Properties of Generalized Offset Curves and Surfaces

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 124240, 13 pages http://dx.doi.org/10.1155/2014/124240

Research Article Properties of Generalized Offset Curves and Surfaces Xuejuan Chen1 and Qun Lin2 1 2

School of Science, Jimei University, Xiamen 361021, China School of Mathematical Science, Xiamen University, Xiamen 361005, China

Correspondence should be addressed to Xuejuan Chen; [email protected] Received 25 October 2013; Accepted 11 March 2014; Published 21 May 2014 Academic Editor: Jacek Rokicki Copyright © 2014 X. Chen and Q. Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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

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.

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: 𝑑e = −𝑘n, 𝑑𝑠

𝑑n = 𝑘e, 𝑑𝑠

(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 r󸀠 e = 󵄨󵄨 󸀠 󵄨󵄨 , 󵄨󵄨r 󵄨󵄨

z = n × e,

e = z × n,

n = e × z.

(2)

So we have 󸀠

𝑘=

󸀠󸀠

(r × r ) ⋅ z . 󵄨󵄨 󸀠 󵄨󵄨3 󵄨󵄨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 curve r = 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 r𝑜 (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 √𝑑12 (𝑡) + 𝑑22 (𝑡). From the above

definition, the related parametric derivatives of r𝑜 can be obtained by r󸀠𝑜 = r󸀠 + 𝑑1󸀠 ⋅ e + 𝑑1 ⋅ e󸀠 + 𝑑2󸀠 ⋅ n + 𝑑2 ⋅ n󸀠 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 = (󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 + 𝑑1󸀠 + 𝑑2 ⋅ 𝑘 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨) e + (𝑑2󸀠 − 𝑑1 ⋅ 𝑘 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨) n.

(5)

Let 𝛼 = |r󸀠 | + 𝑑1󸀠 + 𝑑2 ⋅ 𝑘 ⋅ |r󸀠 | and let 𝛽 = 𝑑2󸀠 − 𝑑1 ⋅ 𝑘 ⋅ |r󸀠 |; we get r󸀠𝑜 = 𝛼 ⋅ e + 𝛽 ⋅ n, r󸀠󸀠𝑜 = 𝛼󸀠 ⋅ e + 𝛼 ⋅ e󸀠 + 𝛽󸀠 ⋅ n + 𝛽 ⋅ n󸀠

(6)

󵄨 󵄨 󵄨 󵄨 = (𝛼󸀠 + 𝛽 ⋅ 𝑘 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨) e + (𝛽󸀠 − 𝛼 ⋅ 𝑘 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨) n. Therefore we have r󸀠 𝛼e + 𝛽n e𝑜 = 󵄨󵄨 𝑜󸀠 󵄨󵄨 = , 󵄨󵄨r𝑜 󵄨󵄨 √𝛼2 + 𝛽2 n 𝑜 = e𝑜 × z =

𝛼 (e × z) + 𝛽 (n × z) √𝛼2 + 𝛽2

=

𝛼n − 𝛽e √𝛼2 + 𝛽2

, (7)

(r󸀠𝑜 × r󸀠󸀠𝑜 ) ⋅ z 𝑘𝑜 = 󵄨󵄨 󸀠 󵄨󵄨3 󵄨󵄨r𝑜 󵄨󵄨 =

󵄨 󵄨 (𝛼󸀠 𝛽 − 𝛼𝛽󸀠 ) + (𝛼2 + 𝛽2 ) ⋅ 𝑘 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 3/2

(𝛼2 + 𝛽2 )

.

Regarding the regularity of generalized offset curves, we have the following theorem. Theorem 2. If there exists 𝑡0 ∈ [0, 1] to satisfy the equation 󵄨 󵄨 𝑑1󸀠 (𝑡0 ) 𝑑1 (𝑡0 ) + 𝑑2󸀠 (𝑡0 ) 𝑑2 (𝑡0 ) + 𝑑1 (𝑡0 ) ⋅ 󵄨󵄨󵄨󵄨r󸀠 (𝑡0 )󵄨󵄨󵄨󵄨 = 0, (8) and for any arbitrary small positive number 𝛿, 𝑑1󸀠 (𝑡0 ) 1 󵄨 󵄨 { − [1 + { 󵄨󵄨 󸀠 󵄨 ] , when 󵄨󵄨󵄨𝑑2 󵄨󵄨󵄨 ≥ 𝛿 > 0, { { 𝑑2 (𝑡0 ) 󵄨󵄨r (𝑡0 )󵄨󵄨󵄨 𝑘={ 󸀠 { { 󵄨 󵄨 { 1 [ 𝑑2 (𝑡0 ) ] , when 󵄨󵄨󵄨𝑑1 󵄨󵄨󵄨 ≥ 𝛿 > 0, 󵄨󵄨 󸀠 󵄨󵄨 r 𝑑 (𝑡 ) (𝑡 ) { 1 0 󵄨󵄨 0 󵄨󵄨

(9)

then r𝑜 (𝑡0 ) is a nonregular point of the generalized offset curve r𝑜 . Proof. Let r𝑜 (𝑡0 ) be any non-regular point of the offset curve

r𝑜 (𝑡); then |r󸀠𝑜 (𝑡0 )| = √𝛼2 + 𝛽2 = 0. We get 𝛼 = 𝛽 = 0. Thus 󵄨 󵄨 󵄨 󵄨 𝑑2 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 ⋅ 𝑘 + 𝑑1󸀠 + 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 = 0, 󵄨 󵄨 −𝑑1 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 ⋅ 𝑘 + 𝑑2󸀠 = 0.

(10) (11)

We discuss the following two cases. (i) When |𝑑2 | ≥ 𝛿 > 0, from (10) it follows that 𝑘=−

1 (1 + 𝑑2

𝑑1󸀠 󵄨󵄨 󸀠 󵄨󵄨 ) . 󵄨󵄨r 󵄨󵄨

(12)


3

Substituting it into (11), we have

Our goal is to get the values of 𝐴 and 𝐵 by solving the above differential equations. From (18) and (19), we get

󵄨 󵄨 𝑑1󸀠 𝑑1 + 𝑑2󸀠 𝑑2 + 𝑑1 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 = 0.

(13)

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 (󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 + 𝑑2 ⋅ 𝑘 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 + 𝑑1󸀠 ) 𝐴 = (𝑑1 ⋅ 𝑘 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 − 𝑑2󸀠 ) 𝐵

(ii) When |𝑑1 | ≥ 𝛿 > 0, from (11) it follows that 𝑘=

󸀠 1 𝑑2 ⋅ 󵄨󵄨 󸀠 󵄨󵄨 . 𝑑1 󵄨󵄨r 󵄨󵄨

(14) Note that

Substituting it into (10), we also have 𝑑1󸀠 𝑑1

+

𝑑2󸀠 𝑑2

󵄨 󵄨 + (𝑑1 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 + 𝑑1󸀠 𝑑1 + 𝑑2󸀠 𝑑2 ) . (20)

󵄨 󵄨 + 𝑑1 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 = 0.

(15)

󵄨 󵄨 𝑑1 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 + 𝑑1󸀠 𝑑1 + 𝑑2󸀠 𝑑2

󵄨 󵄨 󵄨 󵄨 = 𝑑1 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 + 𝑑1󸀠 𝑑1 + 𝑑2 (𝛽 + 𝑑1 ⋅ 𝑘 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨)

Hence we prove Theorem 2.

(21)

= 𝑑1 𝛼 + 𝑑2 𝛽,

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 r𝑜 = r + 𝑑1 e + 𝑑2 n 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 .

and we have 𝛼 (𝑑1 − 𝐴) + 𝛽 (𝑑2 − 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,

Proof. Let r𝑜 = r + 𝑑1 e + 𝑑2 n

2

= (r + 𝐴e + 𝐵n) + (𝑑1 − 𝐴) e + (𝑑2 − 𝐵) n ≜ r1 + 𝑑n1 ,

(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

where 𝐴 and 𝐵 are the functions of 𝑡, 2

2

(𝑑1 − 𝐴) + (𝑑2 − 𝐵) = 𝐶1 ,

(16)

2

𝑑 = √(𝑑1 − 𝐴) + (𝑑2 − 𝐵) ,

2

𝐵 = 𝑑2 ± √𝐶1 − (𝑑1 − 𝑔 (𝑡)) .

𝑑 −𝐴 𝑑 −𝐵 n1 = 1 e+ 2 n, 𝑑 𝑑 r󸀠1 = r󸀠 + 𝐴󸀠 e + 𝐴e󸀠 + 𝐵󸀠 n + 𝐵n󸀠

(17)

󵄨 󵄨 󵄨 󵄨 = (󵄨󵄨󵄨󵄨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 vectors 𝑑n1 and r󸀠1 must be zero. That is, 󵄨 󵄨 󵄨 󵄨 (𝑑1 − 𝐴) (󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 + 𝐴󸀠 + 𝐵 ⋅ 𝑘 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨)

󵄨 󵄨 + (𝑑2 − 𝐵) (𝐵󸀠 − 𝐴 ⋅ 𝑘 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨) = 0.

(2) When 𝛼 ≠ 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) When 𝛼 = 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) When 𝛼 ≠ 0 and 𝛽 ≠ 0, by solving (19) and (22), we get

(18) 𝐴 = 𝑑1 −

(ii) (𝑑1 − 𝐴)2 + (𝑑2 − 𝐵)2 must be constant. That is, (𝑑1 − 𝐴) (𝑑1󸀠 − 𝐴󸀠 ) + (𝑑2 − 𝐵) (𝑑2󸀠 − 𝐵󸀠 ) = 0.

(24)

(19)

𝛽𝐶 √𝛼2 + 𝛽2

,

𝐵 = 𝑑2 +

where 𝐶 is an arbitrary constant.

𝛼𝐶 √𝛼2 + 𝛽2

,

(25)

4


Therefore, in any case there exist two functions 𝐴 and 𝐵 to guarantee that the generalized offset r𝑜 = r + 𝑑1 e + 𝑑2 n can be expressed as the standard offset r𝑜 = r1 + 𝑑n1 , where r1 = r + 𝐴e + 𝐵n = r + (𝑑1 −

𝛽𝐶 √𝛼2 + 𝛽2

) e + (𝑑2 +

𝛼𝐶 √𝛼2 + 𝛽2

(i) Evolute ) n,

𝛽 √ 𝛼2

+

𝛽2

𝛼

e−

√ 𝛼2

𝛽2

+

We construct r𝜀 (𝑡) = r1 (𝑡) − 𝜌1 (𝑡) ⋅ n1 ,

r𝑜 (𝜏) = r1 (𝜏) + 𝑑n1 = r1 (𝜏) − 𝜌1 n1 = r𝜀 (𝜏) .

n.

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. Let 𝜆 = |𝑟󸀠 |+𝐴󸀠 + 𝐵⋅𝑘⋅|𝑟󸀠 | and 𝜔 = 𝐵󸀠 −𝐴⋅𝑘⋅|𝑟󸀠 |. From the expression of standard offsets r𝑜 = r1 + 𝑑n1 , where n1 =

𝑑1 − 𝐴 𝑑 −𝐵 e+ 2 n, 𝑑 𝑑

(27)

𝑑 = const, and 𝑑1 , 𝑑2 , 𝐴, 𝐵 are the functions of 𝑡, we have r󸀠1 = 𝜆e + 𝜔n, 𝑘1 =

󵄨 󵄨 (𝜆󸀠 𝜔 − 𝜆𝜔󸀠 ) + (𝜆2 + 𝜔2 ) ⋅ 𝑘 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 (𝜆2

+

r󸀠 𝜆e + 𝜔n , e1 = 󵄨󵄨 1󸀠 󵄨󵄨 = 󵄨󵄨r1 󵄨󵄨 √𝜆2 + 𝜔2

(32)

On the other hand, we have (26)

r1 = r + 𝐴e + 𝐵n,

(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

2 2 𝑑 = √(𝑑1 − 𝐴) + (𝑑2 − 𝐵) = |𝐶| ,

n1 =

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.

3/2 𝜔2 )

n1 =

,

(28)

𝜆n − 𝜔e . √𝜆2 + 𝜔2

Moreover r󸀠𝑜 = (1 + 𝑑𝑘1 ) ⋅ √𝜆2 + 𝜔2 ⋅ e1 , 1 𝑘𝑜 = 󵄨󵄨 󵄨 𝑘1 , 󵄨󵄨1 + 𝑑𝑘1 󵄨󵄨󵄨

𝑑1 − 𝐴 (1 + 𝑑𝑘1 ) 𝑑 − 𝐵 (1 + 𝑑𝑘1 ) e− 2 n. 𝑑𝑘1 𝑑𝑘1

(33)

Moreover, from r𝑜 −

1 1 n𝑜 = r1 − n1 , 𝑘𝑜 𝑘1

(34)

we can get the relations as follows: 𝑑1 + 𝑑2 −

𝛽 𝑘𝑜 √𝛼2 + 𝛽2 𝛼 𝑘𝑜 √𝛼2 + 𝛽2

=𝐴+

=𝐵−

𝜔 , √ 𝑘1 𝜆2 + 𝜔2 𝜆 𝑘1 √𝜆2 + 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 r𝑜 (𝑡) are, respectively, in one-to-one correspondence to those on r1 (𝑡); (2) the turning point on r𝑜 (𝑡) is in one-to-one correspondence to that on r(𝑡) as 𝜔 = 0; (3) the inflection on r𝑜 (𝑡) is in one-to-one correspondence to that on r(𝑡) as 𝜆󸀠 𝜔 = 𝜆𝜔󸀠 . Proof. Based upon the following relationships

(29)

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 󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨 󵄨󵄨r𝑜 󵄨󵄨 = 󵄨󵄨1 + 𝑑𝑘1 󵄨󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨r󸀠1 󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨1 + 𝑑𝑘1 󵄨󵄨󵄨󵄨 ⋅ √𝜆2 + 𝜔2 , 󵄨 󵄨 󵄨 󵄨

r𝜀 = r −

(30)

e𝑜 = sgn (1 + 𝑑𝑘1 ) ⋅

𝜆e + 𝜔n , √𝜆2 + 𝜔2

󵄨 󵄨 (𝜆󸀠 𝜔 − 𝜆𝜔󸀠 ) + (𝜆2 + 𝜔2 ) ⋅ 𝑘 ⋅ 󵄨󵄨󵄨󵄨𝑟󸀠 󵄨󵄨󵄨󵄨 1 𝑘𝑜 = 󵄨󵄨 , 󵄨⋅ 3/2 󵄨󵄨1 + 𝑑𝑘1 󵄨󵄨󵄨 (𝜆2 + 𝜔2 ) 𝑑𝑘𝑜 −3 𝑑𝑘 = (1 + 𝑑𝑘1 ) ⋅ 1 , 𝑑𝑠𝑜 𝑑𝑠1 we can easily prove Theorem 4.

(36)


5 Therefore 𝑑𝑀 = 𝑑𝑆1 + 𝑑𝑆2 , and ro (t + Δt)

0

dS2

ro (t)

1

𝑀 = ∫ 𝑑𝑀 =

1 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󸀠 󵄨󵄨 [∫ 󵄨𝑑 󵄨 ⋅ 󵄨󵄨r 󵄨󵄨 𝑑𝑡 2 0 󵄨 2󵄨 󵄨 󵄨 1 󵄨 󵄨 󵄨 + ∫ 󵄨󵄨󵄨󵄨𝑑1󸀠 𝑑2 − 𝑑2󸀠 𝑑1 + 𝑑2 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 0

dS1 r(t)

(41)

󵄨 󵄨󵄨 + (𝑑12 + 𝑑22 ) ⋅ 𝑘 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑑𝑡] .

r(t + Δt)

Figure 1: The area element between r(𝑡) and r𝑜 (𝑡).

(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: 󵄨 󵄨 𝛿 (𝑄, r1 ) = inf 󵄨󵄨󵄨𝑄 − r1 (𝑡)󵄨󵄨󵄨 . 𝑡∈[0,1]

(iii) Length and area We can calculate the lengths 𝑙1 and 𝑙𝑜 of the curves r1 and r𝑜 , respectively. Since 𝑑𝑙1 = |r󸀠1 |𝑑𝑡, then

(42)

For the standard offset r𝑜 = r1 + 𝑑n1 , we have the following theorm. Theorem 5. The distance 𝛿(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:

1

1 󵄨 󵄨 𝑙1 = ∫ 󵄨󵄨󵄨󵄨r󸀠1 󵄨󵄨󵄨󵄨 𝑑𝑡 = ∫ √𝜆2 + 𝜔2 𝑑𝑡, 0 0 1

1 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑙𝑜 = ∫ 󵄨󵄨󵄨󵄨r󸀠𝑜 󵄨󵄨󵄨󵄨 𝑑𝑡 = ∫ 󵄨󵄨󵄨1 + 𝑑𝑘1 󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨󵄨r󸀠1 󵄨󵄨󵄨󵄨 𝑑𝑡. 0 0

(37)

The area between r(𝑡) and its generalized offset r𝑜 (𝑡) is denoted by 𝑀 (Figure 1). 𝑑𝑆1 and 𝑑𝑆1 are the area elements. See Figure 4. At first, we compute 󵄨 󵄨󵄨 󵄨[r (𝑡) − r (𝑡)] × [r (𝑡 + Δ𝑡) − r (𝑡)]󵄨󵄨󵄨 , 𝑑𝑆1 = 󵄨 𝑜 2 󵄨 󵄨󵄨 󵄨[r (𝑡) − r𝑜 (𝑡 + Δ𝑡)] × [r (𝑡 + Δ𝑡) − r𝑜 (𝑡 + Δ𝑡)]󵄨󵄨󵄨 𝑑𝑆2 = 󵄨 𝑜 . 2 (38)

𝛿 (r𝑜 (𝜏) , r) = |𝑑| + √𝐴2 + 𝐵2 ,

𝜏 ∈ (𝑖𝑘 , 𝑖𝑘+1 ) ,

𝛿 (r𝑜 (𝜏) , r) < |𝑑| + √𝐴2 + 𝐵2 ,

𝜏 ∈ (𝑖𝑘 , 𝑖𝑘+1 ) ,

(43)

𝑁 ∈ 𝑍+ .

𝑘 = 0, . . . , 𝑁,

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

Since 󸀠

2

r (𝑡 + Δ𝑡) = r (𝑡) + Δ𝑡 ⋅ r (𝑡) + 𝑂 (Δ𝑡 ) , r𝑜 (𝑡 + Δ𝑡) = r𝑜 (𝑡) + Δ𝑡 ⋅ r󸀠𝑜 (𝑡) + 𝑂 (Δ𝑡2 ) ,

(39)

it follows that 󵄨 󵄨󵄨 󵄨󵄨[𝑑1 e + 𝑑2 n] × [Δ𝑡 ⋅ r󸀠 (𝑡)]󵄨󵄨󵄨 1 󵄨 󵄨 󵄨 󸀠 󵄨 󵄨 = 󵄨󵄨𝑑 󵄨󵄨 ⋅ 󵄨󵄨r 󵄨󵄨 𝑑𝑡, 󵄨 𝑑𝑆1 = 󵄨 󵄨 2 2 󵄨 2󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨[Δ𝑡 ⋅ r󸀠𝑜 (𝑡)] × [r (𝑡) − r𝑜 (𝑡) + Δ𝑡 (r󸀠 (𝑡) − r󸀠𝑜 (𝑡))]󵄨󵄨󵄨 󵄨 󵄨 𝑑𝑆2 = 2 1 󵄨 󵄨 󵄨 󵄨󵄨 󵄨 = ⋅ Δ𝑡 ⋅ 󵄨󵄨󵄨󵄨𝑑1󸀠 𝑑2 − 𝑑2󸀠 𝑑1 + 𝑑2 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨 + (𝑑12 + 𝑑22 ) ⋅ 𝑘 ⋅ 󵄨󵄨󵄨󵄨r󸀠 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 . 2 (40)

𝛿 (r𝑜 (𝜏) , r1 ) ≡ |𝑑| ,

𝜏 ∈ (𝑖𝑘 , 𝑖𝑘+1 ) ,

𝛿 (r𝑜 (𝜏) , r1 ) < |𝑑| ,

𝜏 ∈ (𝑖𝑘 , 𝑖𝑘+1 ) ,

(44)

𝑘 = 0, . . . , 𝑁 For the generalized offset r0 (𝑡), we have 󵄨 󵄨󵄨 󵄨󵄨r1 (𝑡) − r (𝑡)󵄨󵄨󵄨 = √𝐴2 + 𝐵2 ,

𝑡 ∈ [0, 1] .

(45)

Considering 𝜏 ∈ (𝑖𝑘 , 𝑖𝑘+1 ), let 𝑝 ∈ r1 such that 󵄨󵄨 󵄨 󵄨󵄨r𝑜 (𝜏) − 𝑝󵄨󵄨󵄨 = 𝛿 (r𝑜 (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)

6

Journal of Applied Mathematics 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]

ro ro (𝜏)

𝛿

r (𝑢, V) e1 = 󵄨󵄨 𝑢 󵄨, 󵄨󵄨r𝑢 (𝑢, V)󵄨󵄨󵄨

p

r (𝑢, V) × rV (𝑢, V) n = 󵄨󵄨 𝑢 󵄨, 󵄨󵄨r𝑢 (𝑢, V) × rV (𝑢, V)󵄨󵄨󵄨

√A2 + B2

q r

Figure 2: Topological property of the curves.

Definition 6. For a regular smooth parametric surface r(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

Thus 󵄨 󵄨 𝛿 (r𝑜 (𝜏) , r) ≤ 󵄨󵄨󵄨r𝑜 (𝜏) − r󵄨󵄨󵄨 ≤ 𝛿 (r𝑜 (𝜏) , r1 ) + √𝐴2 + 𝐵2 . (48) Therefore we prove Theorem 5, which is shown in Figure 2. According to Theorem 5, 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 curve r(𝑡) are 𝑑n = 𝑘e + 𝜏z, 𝑑𝑠

𝑑z = −𝜏n, 𝑑𝑠

r𝑜 (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 𝑑1 e1 , 𝑑2 e2 , and 𝑑3 n. For a regular smooth parametric surface r(𝑢, V), the wellknown first and second fundamental quantities and the Gauss curvature [14] are given as follows: 𝐸 = r2𝑢 = 𝐴21 , 𝐷 = 𝑛 ⋅ r𝑢𝑢 ,

𝐺 = r2V = 𝐴22 , 𝐷󸀠 = 𝑛 ⋅ r𝑢V , 𝑘=

(49)

󸀠󸀠

𝐹 = r𝑢 ⋅ rV , 𝐷󸀠󸀠 = 𝑛 ⋅ rVV ,

(53)

󸀠2

𝐷𝐷 − 𝐷 . 𝐸𝐺 − 𝐹2

Let e1 × n = e󸀠1 , e2 × n = e󸀠2 , and the angle between e1 and e2 is 𝜃; then

where 𝜏 is the torsion, and we have (r󸀠 × r󸀠󸀠 ) ⋅ r󸀠󸀠󸀠 . 𝜏= 󵄨 󵄨󵄨r󸀠 × r󸀠󸀠 󵄨󵄨󵄨2 󵄨 󵄨

(51)

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.

r1

𝑑e = −𝑘n, 𝑑𝑠

r (𝑢, V) e2 = 󵄨󵄨 V 󵄨, 󵄨󵄨rV (𝑢, V)󵄨󵄨󵄨

(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.

e1 × e2 = n sin 𝜃, e󸀠1 ⋅ e2 = − sin 𝜃,

e󸀠2 ⋅ e1 = sin 𝜃.

(54)

Thus the related parametric partial derivatives of generalized offset surface r𝑜 (𝑢, V) can be obtained by r𝑜𝑢 (𝑢, V) = r𝑢 + 𝑑1𝑢 e1 + 𝑑1 e1𝑢 + 𝑑2𝑢 e2 + 𝑑2 e2𝑢 + 𝑑3𝑢 n + 𝑑3 n𝑢 = [𝐴 1 + 𝑑1𝑢 +

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.

e󸀠1 × e󸀠2 = n sin 𝜃,

+ [𝑑2𝑢 −

𝑑2 𝐴 1V 𝑑3 𝐴 1 − ] e1 𝐴2 𝑅1

𝑑1 𝐴 1V 𝑑𝐴 ] e2 + [𝑑3𝑢 + 1 1 ] n 𝐴2 𝑅1

≜ 𝐵1 e1 + 𝐵2 e2 + 𝐵3 n,

(55)


7

r𝑜V (𝑢, V) = rV + 𝑑1V e1 + 𝑑1 e1V + 𝑑2V e2

Thus the unit tangent vectors and unit normal vector of surface offsets r𝑜 (𝑢, V) are given as follows:

+ 𝑑2 e2V + 𝑑3V n + 𝑑3 nV = [𝑑1V −

r𝑜 (𝑢, V) 𝐵1 e1 + 𝐵2 e2 + 𝐵3 n 𝑒1𝑜 = 󵄨󵄨 𝑢𝑜 , 󵄨󵄨 = 󵄨󵄨r𝑢 (𝑢, V)󵄨󵄨 √𝐸𝑜

𝑑2 𝐴 2𝑢 ] e1 𝐴1

+ [𝐴 2 +

𝑑1 𝐴 2𝑢 𝑑𝐴 + 𝑑2V − 3 2 ] e2 𝐴1 𝑅2

+ [𝑑3V +

𝑑2 𝐴 2 ]n 𝑅2

(56)

r𝑜𝑢 × r𝑜V r𝑜 × r𝑜V = n𝑜 = 󵄨󵄨 𝑢𝑜 󵄨 󵄨󵄨r𝑢 × r𝑜V 󵄨󵄨󵄨 √𝐸 𝐺 − 𝐹2 𝑜 𝑜 𝑜

≜ 𝐶1 e1 + 𝐶2 e2 + 𝐶3 n, where r𝑢 , 𝑑1𝑢 , 𝑑2𝑢 , 𝐴 2𝑢 , e1𝑢 , e2𝑢 , n𝑢 are the corresponding partial derivatives of r, 𝑑1 , 𝑑2 , 𝐴 2 , e1 , e2 , n with respect to 𝑢 and rV , 𝑑1V , 𝑑2V , 𝐴 1V , e1V , e2V , nV are the corresponding partial derivatives of r, 𝑑1 , 𝑑2 , 𝐴 1 , 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: r𝑜𝑢𝑢 (𝑢, V) = 𝐵1𝑢 e1 + 𝐵1 e1𝑢 + 𝐵2𝑢 e2 + 𝐵2 e2𝑢 + 𝐵3𝑢 n + 𝐵3 n𝑢 𝐴 𝐴 = (𝐵1𝑢 + 𝐵2 1V − 𝐵3 1 ) e1 𝐴2 𝑅1 𝐴 + (−𝐵1 1V + 𝐵2𝑢 ) 𝐴2 𝐴 + (𝐵1 1 + 𝐵3𝑢 ) n, 𝑅1 𝑜 𝑜 r𝑢V (𝑢, V) = rV𝑢 (𝑢, V) = 𝐵1V e1 + 𝐵1 e1V + 𝐵2V e2 + 𝐵2 e2V + 𝐵3V n + 𝐵3 nV 𝐴 = (𝐵1V − 𝐵2 2𝑢 ) e1 𝐴1 𝐴 2𝑢 𝐴 + (𝐵1 + 𝐵2V − 𝐵3 2 ) e2 𝐴1 𝑅2 𝐴2 + (𝐵2 + 𝐵3V ) n, 𝑅2 r𝑜VV (𝑢, V) = 𝐶1V e1 + 𝐶1 e1V + 𝐶2V e2 + 𝐶2 e2V + 𝐶3V n + 𝐶3 nV 𝐴 = (𝐶1V − 𝐶2 2𝑢 ) e1 𝐴1

r𝑜 (𝑢, V) 𝐶1 e1 + 𝐶2 e2 + 𝐶3 n e𝑜2 = 󵄨󵄨 V𝑜 , 󵄨= 󵄨󵄨rV (𝑢, V)󵄨󵄨󵄨 √𝐺𝑜

=

𝐺𝑜 = 𝐹𝑜 =

r𝑜𝑢

=

×

√𝐸𝑜 𝐺𝑜 − 𝐹𝑜2 +

+ [𝑀1 (−𝐵1

+ 𝑀2 (𝐵1

𝐴 2𝑢 ) 𝐴1

𝐴 2𝑢 𝐴 𝐴 + 𝐵2V − 𝐵3 2 ) + 𝑀3 (𝐵2 2 + 𝐵3V ) 𝐴1 𝑅2 𝑅2

+ [𝑀1 (𝐵1

(57)

𝐴 2𝑢 𝐴 + 𝐵2V − 𝐵3 2 ) 𝐴1 𝑅2

+𝑀2 (𝐵1V − 𝐵2

𝐴 2𝑢 )] cos 𝜃, 𝐴1

𝐷𝑜󸀠󸀠 = n𝑜 r𝑜VV = 𝑀1 (𝐶1V − 𝐶2

𝑘𝑜 =

𝐴 2𝑢 ) 𝐴1

𝐴 2𝑢 𝐴 𝐴 + 𝐶2V − 𝐶3 2 ) + 𝑀3 (𝐶2 2 + 𝐶3V ) 𝐴1 𝑅2 𝑅2 𝐴 2𝑢 𝐴 + 𝐶2V − 𝐶3 2 ) 𝐴1 𝑅2

+𝑀2 (𝐶1V − 𝐶2

+ 2𝐶1 𝐶2 cos 𝜃,

= 𝐵1 𝐶1 + 𝐵2 𝐶2 + 𝐵3 𝐶3 + (𝐵1 𝐶2 + 𝐵2 𝐶1 ) cos 𝜃.

𝐴 1V 𝐴 − 𝐵3 1 )] cos 𝜃, 𝐴2 𝑅1

𝐷𝑜󸀠 = n𝑜 r𝑜𝑢V = 𝑀1 (𝐵1V − 𝐵2

+ [𝑀1 (𝐶1

+

𝐴 1V + 𝐵2𝑢 ) 𝐴2

+𝑀2 (𝐵1𝑢 + 𝐵2

𝐴2 + 𝐶3V ) n, 𝑅2

+

𝐴 1V 𝐴 − 𝐵3 1 ) 𝐴2 𝑅1

𝐴 1V 𝐴 + 𝐵2𝑢 ) + 𝑀3 (𝐵1 1 + 𝐵3𝑢 ) 𝐴2 𝑅1

+ 𝑀2 (−𝐵1

+ (𝐶2

r𝑜V

e󸀠2

n

𝐷𝑜 = n𝑜 r𝑜𝑢𝑢 = 𝑀1 (𝐵1𝑢 + 𝐵2

+ 𝑀2 (𝐶1

𝐶32

√𝐸𝑜 𝐺𝑜 − 𝐹𝑜2

√𝐸𝑜 𝐺𝑜 − 𝐹𝑜 2

𝐴 2𝑢 𝐴 + 𝐶2V − 𝐶3 2 ) e2 𝐴1 𝑅2

𝐶22

𝐵2 𝐶3 − 𝐵3 𝐶2

(𝐵1 𝐶2 − 𝐵2 𝐶1 ) sin 𝜃

+ (𝐶1

𝐶12

e󸀠1 +

≜ 𝑀1 e1 + 𝑀2 e2 + 𝑀3 n,

2 2 2 𝐸𝑜 = r𝑜2 𝑢 = 𝐵1 + 𝐵2 + 𝐵3 + 2𝐵1 𝐵2 cos 𝜃, 2 r𝑜V

𝐵1 𝐶3 − 𝐵3 𝐶1

𝐷𝑜 𝐷𝑜󸀠󸀠 − 𝐷𝑜󸀠2 , 𝐸𝑜 𝐺𝑜 − 𝐹𝑜2

𝐴 2𝑢 )] cos 𝜃, 𝐴1

(58)

8


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). Let r𝑜 (𝑢0 , V0 ) be a nonregular point of r𝑜 (𝑢, V); then 󵄨󵄨 𝑜 󵄨 𝑜 󵄨󵄨r𝑢 (𝑢0 , V0 ) × rV (𝑢0 , V0 )󵄨󵄨󵄨 = 0.

(59)

Proof. Let r𝑜 = r + 𝑑1 e1 + 𝑑2 e2 + 𝑑3 n = (𝑟 + 𝑀e1 + 𝑁e2 + 𝑃n) + (𝑑1 − 𝑀) e1 + (𝑑2 − 𝑁) e2 + (𝑑3 − 𝑃) n

(62)

≜ r1 + 𝑑n1 , where 𝑀, 𝑁, and 𝑃 are the functions of 𝑢 and V,

Note that

2

2

2

𝑑 = √(𝑑1 − 𝑀) + (𝑑2 − 𝑁) + (𝑑3 − 𝑃) ,

r𝑜𝑢 × r𝑜V = (𝐵1 𝐶3 − 𝐵3 𝐶1 ) e󸀠1 + (𝐵2 𝐶3 − 𝐵3 𝐶2 ) e󸀠2 + (𝐵1 𝐶2 − 𝐵2 𝐶1 ) n sin 𝜃

(60)

≜ 𝜌1 e󸀠1 + 𝜌2 e󸀠2 + 𝜌3 n sin 𝜃.

Theorem 7. If (𝑢0 , V0 ) +

𝜌22

(𝑢0 , V0 ) +

𝜌32

(63)

r1𝑢 (𝑢, V) = r𝑢 + 𝑀𝑢 e1 + 𝑀e1𝑢 + 𝑁𝑢 e2

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

(𝜌12

(𝑑 − 𝑀) (𝑑 − 𝑁) (𝑑 − 𝑃) n1 = 1 e1 + 2 e2 + 3 . 𝑑 𝑑 𝑑 The parametric partial derivatives of surface r1 (𝑢, V) are

= [𝐴 1 + 𝑀𝑢 + 𝑁 + [𝑁𝑢 − 𝑀

(𝑢0 , V0 )

1/2 󵄨 󵄨 +2 󵄨󵄨󵄨𝜌1 (𝑢0 , V0 ) 𝜌2 (𝑢0 , V0 )󵄨󵄨󵄨 cos 𝜃) = 0,

+ 𝑁e2𝑢 + 𝑃𝑢 n + 𝑃n𝑢

(61)

then r𝑜 (𝑢0 , V0 ) is a nonregular point of r𝑜 (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, ̃V) to make the new local natural coordinate system (eũ , ẽv , 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 r𝑜 = r + 𝑑1 e1 + 𝑑2 e2 + 𝑑3 n 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 .

𝐴 1V 𝐴 − 𝑃 1 ] e1 𝐴2 𝑅1

(64)

𝐴 1V 𝐴 ] e2 + [𝑃𝑢 + 𝑀 1 ] n, 𝐴2 𝑅1

r1V (𝑢, V) = rV + 𝑀V e1 + 𝑀e1V + 𝑁V e2 + 𝑁e2V + 𝑃V n + 𝑃nV = [𝑀V − 𝑁

𝐴 2𝑢 ]e 𝐴1 1

(65)

𝐴 𝐴 + [𝐴 2 + 𝑀 2𝑢 + 𝑁V − 𝑃 2 ] e2 𝐴1 𝑅2 + [𝑃V + 𝑁

𝐴2 ] n, 𝑅2

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, (𝐴 1 + 𝑀𝑢 + 𝑁

𝐴 1V 𝐴 − 𝑃 1 ) (𝑑1 − 𝑀) 𝐴2 𝑅1

+ (𝑁𝑢 − 𝑀 + (𝑃𝑢 + 𝑀 (𝑀V − 𝑁

𝐴 1V ) (𝑑2 − 𝑁) 𝐴2

(66)

𝐴1 ) (𝑑3 − 𝑃) = 0, 𝑅1

𝐴 2𝑢 ) (𝑑1 − 𝑀) 𝐴1

+ (𝐴 2 + 𝑀 + (𝑃V + 𝑁

𝐴 2𝑢 𝐴 + 𝑁V − 𝑃 2 ) (𝑑2 − 𝑁) 𝐴1 𝑅2

𝐴2 ) (𝑑3 − 𝑃) = 0. 𝑅2

(67)


9

(ii) 𝑑 is constant. That is, 2

ro (u + Δu, )

2

2

(𝑑1 − 𝑀) + (𝑑2 − 𝑁) + (𝑑3 − 𝑃) = const.

ro (u + Δu, + Δ)

(68)

It follows that (𝑑1 − 𝑀) (𝑑1𝑢 − 𝑀𝑢 ) + (𝑑2 − 𝑁) (𝑑2𝑢 − 𝑁𝑢 ) + (𝑑3 − 𝑃) (𝑑3𝑢 − 𝑃𝑢 ) = 0, (𝑑1 − 𝑀) (𝑑1V − 𝑀V ) + (𝑑2 − 𝑁) (𝑑2V − 𝑁V ) + (𝑑3 − 𝑃) (𝑑3V − 𝑃V ) = 0.

ro (u, + Δ)

(69)

Figure 3: The area element of r𝑜 (𝑢, V).

(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 (−𝐴 1 −

ro (u, )

where 𝐶 is an arbitrary constant, 𝛼, 𝛽, and 𝛾 can not all be zero. Thus 2

𝐴 1V 𝑑2 𝐴 𝑑 𝐴 𝑑 − 𝑑1𝑢 + 1 3 ) 𝑀 + ( 1V 1 − 𝑑2𝑢 ) 𝑁 𝐴2 𝑅1 𝐴2

n1 =

𝐴 𝑑 + (− 1 1 − 𝑑3𝑢 ) 𝑃 𝑅1

(71) From (67) and (70), we get (

𝐴 2𝑢 𝑑2 𝐴 𝑑 𝐴 𝑑 − 𝑑1V ) 𝑀 + (− 2𝑢 1 − 𝐴 2 − 𝑑2V + 2 3 ) 𝑁 𝐴1 𝐴1 𝑅2 + (−

𝐴 2 𝑑2 − 𝑑3V ) 𝑃 𝑅2

+ (𝐴 2 𝑑2 + 𝑑1V 𝑑1 + 𝑑2V 𝑑2 + 𝑑3V 𝑑3 ) ≜ 𝛼2 𝑀 + 𝛽2 𝑁 + 𝛾2 𝑃 + 𝜔2 = 0. (72) Let 𝛼1 𝛽2 − 𝛼2 𝛽1 ≜ 𝛼,

𝛽2 𝛾1 − 𝛽1 𝛾2 ≜ 𝛽,

𝛼2 𝛾1 − 𝛼1 𝛾2 ≜ 𝛾.

(73)

From (71) and (72), we get 𝛼 (𝑑1 − 𝑀) + 𝛽 (𝑑3 − 𝑃) = 0, −𝛼 (𝑑2 − 𝑁) + 𝛾 (𝑑3 − 𝑃) = 0.

𝑀 = 𝑑1 −

𝑁 = 𝑑2 +

𝑃 = 𝑑3 +

√𝛼2 + 𝛽2 + 𝛾2 𝛾𝐶

(74)

√𝛼2 + 𝛽2 + 𝛾2 𝛼𝐶 √𝛼2 + 𝛽2 + 𝛾2

√𝛼2 + 𝛽2 + 𝛾2

e1 −

𝛼 √𝛼2 + 𝛽2 + 𝛾2

𝛾 √𝛼2 + 𝛽2 + 𝛾2

e2

(76)

n.

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 r𝑜 = r+𝑑1 e1 +𝑑2 e2 +𝑑3 n 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.

The area of generalized offset surface {r𝑜 (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 = 󵄨󵄨󵄨(r𝑜𝑢 × r𝑜V )󵄨󵄨󵄨 𝑑𝑢 𝑑V. (78)

,

,

2

(i) The area of an offset surface

By solving (69), (70), and (74), we get 𝛽𝐶

𝛽

−

+ (𝐴 1 𝑑1 + 𝑑1𝑢 𝑑1 + 𝑑2𝑢 𝑑2 + 𝑑3𝑢 𝑑3 ) ≜ 𝛼1 𝑀 + 𝛽1 𝑁 + 𝛾1 𝑃 + 𝜔1 = 0.

2

𝑑 = √(𝑑1 − 𝑀) + (𝑑2 − 𝑁) + (𝑑3 − 𝑃) = 𝐶,

Since (75)

2

2

𝑜2 𝑜 𝑜 2 (r𝑜𝑢 × r𝑜V ) = r𝑜2 𝑢 rV − (r𝑢 rV ) = 𝐸𝑜 𝐺𝑜 − 𝐹𝑜 ,

(79)

then ,

𝑆=∬

[0,1]×[0,1]

√𝐸𝑜 𝐺𝑜 − 𝐹𝑜2 𝑑𝑢 𝑑V.

(80)

10

Journal of Applied Mathematics ro (u + Δu, + Δ)

ro (u, + Δ)

𝑑𝑉4 =

ro (u, ) ro (u + Δu, ) r(u, + Δ)

𝑑𝑉5 = r(u + Δu, + Δ) r(u + Δu, )

r(u, )

1 󵄨󵄨 󵄨𝑑 (𝐵 𝐶 − 𝐵3 𝐶2 ) + 𝑑2 (𝐵3 𝐶1 − 𝐵1 𝐶3 ) 6󵄨 1 2 3 󵄨 + 𝑑3 (𝐵1 𝐶2 − 𝐵2 𝐶1 )󵄨󵄨󵄨 𝑑𝑢 𝑑V ≜ 𝑤2 𝑑𝑢 𝑑V, 1 󵄨󵄨 󵄨𝐴 𝐶 𝑑 − 𝐴 1 𝐶2 𝑑3 + 𝐴 2 𝐵3 𝑑1 6󵄨 1 3 2 󵄨 −𝐴 2 𝐵1 𝑑3 󵄨󵄨󵄨 𝑑𝑢 𝑑V ≜ 𝑤5 𝑑𝑢 𝑑V. (83)

𝑜

Figure 4: The volume element between r(𝑢, V) and r (𝑢, V).

Therefore, we have 1

𝑉 = ∬ (𝑑𝑉1 + 𝑑𝑉2 + ⋅ ⋅ ⋅ + 𝑑𝑉5 ) 0

=∬

(ii) The volume between r(𝑢, V) and r𝑜 (𝑢, V) The volume between r(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: 𝑑𝑉 = 𝑑𝑉1 + 𝑑𝑉2 + ⋅ ⋅ ⋅ + 𝑑𝑉5 ,

(81)

𝑑𝑉1 = [r (𝑢, V) , r𝑜 (𝑢, V) , r (𝑢 + Δ𝑢, V) , r (𝑢, V + ΔV)] , 𝑑𝑉2 = [r (𝑢 + Δ𝑢, V) , r (𝑢, V) , r (𝑢 + Δ𝑢, V) , r𝑜 (𝑢 + Δ𝑢, V + ΔV)] ,

(ii) Topological property For a regular parameter surface 𝐿 1 = {r1 (𝑢, V) : (𝑢, V) ∈ [0, 1] × [0, 1]} ,

(85)

the distance between a point 𝑄 and the surface 𝐿 1 is defined as follows: 󵄨󵄨󵄨𝑄 − r1 (𝑢, V)󵄨󵄨󵄨 . inf 𝛿 (𝑄, 𝐿 1 ) = (86) 󵄨 󵄨 (𝑢,V)∈[0,1]×[0,1]

𝛿 (r𝑜 (𝜏, 𝜂) , 𝐿) = |𝑑| + √𝑀2 + 𝑁2 + 𝑃2 ,

𝑑𝑉3 = [r (𝑢 + Δ𝑢, V + ΔV) , r (𝑢 + Δ𝑢, V) , 𝑜

(2𝑤1 + 2𝑤2 + 𝑤5 ) 𝑑𝑢 𝑑V.

Theorem 9. The distance 𝛿(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:

𝑜

r𝑜 (𝑢 + Δ𝑢, V + ΔV) , r (𝑢, V + ΔV)] ,

[0,1]×[0,1]

For the standard offset r𝑜 = r1 + 𝑑n1 , 𝑑 = const, we have the following theorem.

where

𝑜

(84)

(82)

𝑜

𝑑𝑉4 = [r (𝑢, V + ΔV) , r (𝑢, V) , r (𝑢, V + ΔV) , r𝑜 (𝑢 + Δ𝑢, V + ΔV)] , 𝑑𝑉5 = [r (𝑢, V + ΔV) , r𝑜 (𝑢, V) , r (𝑢 + Δ𝑢, V) , r𝑜 (𝑢 + Δ𝑢, V + ΔV)] ,

𝛿 (r𝑜 (𝜏, 𝜂) , 𝐿) < |𝑑| + √𝑀2 + 𝑁2 + 𝑃2 ,

(87)

(𝜏, 𝜂) ∈ (𝑖𝑘 , 𝑖𝑘+1 ) × (𝑗𝑘󸀠 , 𝑗𝑘󸀠 +1 ) , 𝑘, 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:

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 = 𝑑𝑉2 =

1 󵄨󵄨 󵄨 󵄨𝐴 𝐴 𝑑 󵄨󵄨 𝑑𝑢 𝑑V ≜ 𝑤1 𝑑𝑢 𝑑V, 6 󵄨 1 2 3󵄨 1 󵄨󵄨 󵄨𝑑 (𝐵 𝐶 − 𝐵2 𝐶3 ) + 𝑑2 (𝐵1 𝐶3 − 𝐵3 𝐶1 ) 6󵄨 1 3 2 󵄨 +𝑑3 (𝐵2 𝐶1 − 𝐵1 𝐶2 )󵄨󵄨󵄨 𝑑𝑢 𝑑V ≜ 𝑤2 𝑑𝑢 𝑑V,

1󵄨 󵄨 𝑑𝑉3 = 󵄨󵄨󵄨𝐴 1 𝐴 2 𝑑3 󵄨󵄨󵄨 𝑑𝑢 𝑑V ≜ 𝑤1 𝑑𝑢 𝑑V, 6

(1) 𝛿(r𝑜 (𝜏, 𝜂), 𝐿 1 ) ≤ |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 𝛿 (r𝑜 (𝜏, 𝜂) , 𝐿 1 ) = |𝑑| ,

(𝜏, 𝜂) ∈ (𝑖𝑘 , 𝑖𝑘+1 ) × (𝑗𝑘󸀠 , 𝑗𝑘󸀠 +1 ) ,

𝛿 (r𝑜 (𝜏, 𝜂) , 𝐿 1 ) < |𝑑| ,

(𝜏, 𝜂) ∈ (𝑖𝑘 , 𝑖𝑘+1 ) × (𝑗𝑘󸀠 , 𝑗𝑘󸀠 +1 ) , 𝑘, 𝑘󸀠 = 0, 1, . . . , 𝑁. (88)


11

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.

ro (u, )

r1 (u, )

r(u, )

Example 10. Generalized offset curves.

ro (𝜏, 𝜂)

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

𝛿 p

√M2 + N2 + P2

r (𝑡) = {cos (𝑡) , sin (𝑡)} ,

q

Figure 5: Topological property of the surfaces.

r1 = r + 𝑀e1 + 𝑁e2 + 𝑃n,

(89)

(90)

Considering (𝜏, 𝜂) ∈ (𝑖𝑘 , 𝑖𝑘+1 ) × (𝑗𝑘󸀠 , 𝑗𝑘󸀠 +1 ), let 𝑝 ∈ 𝐿 1 such that (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 (92)

Thus 󵄨 󵄨 𝛿 (r𝑜 (𝜏, 𝜂) , 𝐿) ≤ 󵄨󵄨󵄨r𝑜 (𝜏, 𝜂) − 𝐿󵄨󵄨󵄨 ≤ 𝛿 (r𝑜 (𝜏, 𝜂) , 𝐿 1 ) + √𝑀2 + 𝑁2 + 𝑃2 .

= {cos (𝑡) − sin (𝑡) ⋅ |sin (2𝑡)| + cos (𝑡) ⋅ |8 ⋅ cos (2𝑡)| ,

(𝑢, V) ∈ [0, 1] × [0, 1] .

󵄨 󵄨󵄨 󵄨󵄨𝑝 − 𝑞󵄨󵄨󵄨 = √𝑀2 + 𝑁2 + 𝑃2 .

(95)

r𝑜 (𝑡) = r1 (𝑡) + 𝑑n1 (𝑡)

r1 (𝑢, V) − r (𝑢, V) = 𝑀e1 + 𝑁e2 + 𝑃n,

󵄨󵄨 𝑜 󵄨 𝑜 󵄨󵄨r (𝜏, 𝜂) − 𝑝󵄨󵄨󵄨 = 𝛿 (r (𝜏, 𝜂) , 𝐿 1 ) ,

r𝑜 (𝑡) = r (𝑡) + 𝑑1 (𝑡) e (𝑡) + 𝑑2 n (𝑡) , where we take 𝑑1 = | sin(2𝑡)| and 𝑑1 = |8 ⋅ cos(2𝑡)|. Thus the generalized offset r𝑜 (𝑡) is

we have

󵄨 󵄨󵄨 󵄨󵄨r1 (𝑢, V) − r (𝑢, V)󵄨󵄨󵄨 = √𝑀2 + 𝑁2 + 𝑃2 ,

(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

For the generalized offset r𝑜 = r + 𝑑1 e1 + 𝑑2 e2 + 𝑑3 n = r1 + 𝑑n1 ,

𝑡 ∈ [0, 2𝜋] .

(93)

Therefore we prove the above theorem, which is illustrated in Figure 5. According to Theorem 9, 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.

(96)

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. 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. 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

12


5

5

−5

−5

Figure 6: Generalized offset curve. Figure 8: Generalized offset surface with the fixed offset direction.

Figure 9: Generalized offset surface with the variable offset direction.

Figure 7: Flower cloth.

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.

References [1] H. Shen, J. Fu, Z. Chen, and Y. Fan, “Generation of o set surface for tool path in nc machining through level set methods,”


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