# Properties of Generalized Offset Curves and Surfaces

Authors: Xuejuan Chen; Qun Lin
Journal: Journal of Applied Mathematics
 (2014)

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## 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.


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## Body

## 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 curver=r(t), the well-known Frenet [14, 15] equations are given as follows:
(1)deds=-kn,dnds=ke,
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
(2)e=r′|r′|,z=n×e,e=z×n,n=e×z.
So we have
(3)k=(r′×r′′)·z|r′|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 curver=r(t) with the regular parameter t∈[0,1], its generalized offset curve ro(t) with the variable offset distance and offset direction is defined by
(4)ro(t)=r(t)+d1(t)e+d2(t)n,
where d1(t) and d2(t) are the functions of t.Thus the offset direction depends ond2(t)n and d1(t)e, and the offset distance is d12(t)+d22(t). From the above definition, the related parametric derivatives of ro can be obtained by
(5)ro′=r′+d1′·e+d1·e′+d2′·n+d2·n′=(|r′|+d1′+d2·k·|r′|)e+(d2′-d1·k·|r′|)n.
Let α=|r′|+d1′+d2·k·|r′| and let β=d2′-d1·k·|r′|; we get
(6)ro′=α·e+β·n,ro′′=α′·e+α·e′+β′·n+β·n′=(α′+β·k·|r′|)e+(β′-α·k·|r′|)n.
Therefore we have
(7)eo=ro′|ro′|=αe+βnα2+β2,no=eo×z=α(e×z)+β(n×z)α2+β2=αn-βeα2+β2,ko=(ro′×ro′′)·z|ro′|3=(α′β-αβ′)+(α2+β2)·k·|r′|(α2+β2)3/2.Regarding the regularity of generalized offset curves, we have the following theorem.Theorem 2.
If there existst0∈[0,1] to satisfy the equation
(8)d1′(t0)d1(t0)+d2′(t0)d2(t0)+d1(t0)·|r′(t0)|=0,
and for any arbitrary small positive number δ,
(9)k={-1d2(t0)[1+d1′(t0)|r′(t0)|],when|d2|≥δ>0,1d1(t0)[d2′(t0)|r′(t0)|],when|d1|≥δ>0,
then ro(t0) is a nonregular point of the generalized offset curve ro.Proof.
Letro(t0) be any non-regular point of the offset curve ro(t); then |ro′(t0)|=α2+β2=0. We get α=β=0. Thus
(10)d2·|r′|·k+d1′+|r′|=0,(11)-d1·|r′|·k+d2′=0.
We discuss the following two cases. (i)
When|d2|≥δ>0, from (10) it follows that
(12)k=-1d2(1+d1′|r′|).Substituting it into (11), we have
(13)d1′d1+d2′d2+d1|r′|=0.(ii)
When|d1|≥δ>0, from (11) it follows that
(14)k=1d1·d2′|r′|.Substituting it into (10), we also have
(15)d1′d1+d2′d2+d1|r′|=0.
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 offsetro=r+d1e+d2n can be represented as a standard offset: ro=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(16)ro=r+d1e+d2n=(r+Ae+Bn)+(d1-A)e+(d2-B)n≜r1+dn1,
where A and B are the functions of t,
(17)d=(d1-A)2+(d2-B)2,n1=d1-Ade+d2-Bdn,r1′=r′+A′e+Ae′+B′n+Bn′=(|r′|+A′+B·k·|r′|)e+(B′-A·k·|r′|)n.
In order to establish the above relationship, the following two conditions must be satisfied. (i)
The inner product of vectorsdn1 and r1′ must be zero. That is,
(18)(d1-A)(|r′|+A′+B·k·|r′|)+(d2-B)(B′-A·k·|r′|)=0.(ii)
(d1-A)2+(d2-B)2 must be constant. That is,
(19)(d1-A)(d1′-A′)+(d2-B)(d2′-B′)=0.
Our goal is to get the values ofA and B by solving the above differential equations.
From (18) and (19), we get
(20)(|r′|+d2·k·|r′|+d1′)A=(d1·k·|r′|-d2′)B+(d1|r′|+d1′d1+d2′d2).
Note that
(21)d1|r′|+d1′d1+d2′d2=d1|r′|+d1′d1+d2(β+d1·k·|r′|)=d1α+d2β,
and we have
(22)α(d1-A)+β(d2-B)=0.
Analyzing the following four cases, we can get the values of A and B. (1)
Whenα=β=0, A and B only need to satisfy (19). That is,
(23)(d1-A)2+(d2-B)2=C1,whereC1>0 is an arbitrary constant. Thus one of A and B can be determined arbitrarily. For instance, if A=g(t) is given, then
(24)B=d2±C1-(d1-g(t))2.(2)
Whenα≠0 and β=0, from (22) it follows that A=d1 and A′=d1′. Substituting it into (19), we have (d2-B)(d2′-B′)=0. Since d2≠B, then d2′=B′. Therefore B=d2(t)+C2, where C2 is an arbitrary nonzero constant.(3)
Whenα=0 and β≠0, from (22) it follows that B=d2 and B′=d2′. Substituting it into (19), we have (d1-A)(d1′-A′)=0. Since d1≠A, then d1′=A′. Therefore A=d1(t)+C3, where C3 is an arbitrary nonzero constant.(4)
Whenα≠0 and β≠0, by solving (19) and (22), we get
(25)A=d1-βCα2+β2,B=d2+αCα2+β2,whereC is an arbitrary constant.
Therefore, in any case there exist two functionsA and B to guarantee that the generalized offset ro=r+d1e+d2n can be expressed as the standard offset ro=r1+dn1, where
(26)r1=r+Ae+Bn=r+(d1-βCα2+β2)e+(d2+αCα2+β2)n,d=(d1-A)2+(d2-B)2=|C|,n1=βα2+β2e-αα2+β2n.
That is, we can find r1(t) so that ro(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

Letλ=|r′|+A′+B·k·|r′|  and ω=B′-A·k·|r′|. From the expression of standard offsets ro=r1+dn1, where
(27)r1=r+Ae+Bn,n1=d1-Ade+d2-Bdn,d=const, and d1,d2,A,B are the functions of t, we have
(28)r1′=λe+ωn,k1=(λ′ω-λω′)+(λ2+ω2)·k·|r′|(λ2+ω2)3/2,e1=r1′|r1′|=λe+ωnλ2+ω2,n1=λn-ωeλ2+ω2.
Moreover
(29)ro′=(1+dk1)·λ2+ω2·e1,ko=1|1+dk1|k1,eo=sgn⁡(1+dk1)e1,no=sgn⁡(1+dk1)n1,
where ko and k1 are the curvatures of ro and r1 at each point, respectively. In the above case, there is another expression for the nonregular point of ro. Since
(30)|ro′|=|1+dk1|·|r1′|=|1+dk1|·λ2+ω2,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)
EvoluteWe construct(31)rε(t)=r1(t)-ρ1(t)·n1,
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
(32)ro(τ)=r1(τ)+dn1=r1(τ)-ρ1n1=rε(τ).
On the other hand, we have
(33)rε=r-d1-A(1+dk1)dk1e-d2-B(1+dk1)dk1n.
Moreover, from
(34)ro-1kono=r1-1k1n1,
we can get the relations as follows:
(35)d1+βkoα2+β2=A+ωk1λ2+ω2,d2-αkoα2+β2=B-λk1λ2+ω2.(ii)
Turning point, inflection, and vertexLetr(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.
Ifk1≠-1/d, and λ,ω are not both zero, then (1)
the turning point, inflection, and vertex onro(t) are, respectively, in one-to-one correspondence to those on r1(t);(2)
the turning point onro(t) is in one-to-one correspondence to that on r(t) as ω=0;(3)
the inflection onro(t) is in one-to-one correspondence to that on r(t) as λ′ω=λω′.Proof.
Based upon the following relationships(36)eo=sgn⁡(1+dk1)·λe+ωnλ2+ω2,ko=1|1+dk1|·(λ′ω-λω′)+(λ2+ω2)·k·|r′|(λ2+ω2)3/2,dkodso=(1+dk1)-3·dk1ds1,
we can easily prove Theorem 4.(iii)
Length and areaWe can calculate the lengthsl1 and lo of the curves r1 and ro, respectively. Since dl1=|r1′|dt, then
(37)l1=∫01‍|r1′|dt=∫01‍λ2+ω2dt,lo=∫01‍|ro′|dt=∫01‍|1+dk1|·|r1′|dt.The area betweenr(t) and its generalized offset ro(t) is denoted by M (Figure 1). dS1 and dS1 are the area elements. See Figure 4.Figure 1
The area element betweenr(t) and ro(t).At first, we compute(38)dS1=|[ro(t)-r(t)]×[r(t+Δt)-r(t)]|2,dS2=|[ro(t)-ro(t+Δt)]×[r(t+Δt)-ro(t+Δt)]|2.
Since
(39)r(t+Δt)=r(t)+Δt·r′(t)+O(Δt2),ro(t+Δt)=ro(t)+Δt·ro′(t)+O(Δt2),
it follows that
(40)dS1=|[d1e+d2n]×[Δt·r′(t)]|2=12|d2|·|r′|dt,dS2=|[Δt·ro′(t)]×[r(t)-ro(t)+Δt(r′(t)-ro′(t))]|2=12·Δt·|d1′d2-d2′d1+d2·|r′|+(d12+d22)·k·|r′||.
Therefore dM=dS1+dS2, and
(41)M=∫01‍dM=12[∫01‍|d2|·|r′|dtcccccccccccc+∫01|d1′d2-d2′d1+d2·|r′|ccccccccccccccccccc+(d12+d22)·k·|r′||dt∫01‍].(iv)
Topological propertyThe distance between a regular curver1=r1(t),t∈[0,1] and a point Q in the same plane is defined as follows:
(42)δ(Q,r1)=inf⁡t∈[0,1]|Q-r1(t)|.
For the standard offset ro=r1+dn1, we have the following theorm.Theorem 5.
The distanceδ(ro(τ),r) between the point ro(τ) of the generalized offset and the curve r=r(t),t∈[0,1] satisfies one of the following conditions:
(43)δ(ro(τ),r)=|d|+A2+B2,τ∈(ik,ik+1),δ(ro(τ),r)<|d|+A2+B2,τ∈(ik,ik+1),cccccccccccccccccccck=0,…,N,N∈Z+.
Each of the open intervals (ik,ik+1) and k=0,…,N. is delineated by the self-intersections.Proof.
We have the following:(1)
δ(ro(τ),r1)≤|d|,τ∈[0,1];(2)
i0=0, iN+1=1, i1,…,iN∈(0,1), N∈Z+ are the self-intersections of ro.
Then one of the following propositions holds(44)δ(ro(τ),r1)≡|d|,τ∈(ik,ik+1),δ(ro(τ),r1)<|d|,τ∈(ik,ik+1),ccccccccccccccccccccck=0,…,N
For the generalized offset r0(t), we have
(45)|r1(t)-r(t)|=A2+B2,t∈[0,1].
Considering τ∈(ik,ik+1), let p∈r1 such that
(46)|ro(τ)-p|=δ(ro(τ),r1),
and q∈r=r(t), t∈[0,1] is the point with the same parameter t of p on r; then we have
(47)|p-q|=A2+B2.
Thus
(48)δ(ro(τ),r)≤|ro(τ)-r|≤δ(ro(τ),r1)+A2+B2.
Therefore we prove Theorem 5, which is shown in Figure 2.Figure 2
Topological property of the curves.According to Theorem5, each of the segments {ro(t),t∈(ik,ik+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 curver(t) are
(49)deds=-kn,dnds=ke+τz,dzds=-τn,
where τ is the torsion, and we have
(50)τ=(r′×r′′)·r′′′|r′×r′′|2.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.


## 2.1. The Definition and Regularity of Generalized Offset Curves

For a planar parametric curver=r(t), the well-known Frenet [14, 15] equations are given as follows:
(1)deds=-kn,dnds=ke,
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
(2)e=r′|r′|,z=n×e,e=z×n,n=e×z.
So we have
(3)k=(r′×r′′)·z|r′|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 curver=r(t) with the regular parameter t∈[0,1], its generalized offset curve ro(t) with the variable offset distance and offset direction is defined by
(4)ro(t)=r(t)+d1(t)e+d2(t)n,
where d1(t) and d2(t) are the functions of t.Thus the offset direction depends ond2(t)n and d1(t)e, and the offset distance is d12(t)+d22(t). From the above definition, the related parametric derivatives of ro can be obtained by
(5)ro′=r′+d1′·e+d1·e′+d2′·n+d2·n′=(|r′|+d1′+d2·k·|r′|)e+(d2′-d1·k·|r′|)n.
Let α=|r′|+d1′+d2·k·|r′| and let β=d2′-d1·k·|r′|; we get
(6)ro′=α·e+β·n,ro′′=α′·e+α·e′+β′·n+β·n′=(α′+β·k·|r′|)e+(β′-α·k·|r′|)n.
Therefore we have
(7)eo=ro′|ro′|=αe+βnα2+β2,no=eo×z=α(e×z)+β(n×z)α2+β2=αn-βeα2+β2,ko=(ro′×ro′′)·z|ro′|3=(α′β-αβ′)+(α2+β2)·k·|r′|(α2+β2)3/2.Regarding the regularity of generalized offset curves, we have the following theorem.Theorem 2.
If there existst0∈[0,1] to satisfy the equation
(8)d1′(t0)d1(t0)+d2′(t0)d2(t0)+d1(t0)·|r′(t0)|=0,
and for any arbitrary small positive number δ,
(9)k={-1d2(t0)[1+d1′(t0)|r′(t0)|],when|d2|≥δ>0,1d1(t0)[d2′(t0)|r′(t0)|],when|d1|≥δ>0,
then ro(t0) is a nonregular point of the generalized offset curve ro.Proof.
Letro(t0) be any non-regular point of the offset curve ro(t); then |ro′(t0)|=α2+β2=0. We get α=β=0. Thus
(10)d2·|r′|·k+d1′+|r′|=0,(11)-d1·|r′|·k+d2′=0.
We discuss the following two cases. (i)
When|d2|≥δ>0, from (10) it follows that
(12)k=-1d2(1+d1′|r′|).Substituting it into (11), we have
(13)d1′d1+d2′d2+d1|r′|=0.(ii)
When|d1|≥δ>0, from (11) it follows that
(14)k=1d1·d2′|r′|.Substituting it into (10), we also have
(15)d1′d1+d2′d2+d1|r′|=0.
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 offsetro=r+d1e+d2n can be represented as a standard offset: ro=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(16)ro=r+d1e+d2n=(r+Ae+Bn)+(d1-A)e+(d2-B)n≜r1+dn1,
where A and B are the functions of t,
(17)d=(d1-A)2+(d2-B)2,n1=d1-Ade+d2-Bdn,r1′=r′+A′e+Ae′+B′n+Bn′=(|r′|+A′+B·k·|r′|)e+(B′-A·k·|r′|)n.
In order to establish the above relationship, the following two conditions must be satisfied. (i)
The inner product of vectorsdn1 and r1′ must be zero. That is,
(18)(d1-A)(|r′|+A′+B·k·|r′|)+(d2-B)(B′-A·k·|r′|)=0.(ii)
(d1-A)2+(d2-B)2 must be constant. That is,
(19)(d1-A)(d1′-A′)+(d2-B)(d2′-B′)=0.
Our goal is to get the values ofA and B by solving the above differential equations.
From (18) and (19), we get
(20)(|r′|+d2·k·|r′|+d1′)A=(d1·k·|r′|-d2′)B+(d1|r′|+d1′d1+d2′d2).
Note that
(21)d1|r′|+d1′d1+d2′d2=d1|r′|+d1′d1+d2(β+d1·k·|r′|)=d1α+d2β,
and we have
(22)α(d1-A)+β(d2-B)=0.
Analyzing the following four cases, we can get the values of A and B. (1)
Whenα=β=0, A and B only need to satisfy (19). That is,
(23)(d1-A)2+(d2-B)2=C1,whereC1>0 is an arbitrary constant. Thus one of A and B can be determined arbitrarily. For instance, if A=g(t) is given, then
(24)B=d2±C1-(d1-g(t))2.(2)
Whenα≠0 and β=0, from (22) it follows that A=d1 and A′=d1′. Substituting it into (19), we have (d2-B)(d2′-B′)=0. Since d2≠B, then d2′=B′. Therefore B=d2(t)+C2, where C2 is an arbitrary nonzero constant.(3)
Whenα=0 and β≠0, from (22) it follows that B=d2 and B′=d2′. Substituting it into (19), we have (d1-A)(d1′-A′)=0. Since d1≠A, then d1′=A′. Therefore A=d1(t)+C3, where C3 is an arbitrary nonzero constant.(4)
Whenα≠0 and β≠0, by solving (19) and (22), we get
(25)A=d1-βCα2+β2,B=d2+αCα2+β2,whereC is an arbitrary constant.
Therefore, in any case there exist two functionsA and B to guarantee that the generalized offset ro=r+d1e+d2n can be expressed as the standard offset ro=r1+dn1, where
(26)r1=r+Ae+Bn=r+(d1-βCα2+β2)e+(d2+αCα2+β2)n,d=(d1-A)2+(d2-B)2=|C|,n1=βα2+β2e-αα2+β2n.
That is, we can find r1(t) so that ro(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

Letλ=|r′|+A′+B·k·|r′|  and ω=B′-A·k·|r′|. From the expression of standard offsets ro=r1+dn1, where
(27)r1=r+Ae+Bn,n1=d1-Ade+d2-Bdn,d=const, and d1,d2,A,B are the functions of t, we have
(28)r1′=λe+ωn,k1=(λ′ω-λω′)+(λ2+ω2)·k·|r′|(λ2+ω2)3/2,e1=r1′|r1′|=λe+ωnλ2+ω2,n1=λn-ωeλ2+ω2.
Moreover
(29)ro′=(1+dk1)·λ2+ω2·e1,ko=1|1+dk1|k1,eo=sgn⁡(1+dk1)e1,no=sgn⁡(1+dk1)n1,
where ko and k1 are the curvatures of ro and r1 at each point, respectively. In the above case, there is another expression for the nonregular point of ro. Since
(30)|ro′|=|1+dk1|·|r1′|=|1+dk1|·λ2+ω2,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)
EvoluteWe construct(31)rε(t)=r1(t)-ρ1(t)·n1,
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
(32)ro(τ)=r1(τ)+dn1=r1(τ)-ρ1n1=rε(τ).
On the other hand, we have
(33)rε=r-d1-A(1+dk1)dk1e-d2-B(1+dk1)dk1n.
Moreover, from
(34)ro-1kono=r1-1k1n1,
we can get the relations as follows:
(35)d1+βkoα2+β2=A+ωk1λ2+ω2,d2-αkoα2+β2=B-λk1λ2+ω2.(ii)
Turning point, inflection, and vertexLetr(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.
Ifk1≠-1/d, and λ,ω are not both zero, then (1)
the turning point, inflection, and vertex onro(t) are, respectively, in one-to-one correspondence to those on r1(t);(2)
the turning point onro(t) is in one-to-one correspondence to that on r(t) as ω=0;(3)
the inflection onro(t) is in one-to-one correspondence to that on r(t) as λ′ω=λω′.Proof.
Based upon the following relationships(36)eo=sgn⁡(1+dk1)·λe+ωnλ2+ω2,ko=1|1+dk1|·(λ′ω-λω′)+(λ2+ω2)·k·|r′|(λ2+ω2)3/2,dkodso=(1+dk1)-3·dk1ds1,
we can easily prove Theorem 4.(iii)
Length and areaWe can calculate the lengthsl1 and lo of the curves r1 and ro, respectively. Since dl1=|r1′|dt, then
(37)l1=∫01‍|r1′|dt=∫01‍λ2+ω2dt,lo=∫01‍|ro′|dt=∫01‍|1+dk1|·|r1′|dt.The area betweenr(t) and its generalized offset ro(t) is denoted by M (Figure 1). dS1 and dS1 are the area elements. See Figure 4.Figure 1
The area element betweenr(t) and ro(t).At first, we compute(38)dS1=|[ro(t)-r(t)]×[r(t+Δt)-r(t)]|2,dS2=|[ro(t)-ro(t+Δt)]×[r(t+Δt)-ro(t+Δt)]|2.
Since
(39)r(t+Δt)=r(t)+Δt·r′(t)+O(Δt2),ro(t+Δt)=ro(t)+Δt·ro′(t)+O(Δt2),
it follows that
(40)dS1=|[d1e+d2n]×[Δt·r′(t)]|2=12|d2|·|r′|dt,dS2=|[Δt·ro′(t)]×[r(t)-ro(t)+Δt(r′(t)-ro′(t))]|2=12·Δt·|d1′d2-d2′d1+d2·|r′|+(d12+d22)·k·|r′||.
Therefore dM=dS1+dS2, and
(41)M=∫01‍dM=12[∫01‍|d2|·|r′|dtcccccccccccc+∫01|d1′d2-d2′d1+d2·|r′|ccccccccccccccccccc+(d12+d22)·k·|r′||dt∫01‍].(iv)
Topological propertyThe distance between a regular curver1=r1(t),t∈[0,1] and a point Q in the same plane is defined as follows:
(42)δ(Q,r1)=inf⁡t∈[0,1]|Q-r1(t)|.
For the standard offset ro=r1+dn1, we have the following theorm.Theorem 5.
The distanceδ(ro(τ),r) between the point ro(τ) of the generalized offset and the curve r=r(t),t∈[0,1] satisfies one of the following conditions:
(43)δ(ro(τ),r)=|d|+A2+B2,τ∈(ik,ik+1),δ(ro(τ),r)<|d|+A2+B2,τ∈(ik,ik+1),cccccccccccccccccccck=0,…,N,N∈Z+.
Each of the open intervals (ik,ik+1) and k=0,…,N. is delineated by the self-intersections.Proof.
We have the following:(1)
δ(ro(τ),r1)≤|d|,τ∈[0,1];(2)
i0=0, iN+1=1, i1,…,iN∈(0,1), N∈Z+ are the self-intersections of ro.
Then one of the following propositions holds(44)δ(ro(τ),r1)≡|d|,τ∈(ik,ik+1),δ(ro(τ),r1)<|d|,τ∈(ik,ik+1),ccccccccccccccccccccck=0,…,N
For the generalized offset r0(t), we have
(45)|r1(t)-r(t)|=A2+B2,t∈[0,1].
Considering τ∈(ik,ik+1), let p∈r1 such that
(46)|ro(τ)-p|=δ(ro(τ),r1),
and q∈r=r(t), t∈[0,1] is the point with the same parameter t of p on r; then we have
(47)|p-q|=A2+B2.
Thus
(48)δ(ro(τ),r)≤|ro(τ)-r|≤δ(ro(τ),r1)+A2+B2.
Therefore we prove Theorem 5, which is shown in Figure 2.Figure 2
Topological property of the curves.According to Theorem5, each of the segments {ro(t),t∈(ik,ik+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 curver(t) are
(49)deds=-kn,dnds=ke+τz,dzds=-τn,
where τ is the torsion, and we have
(50)τ=(r′×r′′)·r′′′|r′×r′′|2.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 Section3 are all redefined.For a regular parameter surfacer(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]
(51)e1=ru(u,v)|ru(u,v)|,e2=rv(u,v)|rv(u,v)|,n=ru(u,v)×rv(u,v)|ru(u,v)×rv(u,v)|,
where ru(u,v) and rv(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 ro(u,v) with the variable offset direction and the variable offset distance can be defined.Definition 6.
For a regular smooth parametric surfacer(u,v),(u,v)∈[0,1]×[0,1], the generalized offset surface ro(u,v) with the variable offset distance and offset direction is defined by
(52)ro(u,v)=r(u,v)+d1(u,v)e1+d2(u,v)e2+d3(u,v)n,
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 surfacer(u,v), the well-known first and second fundamental quantities and the Gauss curvature [14] are given as follows:
(53)E=ru2=A12,G=rv2=A22,F=ru·rv,D=n·ruu,D′=n·ruv,D′′=n·rvv,k=DD′′-D′2EG-F2.
Let e1×n=e1′,e2×n=e2′, and the angle between e1 and e2 is θ; then
(54)e1×e2=nsinθ,e1′×e2′=nsinθ,e1′·e2=-sinθ,e2′·e1=sinθ.
Thus the related parametric partial derivatives of generalized offset surface ro(u,v) can be obtained by
(55)ruo(u,v)=ru+d1ue1+d1e1u+d2ue2+d2e2u+d3un+d3nu=[A1+d1u+d2A1vA2-d3A1R1]e1+[d2u-d1A1vA2]e2+[d3u+d1A1R1]n≜B1e1+B2e2+B3n,(56)rvo(u,v)=rv+d1ve1+d1e1v+d2ve2+d2e2v+d3vn+d3nv=[d1v-d2A2uA1]e1+[A2+d1A2uA1+d2v-d3A2R2]e2+[d3v+d2A2R2]n≜C1e1+C2e2+C3n,
where ru,d1u,d2u,A2u,e1u,e2u,nu are the corresponding partial derivatives of r,d1,d2,A2,e1,e2,n with respect to u and rv,d1v,d2v,A1v,e1v,e2v,nv 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:
(57)ruuo(u,v)=B1ue1+B1e1u+B2ue2+B2e2uruuo(u,v)=+B3un+B3nuruuo(u,v)=(B1u+B2A1vA2-B3A1R1)e1ruuo(u,v)=+(-B1A1vA2+B2u)ruuo(u,v)=+(B1A1R1+B3u)n,ruvo(u,v)=rvuo(u,v)ruuo(u,v)=B1ve1+B1e1v+B2ve2+B2e2vruuo(u,v)=+B3vn+B3nvruuo(u,v)=(B1v-B2A2uA1)e1ruuo(u,v)=+(B1A2uA1+B2v-B3A2R2)e2ruuo(u,v)=+(B2A2R2+B3v)n,rvvo(u,v)=C1ve1+C1e1v+C2ve2+C2e2vruuo(u,v)=+C3vn+C3nvruuo(u,v)=(C1v-C2A2uA1)e1ruuo(u,v)=+(C1A2uA1+C2v-C3A2R2)e2ruuo(u,v)=+(C2A2R2+C3v)n,Eo=ruo2=B12+B22+B32+2B1B2cos⁡θ,Go=rvo2=C12+C22+C32+2C1C2cos⁡θ,Fo=ruo×rvoFo=B1C1+B2C2+B3C3+(B1C2+B2C1)cos⁡θ.
Thus the unit tangent vectors and unit normal vector of surface offsets ro(u,v) are given as follows:
(58)e1o=ruo(u,v)|ruo(u,v)|=B1e1+B2e2+B3nEo,e2o=rvo(u,v)|rvo(u,v)|=C1e1+C2e2+C3nGo,no=ruo×rvo|ruo×rvo|=ruo×rvoEoGo-Fo2no=B1C3-B3C1EoGo-Fo2e1′+B2C3-B3C2EoGo-Fo2e2′no=+(B1C2-B2C1)sinθEoGo-Fo2nno≜M1e1+M2e2+M3n,Do=noruuo=M1(B1u+B2A1vA2-B3A1R1)Do=+M2(-B1A1vA2+B2u)+M3(B1A1R1+B3u)Do=+[M1(-B1A1vA2+B2u)cccccccccc+M2(B1u+B2A1vA2-B3A1R1)]cos⁡θ,Do′=noruvo=M1(B1v-B2A2uA1)Do′=+M2(B1A2uA1+B2v-B3A2R2)+M3(B2A2R2+B3v)Do′=+[M1(B1A2uA1+B2v-B3A2R2)ccccccccc+M2(B1v-B2A2uA1)]cos⁡θ,Do′′=norvvo=M1(C1v-C2A2uA1)Do′′=+M2(C1A2uA1+C2v-C3A2R2)+M3(C2A2R2+C3v)Do′′=+[M1(C1A2uA1+C2v-C3A2R2)cccccccccc+M2(C1v-C2A2uA1)]cos⁡θ,ko=DoDo′′-Do′2EoGo-Fo2,
where Eo,Go,Fo,Do,Do′,Do′′, and ko are, respectively, the basic quantities and Gauss curvature of surface offset ro(u,v). Moreover, we can get the tangent plane and normal line at one particular point of surface ro(u,v).Letro(u0,v0) be a nonregular point of ro(u,v); then
(59)|ruo(u0,v0)×rvo(u0,v0)|=0.
Note that
(60)ruo×rvo=(B1C3-B3C1)e1′+(B2C3-B3C2)e2′+(B1C2-B2C1)nsinθ≜ρ1e1′+ρ2e2′+ρ3nsinθ.
Regarding the regularity of generalized offset surfaces, we have the following theorem.Theorem 7.
If(61)(ρ12(u0,v0)+ρ22(u0,v0)+ρ32(u0,v0)+2|ρ1(u0,v0)ρ2(u0,v0)|cos⁡θρ12)1/2=0,
then ro(u0,v0) is a nonregular point of ro(u,v).From the above explanation, we can easily prove Theorem7.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~,v~) to make the new local natural coordinate system (eu~,ev~,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 offsetro=r+d1e1+d2e2+d3n can be represented as a standard offset: ro=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(62)ro=r+d1e1+d2e2+d3n=(r+Me1+Ne2+Pn)+(d1-M)e1+(d2-N)e2+(d3-P)n≜r1+dn1,
where M, N, and P are the functions of u and v,
(63)d=(d1-M)2+(d2-N)2+(d3-P)2,n1=(d1-M)de1+(d2-N)de2+(d3-P)d.
The parametric partial derivatives of surface r1(u,v) are
(64)r1u(u,v)=ru+Mue1+Me1u+Nue2+Ne2u+Pun+Pnu=[A1+Mu+NA1vA2-PA1R1]e1+[Nu-MA1vA2]e2+[Pu+MA1R1]n,(65)r1v(u,v)=rv+Mve1+Me1v+Nve2+Ne2v+Pvn+Pnv=[Mv-NA2uA1]e1+[A2+MA2uA1+Nv-PA2R2]e2+[Pv+NA2R2]n,
where Mu,Mv,Nu,Nv,Pu,Pv 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,
(66)(A1+Mu+NA1vA2-PA1R1)(d1-M)+(Nu-MA1vA2)(d2-N)+(Pu+MA1R1)(d3-P)=0,(67)(Mv-NA2uA1)(d1-M)+(A2+MA2uA1+Nv-PA2R2)(d2-N)+(Pv+NA2R2)(d3-P)=0.(ii)
d is constant. That is,
(68)(d1-M)2+(d2-N)2+(d3-P)2=const.It follows that
(69)(d1-M)(d1u-Mu)+(d2-N)(d2u-Nu)+(d3-P)(d3u-Pu)=0,(70)(d1-M)(d1v-Mv)+(d2-N)(d2v-Nv)+(d3-P)(d3v-Pv)=0.
Our goal is to get the values of M,N, and P by solving the above differential equations. From (66) and (69), we get
(71)(-A1-A1vd2A2-d1u+A1d3R1)M+(A1vd1A2-d2u)N+(-A1d1R1-d3u)P+(A1d1+d1ud1+d2ud2+d3ud3)≜α1M+β1N+γ1P+ω1=0.
From (67) and (70), we get
(72)(A2ud2A1-d1v)M+(-A2ud1A1-A2-d2v+A2d3R2)N+(-A2d2R2-d3v)P+(A2d2+d1vd1+d2vd2+d3vd3)≜α2M+β2N+γ2P+ω2=0.
Let
(73)α1β2-α2β1≜α-,β2γ1-β1γ2≜β-,α2γ1-α1γ2≜γ-.
From (71) and (72), we get
(74)α-(d1-M)+β-(d3-P)=0,-α-(d2-N)+γ-(d3-P)=0.
By solving (69), (70), and (74), we get
(75)M=d1-β-C-α-2+β-2+γ-2,N=d2+γ-C-α-2+β-2+γ-2,P=d3+α-C-α-2+β-2+γ-2,
where C- is an arbitrary constant, α-, β-, and γ- can not all be zero. Thus
(76)d=(d1-M)2+(d2-N)2+(d3-P)2=C-,n1=β-α-2+β-2+γ-2e1-γ-α-2+β-2+γ-2e2-α-α-2+β-2+γ-2n.
When α-=β-=γ-=0, M,N, and P only need to satisfy (69) and (70).
Therefore, in any case there are three functionsM,N, and P to guarantee the generalized offset ro=r+d1e1+d2e2+d3n can be expressed as the standard offset ro(u,v)=r1(u,v)+dn1(u,v),d=const. That is, we can find r1(u,v) so that ro(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 surfaceThe area of generalized offset surface(77){ro(u,v):(u,v)∈[0,1]×[0,1]}
is denoted by S. We consider the area element dS, which is shown in Figure 3. Consider the following:
(78)dS=|(ruo×rvo)|dudv.
Since
(79)(ruo×rvo)2=ruo2rvo2-(ruorvo)2=EoGo-Fo2,
then
(80)S=∬[0,1]×[0,1]EoGo-Fo2dudv.(ii)
The volume betweenr(u,v) and ro(u,v)Figure 3
The area element ofro(u,v).Figure 4
The volume element betweenr(u,v) and ro(u,v).The volume betweenr(u,v) and its generalized offset ro(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:
(81)dV=dV1+dV2+⋯+dV5,
where
(82)dV1=[r(u,v),ro(u,v),r(u+Δu,v),r(u,v+Δv)],dV2=[ro(u+Δu,v),ro(u,v),r(u+Δu,v),ccccccccro(u+Δu,v+Δv)],dV3=[r(u+Δu,v+Δv),r(u+Δu,v),ccccccccro(u+Δu,v+Δv),r(u,v+Δv)],dV4=[ro(u,v+Δv),ro(u,v),ccccccccr(u,v+Δv),ro(u+Δu,v+Δv)],dV5=[r(u,v+Δv),ro(u,v),r(u+Δu,v),ccccccccro(u+Δu,v+Δv)],
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(83)dV1=16|A1A2d3|dudv≜w1dudv,dV2=16|d1(B3C2-B2C3)+d2(B1C3-B3C1)+d3(B2C1-B1C2)|dudv≜w2dudv,dV3=16|A1A2d3|dudv≜w1dudv,dV4=16|d1(B2C3-B3C2)+d2(B3C1-B1C3)+d3(B1C2-B2C1)|dudv≜w2dudv,dV5=16|A1C3d2-A1C2d3+A2B3d1-A2B1d3|dudv≜w5dudv.
Therefore, we have
(84)V=∬01(dV1+dV2+⋯+dV5)=∬[0,1]×[0,1](2w1+2w2+w5)dudv.(ii) Topological propertyFor a regular parameter surface(85)L1={r1(u,v):(u,v)∈[0,1]×[0,1]},
the distance between a point Q and the surface L1 is defined as follows:
(86)δ(Q,L1)=inf⁡(u,v)∈[0,1]×[0,1]|Q-r1(u,v)|.
For the standard offset ro=r1+dn1,d=const, we have the following theorem.Theorem 9.
The distanceδ(ro(τ,η),C) between the point ro(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:
(87)δ(ro(τ,η),L)=|d|+M2+N2+P2,δ(ro(τ,η),L)<|d|+M2+N2+P2,cccccccc(τ,η)∈(ik,ik+1)×(jk′,jk′+1),cccccccccck,k′=0,1,…,N-,N-∈Z+.
Each of the open fields (ik,ik+1)×(jk′,jk′+1),k,k′=0,1,…,N- is delineated by the self-intersections.Proof.
We have the following:(1)
δ(ro(τ,η),L1)≤|d|,(τ,η)∈[0,1]×[0,1];(2)
(i0,j0)=(0,0),(i0,jN-+1)=(0,1), (iN-+1,j0)=(1,0), (iN-+1,jN-+1)=(1,1), i1,…,iN-,j1,…,jN-∈(0,1), N-∈Z+ are the self-intersections of ro.
Then one of the following propositions holds(88)δ(ro(τ,η),L1)=|d|,(τ,η)∈(ik,ik+1)×(jk′,jk′+1),δ(ro(τ,η),L1)<|d|,(τ,η)∈(ik,ik+1)×(jk′,jk′+1),cccccccccccccccccccccccccccccccccck,k′=0,1,…,N-.
For the generalized offset
(89)ro=r+d1e1+d2e2+d3n=r1+dn1,r1=r+Me1+Ne2+Pn,
we have
(90)r1(u,v)-r(u,v)=Me1+Ne2+Pn,|r1(u,v)-r(u,v)|=M2+N2+P2,cccccccccccccccc(u,v)∈[0,1]×[0,1].
Considering (τ,η)∈(ik,ik+1)×(jk′,jk′+1), let p∈L1 such that
(91)|ro(τ,η)-p|=δ(ro(τ,η),L1),
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
(92)|p-q|=M2+N2+P2.
Thus
(93)δ(ro(τ,η),L)≤|ro(τ,η)-L|≤δ(ro(τ,η),L1)+M2+N2+P2.
Therefore we prove the above theorem, which is illustrated in Figure 5.Figure 5
Topological property of the surfaces.According to Theorem9, each of the segments {r0(s,t):(s,t)∈(ik,ik+1)×(jk′,jk′+1)} of the offset surface among its self-intersections should either be retained or rejected in its entirety when forming the trimmed offset.


## 3.1.  The Definition and Regularity of Generalized Offset Surfaces

Note that the symbols used in Section3 are all redefined.For a regular parameter surfacer(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]
(51)e1=ru(u,v)|ru(u,v)|,e2=rv(u,v)|rv(u,v)|,n=ru(u,v)×rv(u,v)|ru(u,v)×rv(u,v)|,
where ru(u,v) and rv(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 ro(u,v) with the variable offset direction and the variable offset distance can be defined.Definition 6.
For a regular smooth parametric surfacer(u,v),(u,v)∈[0,1]×[0,1], the generalized offset surface ro(u,v) with the variable offset distance and offset direction is defined by
(52)ro(u,v)=r(u,v)+d1(u,v)e1+d2(u,v)e2+d3(u,v)n,
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 surfacer(u,v), the well-known first and second fundamental quantities and the Gauss curvature [14] are given as follows:
(53)E=ru2=A12,G=rv2=A22,F=ru·rv,D=n·ruu,D′=n·ruv,D′′=n·rvv,k=DD′′-D′2EG-F2.
Let e1×n=e1′,e2×n=e2′, and the angle between e1 and e2 is θ; then
(54)e1×e2=nsinθ,e1′×e2′=nsinθ,e1′·e2=-sinθ,e2′·e1=sinθ.
Thus the related parametric partial derivatives of generalized offset surface ro(u,v) can be obtained by
(55)ruo(u,v)=ru+d1ue1+d1e1u+d2ue2+d2e2u+d3un+d3nu=[A1+d1u+d2A1vA2-d3A1R1]e1+[d2u-d1A1vA2]e2+[d3u+d1A1R1]n≜B1e1+B2e2+B3n,(56)rvo(u,v)=rv+d1ve1+d1e1v+d2ve2+d2e2v+d3vn+d3nv=[d1v-d2A2uA1]e1+[A2+d1A2uA1+d2v-d3A2R2]e2+[d3v+d2A2R2]n≜C1e1+C2e2+C3n,
where ru,d1u,d2u,A2u,e1u,e2u,nu are the corresponding partial derivatives of r,d1,d2,A2,e1,e2,n with respect to u and rv,d1v,d2v,A1v,e1v,e2v,nv 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:
(57)ruuo(u,v)=B1ue1+B1e1u+B2ue2+B2e2uruuo(u,v)=+B3un+B3nuruuo(u,v)=(B1u+B2A1vA2-B3A1R1)e1ruuo(u,v)=+(-B1A1vA2+B2u)ruuo(u,v)=+(B1A1R1+B3u)n,ruvo(u,v)=rvuo(u,v)ruuo(u,v)=B1ve1+B1e1v+B2ve2+B2e2vruuo(u,v)=+B3vn+B3nvruuo(u,v)=(B1v-B2A2uA1)e1ruuo(u,v)=+(B1A2uA1+B2v-B3A2R2)e2ruuo(u,v)=+(B2A2R2+B3v)n,rvvo(u,v)=C1ve1+C1e1v+C2ve2+C2e2vruuo(u,v)=+C3vn+C3nvruuo(u,v)=(C1v-C2A2uA1)e1ruuo(u,v)=+(C1A2uA1+C2v-C3A2R2)e2ruuo(u,v)=+(C2A2R2+C3v)n,Eo=ruo2=B12+B22+B32+2B1B2cos⁡θ,Go=rvo2=C12+C22+C32+2C1C2cos⁡θ,Fo=ruo×rvoFo=B1C1+B2C2+B3C3+(B1C2+B2C1)cos⁡θ.
Thus the unit tangent vectors and unit normal vector of surface offsets ro(u,v) are given as follows:
(58)e1o=ruo(u,v)|ruo(u,v)|=B1e1+B2e2+B3nEo,e2o=rvo(u,v)|rvo(u,v)|=C1e1+C2e2+C3nGo,no=ruo×rvo|ruo×rvo|=ruo×rvoEoGo-Fo2no=B1C3-B3C1EoGo-Fo2e1′+B2C3-B3C2EoGo-Fo2e2′no=+(B1C2-B2C1)sinθEoGo-Fo2nno≜M1e1+M2e2+M3n,Do=noruuo=M1(B1u+B2A1vA2-B3A1R1)Do=+M2(-B1A1vA2+B2u)+M3(B1A1R1+B3u)Do=+[M1(-B1A1vA2+B2u)cccccccccc+M2(B1u+B2A1vA2-B3A1R1)]cos⁡θ,Do′=noruvo=M1(B1v-B2A2uA1)Do′=+M2(B1A2uA1+B2v-B3A2R2)+M3(B2A2R2+B3v)Do′=+[M1(B1A2uA1+B2v-B3A2R2)ccccccccc+M2(B1v-B2A2uA1)]cos⁡θ,Do′′=norvvo=M1(C1v-C2A2uA1)Do′′=+M2(C1A2uA1+C2v-C3A2R2)+M3(C2A2R2+C3v)Do′′=+[M1(C1A2uA1+C2v-C3A2R2)cccccccccc+M2(C1v-C2A2uA1)]cos⁡θ,ko=DoDo′′-Do′2EoGo-Fo2,
where Eo,Go,Fo,Do,Do′,Do′′, and ko are, respectively, the basic quantities and Gauss curvature of surface offset ro(u,v). Moreover, we can get the tangent plane and normal line at one particular point of surface ro(u,v).Letro(u0,v0) be a nonregular point of ro(u,v); then
(59)|ruo(u0,v0)×rvo(u0,v0)|=0.
Note that
(60)ruo×rvo=(B1C3-B3C1)e1′+(B2C3-B3C2)e2′+(B1C2-B2C1)nsinθ≜ρ1e1′+ρ2e2′+ρ3nsinθ.
Regarding the regularity of generalized offset surfaces, we have the following theorem.Theorem 7.
If(61)(ρ12(u0,v0)+ρ22(u0,v0)+ρ32(u0,v0)+2|ρ1(u0,v0)ρ2(u0,v0)|cos⁡θρ12)1/2=0,
then ro(u0,v0) is a nonregular point of ro(u,v).From the above explanation, we can easily prove Theorem7.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~,v~) to make the new local natural coordinate system (eu~,ev~,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 offsetro=r+d1e1+d2e2+d3n can be represented as a standard offset: ro=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(62)ro=r+d1e1+d2e2+d3n=(r+Me1+Ne2+Pn)+(d1-M)e1+(d2-N)e2+(d3-P)n≜r1+dn1,
where M, N, and P are the functions of u and v,
(63)d=(d1-M)2+(d2-N)2+(d3-P)2,n1=(d1-M)de1+(d2-N)de2+(d3-P)d.
The parametric partial derivatives of surface r1(u,v) are
(64)r1u(u,v)=ru+Mue1+Me1u+Nue2+Ne2u+Pun+Pnu=[A1+Mu+NA1vA2-PA1R1]e1+[Nu-MA1vA2]e2+[Pu+MA1R1]n,(65)r1v(u,v)=rv+Mve1+Me1v+Nve2+Ne2v+Pvn+Pnv=[Mv-NA2uA1]e1+[A2+MA2uA1+Nv-PA2R2]e2+[Pv+NA2R2]n,
where Mu,Mv,Nu,Nv,Pu,Pv 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,
(66)(A1+Mu+NA1vA2-PA1R1)(d1-M)+(Nu-MA1vA2)(d2-N)+(Pu+MA1R1)(d3-P)=0,(67)(Mv-NA2uA1)(d1-M)+(A2+MA2uA1+Nv-PA2R2)(d2-N)+(Pv+NA2R2)(d3-P)=0.(ii)
d is constant. That is,
(68)(d1-M)2+(d2-N)2+(d3-P)2=const.It follows that
(69)(d1-M)(d1u-Mu)+(d2-N)(d2u-Nu)+(d3-P)(d3u-Pu)=0,(70)(d1-M)(d1v-Mv)+(d2-N)(d2v-Nv)+(d3-P)(d3v-Pv)=0.
Our goal is to get the values of M,N, and P by solving the above differential equations. From (66) and (69), we get
(71)(-A1-A1vd2A2-d1u+A1d3R1)M+(A1vd1A2-d2u)N+(-A1d1R1-d3u)P+(A1d1+d1ud1+d2ud2+d3ud3)≜α1M+β1N+γ1P+ω1=0.
From (67) and (70), we get
(72)(A2ud2A1-d1v)M+(-A2ud1A1-A2-d2v+A2d3R2)N+(-A2d2R2-d3v)P+(A2d2+d1vd1+d2vd2+d3vd3)≜α2M+β2N+γ2P+ω2=0.
Let
(73)α1β2-α2β1≜α-,β2γ1-β1γ2≜β-,α2γ1-α1γ2≜γ-.
From (71) and (72), we get
(74)α-(d1-M)+β-(d3-P)=0,-α-(d2-N)+γ-(d3-P)=0.
By solving (69), (70), and (74), we get
(75)M=d1-β-C-α-2+β-2+γ-2,N=d2+γ-C-α-2+β-2+γ-2,P=d3+α-C-α-2+β-2+γ-2,
where C- is an arbitrary constant, α-, β-, and γ- can not all be zero. Thus
(76)d=(d1-M)2+(d2-N)2+(d3-P)2=C-,n1=β-α-2+β-2+γ-2e1-γ-α-2+β-2+γ-2e2-α-α-2+β-2+γ-2n.
When α-=β-=γ-=0, M,N, and P only need to satisfy (69) and (70).
Therefore, in any case there are three functionsM,N, and P to guarantee the generalized offset ro=r+d1e1+d2e2+d3n can be expressed as the standard offset ro(u,v)=r1(u,v)+dn1(u,v),d=const. That is, we can find r1(u,v) so that ro(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 surfaceThe area of generalized offset surface(77){ro(u,v):(u,v)∈[0,1]×[0,1]}
is denoted by S. We consider the area element dS, which is shown in Figure 3. Consider the following:
(78)dS=|(ruo×rvo)|dudv.
Since
(79)(ruo×rvo)2=ruo2rvo2-(ruorvo)2=EoGo-Fo2,
then
(80)S=∬[0,1]×[0,1]EoGo-Fo2dudv.(ii)
The volume betweenr(u,v) and ro(u,v)Figure 3
The area element ofro(u,v).Figure 4
The volume element betweenr(u,v) and ro(u,v).The volume betweenr(u,v) and its generalized offset ro(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:
(81)dV=dV1+dV2+⋯+dV5,
where
(82)dV1=[r(u,v),ro(u,v),r(u+Δu,v),r(u,v+Δv)],dV2=[ro(u+Δu,v),ro(u,v),r(u+Δu,v),ccccccccro(u+Δu,v+Δv)],dV3=[r(u+Δu,v+Δv),r(u+Δu,v),ccccccccro(u+Δu,v+Δv),r(u,v+Δv)],dV4=[ro(u,v+Δv),ro(u,v),ccccccccr(u,v+Δv),ro(u+Δu,v+Δv)],dV5=[r(u,v+Δv),ro(u,v),r(u+Δu,v),ccccccccro(u+Δu,v+Δv)],
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(83)dV1=16|A1A2d3|dudv≜w1dudv,dV2=16|d1(B3C2-B2C3)+d2(B1C3-B3C1)+d3(B2C1-B1C2)|dudv≜w2dudv,dV3=16|A1A2d3|dudv≜w1dudv,dV4=16|d1(B2C3-B3C2)+d2(B3C1-B1C3)+d3(B1C2-B2C1)|dudv≜w2dudv,dV5=16|A1C3d2-A1C2d3+A2B3d1-A2B1d3|dudv≜w5dudv.
Therefore, we have
(84)V=∬01(dV1+dV2+⋯+dV5)=∬[0,1]×[0,1](2w1+2w2+w5)dudv.(ii) Topological propertyFor a regular parameter surface(85)L1={r1(u,v):(u,v)∈[0,1]×[0,1]},
the distance between a point Q and the surface L1 is defined as follows:
(86)δ(Q,L1)=inf⁡(u,v)∈[0,1]×[0,1]|Q-r1(u,v)|.
For the standard offset ro=r1+dn1,d=const, we have the following theorem.Theorem 9.
The distanceδ(ro(τ,η),C) between the point ro(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:
(87)δ(ro(τ,η),L)=|d|+M2+N2+P2,δ(ro(τ,η),L)<|d|+M2+N2+P2,cccccccc(τ,η)∈(ik,ik+1)×(jk′,jk′+1),cccccccccck,k′=0,1,…,N-,N-∈Z+.
Each of the open fields (ik,ik+1)×(jk′,jk′+1),k,k′=0,1,…,N- is delineated by the self-intersections.Proof.
We have the following:(1)
δ(ro(τ,η),L1)≤|d|,(τ,η)∈[0,1]×[0,1];(2)
(i0,j0)=(0,0),(i0,jN-+1)=(0,1), (iN-+1,j0)=(1,0), (iN-+1,jN-+1)=(1,1), i1,…,iN-,j1,…,jN-∈(0,1), N-∈Z+ are the self-intersections of ro.
Then one of the following propositions holds(88)δ(ro(τ,η),L1)=|d|,(τ,η)∈(ik,ik+1)×(jk′,jk′+1),δ(ro(τ,η),L1)<|d|,(τ,η)∈(ik,ik+1)×(jk′,jk′+1),cccccccccccccccccccccccccccccccccck,k′=0,1,…,N-.
For the generalized offset
(89)ro=r+d1e1+d2e2+d3n=r1+dn1,r1=r+Me1+Ne2+Pn,
we have
(90)r1(u,v)-r(u,v)=Me1+Ne2+Pn,|r1(u,v)-r(u,v)|=M2+N2+P2,cccccccccccccccc(u,v)∈[0,1]×[0,1].
Considering (τ,η)∈(ik,ik+1)×(jk′,jk′+1), let p∈L1 such that
(91)|ro(τ,η)-p|=δ(ro(τ,η),L1),
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
(92)|p-q|=M2+N2+P2.
Thus
(93)δ(ro(τ,η),L)≤|ro(τ,η)-L|≤δ(ro(τ,η),L1)+M2+N2+P2.
Therefore we prove the above theorem, which is illustrated in Figure 5.Figure 5
Topological property of the surfaces.According to Theorem9, each of the segments {r0(s,t):(s,t)∈(ik,ik+1)×(jk′,jk′+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(94)r(t)={cos⁡(t),sin(t)},t∈[0,2π].
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
(95)ro(t)=r(t)+d1(t)e(t)+d2n(t),
where we take d1=|sin(2t)| and d1=|8·cos⁡(2t)|.Thus the generalized offsetro(t) is
(96)ro(t)=r1(t)+dn1(t)={cos⁡(t)-sin(t)·|sin(2t)|+cos⁡(t)·|8·cos⁡(2t)|,sin(t)+cos⁡(t)·|sin(2t)|+sin(t)·|8·cos⁡(2t)|}
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 Figures8 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.



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