂+𝐶₃𝐧, (55)(56) where 𝐫_𝑢,𝑑_1𝑢,𝑑_2𝑢,𝐴_2𝑢,𝐞_1𝑢,𝐞_2𝑢,𝐧_𝑢 are the corresponding partial derivatives of 𝐫,𝑑₁,𝑑₂,𝐴₂,𝐞₁,𝐞₂,𝐧 with respect to 𝑢 and 𝐫_𝑣,𝑑_1𝑣,𝑑_2𝑣,𝐴_1𝑣,𝐞_1𝑣,𝐞_2𝑣,𝐧_𝑣 are the corresponding partial derivatives of 𝐫,𝑑₁,𝑑₂,𝐴₁,𝐞₁,𝐞₂,𝐧 with respect to 𝑣. 𝑅₁,𝑅₂ are the radii of principal curvature and 𝑘 is the Gauss curvature. We can get the following equations: 𝐫^𝑜_𝑢𝑢(𝑢,𝑣)=𝐵_1𝑢𝐞₁+𝐵₁𝐞_1𝑢+𝐵_2𝑢𝐞₂+𝐵₂𝐞_2𝑢+𝐵_3𝑢𝐧+𝐵₃𝐧_𝑢=𝐵_1𝑢+𝐵₂𝐴_1𝑣𝐴₂−𝐵₃𝐴₁𝑅₁𝐞₁+−𝐵₁𝐴_1𝑣𝐴₂+𝐵_2𝑢+𝐵₁𝐴₁𝑅₁+𝐵_3𝑢𝐫𝐧,_𝑜𝑢𝑣(𝑢,𝑣)=𝐫^𝑜_𝑣𝑢(𝑢,𝑣)=𝐵_1𝑣𝐞₁+𝐵₁𝐞_1𝑣+𝐵_2𝑣𝐞₂+𝐵₂𝐞_2𝑣+𝐵_3𝑣𝐧+𝐵₃𝐧_𝑣=𝐵_1𝑣−𝐵₂𝐴_2𝑢𝐴₁𝐞₁+𝐵₁𝐴_2𝑢𝐴₁+𝐵_2𝑣−𝐵₃𝐴₂𝑅₂𝐞₂+𝐵₂𝐴₂𝑅₂+𝐵_3𝑣𝐫𝐧,_𝑜𝑣𝑣(𝑢,𝑣)=𝐶_1𝑣𝐞₁+𝐶₁𝐞_1𝑣+𝐶_2𝑣𝐞₂+𝐶₂𝐞_2𝑣+𝐶_3𝑣𝐧+𝐶₃𝐧_𝑣=𝐶_1𝑣−𝐶₂𝐴_2𝑢𝐴₁𝐞₁+𝐶₁𝐴_2𝑢𝐴₁+𝐶_2𝑣−𝐶₃𝐴₂𝑅₂𝐞₂+𝐶₂𝐴₂𝑅₂+𝐶_3𝑣𝐸𝐧,_𝑜=𝐫_𝑢^𝑜2=𝐵²₁+𝐵²₂+𝐵²₃+2𝐵₁𝐵₂𝐺cos𝜃,_𝑜=𝐫^𝑜_𝑣²=𝐶²₁+𝐶²₂+𝐶²₃+2𝐶₁𝐶₂𝐹cos𝜃,_𝑜=𝐫^𝑜_𝑢×𝐫^𝑜_𝑣=𝐵₁𝐶₁+𝐵₂𝐶₂+𝐵₃𝐶₃+𝐵₁𝐶₂+𝐵₂𝐶₁cos𝜃. (57) Thus the unit tangent vectors and unit normal vector of surface offsets 𝐫^𝑜(𝑢,𝑣) are given as follows: 𝑒^𝑜₁=𝐫^𝑜_𝑢(𝑢,𝑣)𝐫^𝑜_𝑢(=𝐵𝑢,𝑣)¹𝐞₁+𝐵₂𝐞₂+𝐵₃𝐧𝐸_𝑜,𝐞^𝑜₂=𝐫^𝑜_𝑣(𝑢,𝑣)𝐫^𝑜_𝑣(=𝐶𝑢,𝑣)¹𝐞₁+𝐶₂𝐞₂+𝐶₃𝐧𝐺_𝑜,𝐧^𝑜=𝐫^𝑜_𝑢×𝐫^𝑜_𝑣𝐫^𝑜_𝑢×𝐫^𝑜_𝑣=𝐫^𝑜_𝑢×𝐫^𝑜_𝑣𝐸_𝑜𝐺_𝑜−𝐹²_𝑜=𝐵₁𝐶₃−𝐵₃𝐶₁𝐸_𝑜𝐺_𝑜−𝐹²_𝑜𝐞^′₁+𝐵₂𝐶₃−𝐵₃𝐶₂𝐸_𝑜𝐺_𝑜−𝐹²_𝑜𝐞^′₂+𝐵₁𝐶₂−𝐵₂𝐶₁sin𝜃𝐸_𝑜𝐺_𝑜−𝐹_𝑜²𝐧≜𝑀₁𝐞₁+𝑀₂𝐞₂+𝑀₃𝐷𝐧,_𝑜=𝐧^𝑜𝐫^𝑜_𝑢𝑢=𝑀₁𝐵_1𝑢+𝐵₂𝐴_1𝑣𝐴₂−𝐵₃𝐴₁𝑅₁+𝑀₂−𝐵₁𝐴_1𝑣𝐴₂+𝐵_2𝑢+𝑀₃𝐵₁𝐴₁𝑅₁+𝐵_3𝑢+𝑀₁−𝐵₁𝐴_1𝑣𝐴₂+𝐵_2𝑢+𝑀₂𝐵_1𝑢+𝐵₂𝐴_1𝑣𝐴₂−𝐵₃𝐴₁𝑅₁𝐷cos𝜃,_′𝑜=𝐧^𝑜𝐫^𝑜_𝑢𝑣=𝑀₁𝐵_1𝑣−𝐵₂𝐴_2𝑢𝐴₁+𝑀₂𝐵₁𝐴_2𝑢𝐴₁+𝐵_2𝑣−𝐵₃𝐴₂𝑅₂+𝑀₃𝐵₂𝐴₂𝑅₂+𝐵_3𝑣+𝑀₁𝐵₁𝐴_2𝑢𝐴₁+𝐵_2𝑣−𝐵₃𝐴₂𝑅₂+𝑀₂𝐵_1𝑣−𝐵₂𝐴_2𝑢𝐴₁𝐷cos𝜃,_𝑜′′=𝐧^𝑜𝐫^𝑜_𝑣𝑣=𝑀₁𝐶_1𝑣−𝐶₂𝐴_2𝑢𝐴₁+𝑀₂𝐶₁𝐴_2𝑢𝐴₁+𝐶_2𝑣−𝐶₃𝐴₂𝑅₂+𝑀₃𝐶₂𝐴₂𝑅₂+𝐶_3𝑣+𝑀₁𝐶₁𝐴_2𝑢𝐴₁+𝐶_2𝑣−𝐶₃𝐴₂𝑅₂+𝑀₂𝐶_1𝑣−𝐶₂𝐴_2𝑢𝐴₁𝑘cos𝜃,_𝑜=𝐷_𝑜𝐷_𝑜^′′−𝐷_𝑜^′2𝐸_𝑜𝐺_𝑜−𝐹²_𝑜, (58) where 𝐸_𝑜,𝐺_𝑜,𝐹_𝑜,𝐷_𝑜,𝐷^′_𝑜,𝐷_𝑜^′′, and 𝑘_𝑜 are, respectively, the basic quantities and Gauss curvature of surface offset 𝐫^𝑜(𝑢,𝑣). Moreover, we can get the tangent plane and normal line at one particular point of surface 𝐫^𝑜(𝑢,𝑣).

Let 𝐫^𝑜(𝑢₀,𝑣₀) be a nonregular point of 𝐫^𝑜(𝑢,𝑣); then 𝐫^𝑜_𝑢𝑢₀,𝑣₀×𝐫^𝑜_𝑣𝑢₀,𝑣₀=0. (59) Note that 𝐫^𝑜_𝑢×𝐫^𝑜_𝑣=𝐵₁𝐶₃−𝐵₃𝐶₁𝐞^′₁+𝐵₂𝐶₃−𝐵₃𝐶₂𝐞^′₂+𝐵₁𝐶₂−𝐵₂𝐶₁𝐧sin𝜃≜𝜌₁𝐞^′₁+𝜌₂𝐞^′₂+𝜌₃𝐧sin𝜃. (60) Regarding the regularity of generalized offset surfaces, we have the following theorem.

Theorem 7. If 𝜌²₁𝑢₀,𝑣₀+𝜌²₂𝑢₀,𝑣₀+𝜌²₃𝑢₀,𝑣₀+2𝜌₁𝑢₀,𝑣₀𝜌₂𝑢₀,𝑣₀cos𝜃^1/2=0, (61) then 𝐫^𝑜(𝑢₀,𝑣₀) is a nonregular point of 𝐫^𝑜(𝑢,𝑣).

From the above explanation, we can easily prove Theorem 7.

In most cases the local natural coordinate system (𝐞₁,𝐞₂,𝐧) at each point of a regular parameter surface 𝐫(𝑢,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(𝑢,𝑣) firstly. According to the theorem [14], for every point at the regular parameter surface 𝐫(𝑢,𝑣), we can find a neighbourhood and a new parameter system (𝑢,𝑣̃) to make the new local natural coordinate system (𝐞_𝑢,𝐞_̃𝑣,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 (𝐞₁,𝐞₂,𝐧) of a regular surface 𝐫(𝑢,𝑣) is the orthonormal coordinate system.

3.2. Relationship between Generalized and Standard Offset Surfaces

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

Theorem 8. The generalized offset 𝐫^𝑜=𝐫+𝑑₁𝐞₁+𝑑₂𝐞₂+𝑑₃𝐧 can be represented as a standard offset: 𝐫_𝑜=𝐫₁+𝑑𝐧₁, where 𝐫₁ is a new regular smooth parametric surface, 𝑑 is constant, and 𝐧₁ is the unit normal vector of 𝐫₁.

Proof. Let 𝐫^𝑜=𝐫+𝑑₁𝐞₁+𝑑₂𝐞₂+𝑑₃𝐧=𝑟+𝑀𝐞₁+𝑁𝐞₂+𝑃𝐧+𝑑₁−𝑀𝐞₁+𝑑₂−𝑁𝐞₂+𝑑₃−𝑃𝐧≜𝐫₁+𝑑𝐧₁, (62) where 𝑀, 𝑁, and 𝑃 are the functions of 𝑢 and 𝑣, 𝑑=𝑑₁−𝑀²+𝑑₂−𝑁²+𝑑₃−𝑃²,𝐧₁=𝑑₁−𝑀𝑑𝐞₁+𝑑₂−𝑁𝑑𝐞₂+𝑑₃−𝑃𝑑. (63) The parametric partial derivatives of surface 𝐫₁(𝑢,𝑣) are 𝐫_1𝑢(𝑢,𝑣)=𝐫_𝑢+𝑀_𝑢𝐞₁+𝑀𝐞_1𝑢+𝑁_𝑢𝐞₂+𝑁𝐞_2𝑢+𝑃_𝑢𝐧+𝑃𝐧_𝑢=𝐴₁+𝑀_𝑢𝐴+𝑁^1𝑣𝐴₂𝐴−𝑃¹𝑅₁𝐞₁+𝑁_𝑢𝐴−𝑀^1𝑣𝐴₂𝐞₂+𝑃_𝑢𝐴+𝑀¹𝑅₁𝐧,𝐫_1𝑣(𝑢,𝑣)=𝐫_𝑣+𝑀_𝑣𝐞₁+𝑀𝐞_1𝑣+𝑁_𝑣𝐞₂+𝑁𝐞_2𝑣+𝑃_𝑣𝐧+𝑃𝐧_𝑣=𝑀_𝑣𝐴−𝑁^2𝑢𝐴₁𝐞₁+𝐴₂𝐴+𝑀^2𝑢𝐴₁+𝑁_𝑣𝐴−𝑃²𝑅₂𝐞₂+𝑃_𝑣𝐴+𝑁²𝑅₂𝐧, (64)(65) where 𝑀_𝑢,𝑀_𝑣,𝑁_𝑢,𝑁_𝑣,𝑃_𝑢,𝑃_𝑣 are the corresponding partial derivatives.
In order to establish the above relationship, the following two conditions must be satisfied: (i)𝐧₁ must be the unit normal vector of 𝐫₁. That is, 𝐴₁+𝑀_𝑢𝐴+𝑁^1𝑣𝐴₂𝐴−𝑃¹𝑅₁𝑑₁+𝑁−𝑀_𝑢𝐴−𝑀^1𝑣𝐴₂𝑑₂+𝑃−𝑁_𝑢𝐴+𝑀¹𝑅₁𝑑₃−𝑃=0,𝑀_𝑣𝐴−𝑁^2𝑢𝐴₁𝑑₁+𝐴−𝑀₂𝐴+𝑀^2𝑢𝐴₁+𝑁_𝑣𝐴−𝑃²𝑅₂𝑑₂+𝑃−𝑁_𝑣𝐴+𝑁²𝑅₂𝑑₃−𝑃=0. (66)(67)(ii)𝑑 is constant. That is, 𝑑₁−𝑀²+𝑑₂−𝑁²+𝑑₃−𝑃²=const. (68)It follows that 𝑑₁−𝑀𝑑_1𝑢−𝑀_𝑢+𝑑₂−𝑁𝑑_2𝑢−𝑁_𝑢+𝑑₃−𝑃𝑑_3𝑢−𝑃_𝑢=0,𝑑₁−𝑀𝑑_1𝑣−𝑀_𝑣+𝑑₂−𝑁𝑑_2𝑣−𝑁_𝑣+𝑑₃−𝑃𝑑_3𝑣−𝑃_𝑣=0. (69)(70) Our goal is to get the values of 𝑀,𝑁, and 𝑃 by solving the above differential equations. From (66) and (69), we get −𝐴₁−𝐴_1𝑣𝑑₂𝐴₂−𝑑_1𝑢+𝐴₁𝑑₃𝑅₁𝐴𝑀+^1𝑣𝑑₁𝐴₂−𝑑_2𝑢𝑁+−𝐴₁𝑑₁𝑅₁−𝑑_3𝑢𝑃+𝐴₁𝑑₁+𝑑_1𝑢𝑑₁+𝑑_2𝑢𝑑₂+𝑑_3𝑢𝑑₃≜𝛼₁𝑀+𝛽₁𝑁+𝛾₁𝑃+𝜔₁=0. (71) From (67) and (70), we get 𝐴_2𝑢𝑑₂𝐴₁−𝑑_1𝑣−𝐴𝑀+^2𝑢𝑑₁𝐴₁−𝐴₂−𝑑_2𝑣+𝐴₂𝑑₃𝑅₂𝑁+−𝐴₂𝑑₂𝑅₂−𝑑_3𝑣𝑃+𝐴₂𝑑₂+𝑑_1𝑣𝑑₁+𝑑_2𝑣𝑑₂+𝑑_3𝑣𝑑₃≜𝛼₂𝑀+𝛽₂𝑁+𝛾₂𝑃+𝜔₂=0. (72) Let 𝛼₁𝛽₂−𝛼₂𝛽₁≜𝛼,𝛽₂𝛾₁−𝛽₁𝛾₂≜𝛼𝛽,₂𝛾₁−𝛼₁𝛾₂≜𝛾. (73) From (71) and (72), we get 𝛼𝑑₁−𝑀+𝛽𝑑₃−−𝑃=0,𝛼𝑑₂−𝑁+𝛾𝑑₃−𝑃=0. (74) By solving (69), (70), and (74), we get 𝑀=𝑑₁−𝛽𝐶𝛼²+𝛽²+𝛾²,𝑁=𝑑₂+𝛾𝐶𝛼²+𝛽²+𝛾²,𝑃=𝑑₃+𝛼𝐶𝛼²+𝛽²+𝛾², (75) where 𝐶 is an arbitrary constant, 𝛼, 𝛽, and 𝛾 can not all be zero. Thus 𝑑=𝑑₁−𝑀²+𝑑₂−𝑁²+𝑑₃−𝑃²=𝐧𝐶,₁=𝛽𝛼²+𝛽²+𝛾²𝐞₁−𝛾𝛼²+𝛽²+𝛾²𝐞₂−𝛼𝛼²+𝛽²+𝛾²𝐧. (76) When 𝛼=𝛽=𝛾=0, 𝑀,𝑁, and 𝑃 only need to satisfy (69) and (70).
Therefore, in any case there are three functions 𝑀,𝑁, and 𝑃 to guarantee the generalized offset 𝐫^𝑜=𝐫+𝑑₁𝐞₁+𝑑₂𝐞₂+𝑑₃𝐧 can be expressed as the standard offset 𝐫^𝑜(𝑢,𝑣)=𝐫₁(𝑢,𝑣)+𝑑𝐧₁(𝑢,𝑣),𝑑=const. That is, we can find 𝐫₁(𝑢,𝑣) so that 𝐫^𝑜(𝑢,𝑣) becomes the standard offset of 𝐫₁(𝑢,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.

3.3. 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 𝐫^𝑜(𝑢,𝑣)∶(𝑢,𝑣)∈[0,1]×[0,1] (77) is denoted by 𝑆. We consider the area element 𝑑𝑆, which is shown in Figure 3. Consider the following: 𝑑𝑆=𝐫^𝑜_𝑢×𝐫^𝑜_𝑣𝑑𝑢𝑑𝑣. (78) Since 𝐫^𝑜_𝑢×𝐫^𝑜_𝑣²=𝐫_𝑢^𝑜2𝐫_𝑣^𝑜2−𝐫^𝑜_𝑢𝐫^𝑜_𝑣²=𝐸_𝑜𝐺_𝑜−𝐹²_𝑜, (79) then 𝑆=_[0,1]×[0,1]𝐸_𝑜𝐺_𝑜−𝐹²_𝑜𝑑𝑢𝑑𝑣. (80)(ii)The volume between 𝐫(𝑢,𝑣) and 𝐫^𝑜(𝑢,𝑣)

Figure 3 

The area element of 𝐫𝑜(𝑢,𝑣).

Figure 4 

The volume element between 𝐫(𝑢,𝑣) and 𝐫𝑜(𝑢,𝑣).

The volume between 𝐫(𝑢,𝑣) and its generalized offset 𝐫^𝑜(𝑢,𝑣) is denoted by 𝑉. We consider a volume element 𝑑𝑉, which is shown in Figure 4. The volume element 𝑑𝑉 can be divided into five subvolumes: 𝑑𝑉=𝑑𝑉₁+𝑑𝑉₂+⋯+𝑑𝑉₅, (81) where 𝑑𝑉₁=𝐫(𝑢,𝑣),𝐫^𝑜(𝑢,𝑣),𝐫(𝑢+Δ𝑢,𝑣),𝐫(𝑢,𝑣+Δ𝑣),𝑑𝑉₂=𝐫^𝑜(𝑢+Δ𝑢,𝑣),𝐫^𝑜()(),𝐫𝑢,𝑣,𝐫𝑢+Δ𝑢,𝑣_𝑜(𝑢+Δ𝑢,𝑣+Δ𝑣),𝑑𝑉₃=[𝐫()(),𝐫𝑢+Δ𝑢,𝑣+Δ𝑣,𝐫𝑢+Δ𝑢,𝑣_𝑜(𝑢+Δ𝑢,𝑣+Δ𝑣),𝐫(𝑢,𝑣+Δ𝑣),𝑑𝑉₄=𝐫^𝑜(𝑢,𝑣+Δ𝑣),𝐫^𝑜(),𝐫()𝑢,𝑣𝑢,𝑣+Δ𝑣,𝐫^𝑜(𝑢+Δ𝑢,𝑣+Δ𝑣),𝑑𝑉₅=𝐫(𝑢,𝑣+Δ𝑣),𝐫^𝑜()(),𝐫𝑢,𝑣,𝐫𝑢+Δ𝑢,𝑣_𝑜(𝑢+Δ𝑢,𝑣+Δ𝑣), (82) and the symbol [𝑏1,𝑏2,𝑏3,𝑏4] denotes the volume of tetrahedron with four vertices 𝑏1,𝑏2,𝑏3, and 𝑏4.

After several computations, we get 𝑑𝑉₁=16𝐴₁𝐴₂𝑑₃𝑑𝑢𝑑𝑣≜𝑤₁𝑑𝑢𝑑𝑣,𝑑𝑉₂=16𝑑₁𝐵₃𝐶₂−𝐵₂𝐶₃+𝑑₂𝐵₁𝐶₃−𝐵₃𝐶₁+𝑑₃𝐵₂𝐶₁−𝐵₁𝐶₂𝑑𝑢𝑑𝑣≜𝑤₂𝑑𝑢𝑑𝑣,𝑑𝑉₃=16𝐴₁𝐴₂𝑑₃𝑑𝑢𝑑𝑣≜𝑤₁𝑑𝑢𝑑𝑣,𝑑𝑉₄=16𝑑₁𝐵₂𝐶₃−𝐵₃𝐶₂+𝑑₂𝐵₃𝐶₁−𝐵₁𝐶₃+𝑑₃𝐵₁𝐶₂−𝐵₂𝐶₁𝑑𝑢𝑑𝑣≜𝑤₂𝑑𝑢𝑑𝑣,𝑑𝑉₅=16𝐴₁𝐶₃𝑑₂−𝐴₁𝐶₂𝑑₃+𝐴₂𝐵₃𝑑₁−𝐴₂𝐵₁𝑑₃𝑑𝑢𝑑𝑣≜𝑤₅𝑑𝑢𝑑𝑣. (83) Therefore, we have 𝑉=¹₀𝑑𝑉₁+𝑑𝑉₂+⋯+𝑑𝑉₅=_[]×[]0,10,12𝑤₁+2𝑤₂+𝑤₅𝑑𝑢𝑑𝑣. (84)

(ii) Topological property

For a regular parameter surface 𝐿₁=𝐫₁(𝑢,𝑣)∶(𝑢,𝑣)∈[0,1]×[0,1], (85) the distance between a point 𝑄 and the surface 𝐿₁ is defined as follows: 𝛿𝑄,𝐿₁=inf_{()𝑢,𝑣∈[0,1]×[0,1]}𝑄−𝐫₁(𝑢,𝑣). (86) For the standard offset 𝐫^𝑜=𝐫₁+𝑑𝐧₁,𝑑=const, we have the following theorem.

Theorem 9. The distance 𝛿(𝐫^𝑜(𝜏,𝜂),𝐶) between the point 𝐫^𝑜(𝜏,𝜂) of the generalized offset and the surface 𝐿={𝐫(𝑢,𝑣)∶(𝑢,𝑣)∈[0,1]×[0,1]} satisfies one of the following conditions: 𝛿𝐫^𝑜𝜏,𝜂,𝐿=|𝑑|+𝑀²+𝑁²+𝑃²,𝛿𝐫^𝑜𝜏,𝜂,𝐿<|𝑑|+𝑀²+𝑁²+𝑃²,𝜏,𝜂∈𝑖_𝑘,𝑖_𝑘+1×𝑗_𝑘′,𝑗_𝑘′+1,𝑘,𝑘^′=0,1,…,𝑁,𝑁∈𝑍⁺. (87) Each of the open fields (𝑖_𝑘,𝑖_𝑘+1)×(𝑗_𝑘′,𝑗_𝑘′+1),𝑘,𝑘^′=0,1,…,𝑁 is delineated by the self-intersections.

Proof. We have the following:(1)𝛿(𝐫^𝑜(𝜏,𝜂),𝐿₁)≤|𝑑|,(𝜏,𝜂)∈[0,1]×[0,1];(2)(𝑖₀,𝑗₀)=(0,0),(𝑖₀,𝑗_𝑁+1)=(0,1), (𝑖_𝑁+1,𝑗₀)=(1,0), (𝑖_𝑁+1,𝑗_𝑁+1)=(1,1), 𝑖₁,…,𝑖_𝑁,𝑗₁,…,𝑗_𝑁∈(0,1), 𝑁∈𝑍⁺ are the self-intersections of 𝐫^𝑜.
Then one of the following propositions holds 𝛿𝐫^𝑜𝜏,𝜂,𝐿₁=|𝑑|,𝜏,𝜂∈𝑖_𝑘,𝑖_𝑘+1×𝑗_𝑘′,𝑗_𝑘′+1,𝛿𝐫^𝑜𝜏,𝜂,𝐿₁<|𝑑|,𝜏,𝜂∈𝑖_𝑘,𝑖_𝑘+1×𝑗_𝑘′,𝑗_𝑘′+1,𝑘,𝑘^′=0,1,…,𝑁. (88) For the generalized offset 𝐫^𝑜=𝐫+𝑑₁𝐞₁+𝑑₂𝐞₂+𝑑₃𝐧=𝐫₁+𝑑𝐧₁,𝐫₁=𝐫+𝑀𝐞₁+𝑁𝐞₂+𝑃𝐧, (89) we have 𝐫₁(𝑢,𝑣)−𝐫(𝑢,𝑣)=𝑀𝐞₁+𝑁𝐞₂𝐫+𝑃𝐧,₁(𝑢,𝑣)−𝐫(𝑢,𝑣)=𝑀²+𝑁²+𝑃²,(𝑢,𝑣)∈[0,1]×[0,1]. (90) Considering (𝜏,𝜂)∈(𝑖_𝑘,𝑖_𝑘+1)×(𝑗_𝑘′,𝑗_𝑘′+1), let 𝑝∈𝐿₁ such that 𝐫^𝑜𝜏,𝜂−𝑝=𝛿𝐫^𝑜𝜏,𝜂,𝐿₁, (91) and 𝑞∈𝐿={𝐫(𝑢,𝑣)∶(𝑢,𝑣)∈[0,1]×[0,1]} is the point with the same parameter (𝑢,𝑣) of 𝑝 on 𝐿; then we have 𝑝−𝑞=𝑀²+𝑁²+𝑃². (92) Thus 𝛿𝐫^𝑜𝜏,𝜂,𝐿≤𝐫^𝑜𝐫𝜏,𝜂−𝐿≤𝛿^𝑜𝜏,𝜂,𝐿₁+𝑀²+𝑁²+𝑃². (93) Therefore we prove the above theorem, which is illustrated in Figure 5.

Figure 5 

Topological property of the surfaces.

According to Theorem 9, each of the segments {𝐫⁰(𝑠,𝑡)∶(𝑠,𝑡)∈(𝑖_𝑘,𝑖_𝑘+1)×(𝑗_𝑘′,𝑗_𝑘′+1)} of the offset surface among its self-intersections should either be retained or rejected in its entirety when forming the trimmed offset.

4. 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.

Example 10. 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 𝐫(𝑡)={cos(𝑡),sin(𝑡)},𝑡∈[0,2𝜋]. (94) Then the unit tangent vector at each point of the curve 𝐫(𝑡) is 𝐞(𝑡)={−sin(𝑡),cos(𝑡)}, and the unit normal vector at each point of the curve 𝐫(𝑡) is 𝐧(𝑡)={cos(𝑡),sin(𝑡)}. According to the definition of generalized offset curve, we get the new curve 𝐫_𝑜(𝑡)=𝐫(𝑡)+𝑑₁(𝑡)𝐞(𝑡)+𝑑₂𝐧(𝑡), (95) where we take 𝑑₁=|sin(2𝑡)| and 𝑑₁=|8⋅cos(2𝑡)|.

Thus the generalized offset 𝐫_𝑜(𝑡) is 𝐫_𝑜(𝑡)=𝐫₁(𝑡)+𝑑𝐧₁(𝑡)={(𝑡)(𝑡)⋅|((𝑡)⋅|(,(𝑡)(𝑡)⋅|((𝑡)⋅|(cos−sinsin2𝑡)|+cos8⋅cos2𝑡)|sin+cossin2𝑡)|+sin8⋅cos2𝑡)|} (96) 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.

Figure 6 

Generalized offset curve.

Figure 7 

Flower cloth.

Example 11. 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 8 and 9.

Figure 8 

Generalized offset surface with the fixed offset direction.

Figure 9 

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.

5. 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.

Conflict of Interests

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

Acknowledgments

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.