Study of Second-Order Continuities of a Composite Surface

There are two classes of continuities, parametric continuities and geometric continuities, which are used to illuminate the smoothness of a composite surface in surface construction and reconstruction in computer graphics (CG) and computer aided design (CAD). A parametric continuity is more stiff than its corresponding geometric continuity of the same order. This paper uncovers the geometric properties of parametric and geometric continuities less than and equal to second order and presents the proofs for the corresponding propositions. These propositions can be applied to the existent or promising schemes of surface construction or reconstruction, which can provide a convincing theory for researchers to establish their schemes in surface construction. Three examples are used in this paper to show the applications of these propositions.


Introduction
It is known that there are a lot of applications of surface construction or recon- The process of surface construction or reconstruction usually starts from sampling discrete points (vertices) on a target surface that is studied by researchers. The results of surface construction and reconstruction in computer are often composite surfaces which are patches glued together. Researchers create their schemes to construct or reconstruct a composite surface via computer technology to approximate the target surface. Then the constructed or reconstructed surfaces can be applied to education, researches, and management in medicine, biology, geography, mechanics, electrical engineering, science, business, entertainment, and others.
There are two classes of continuities characterized by many researchers to examine how smooth a composite surface is, these being called parametric continuities and geometric continuities [3] [4] [5] [6] [7]. Following the studies of the predecessors [3] [4] [5] [6] [7], this paper provides a further description on the continuities of r th order, 2 r ≤ , and presents different mathematical features of these continuities.
The required proofs are given along with the discussion, which can verify the study in this paper. The features presented in this paper can be directly applied to examining how smooth a surface can be constructed or reconstructed by an existent or promising scheme. It can save the time and effort for researchers to look for a convincing theory of surface continuities to make their new methods established.
Related studies are given in Section 2. A system of surface construction has to be defined first in the context of algebraic topology and differential geometry in Section 3. After uni-parametric representations of continuities are given in Section 4, bi-parametric formulae of geometric continuities are further presented in Section 5. In Section 6, properties of first-order continuities are discussed. In Section 7, properties of second-order continuities are further extended. In Section 8, three examples are used to show how to apply the theory of this paper. Section 9 is results and conclusions.

Related Studies
In CG, CAD, CAGD and other applications of surface construction and reconstruction, researchers work hard to construct composite surfaces as smooth as needed in their applications [3] [4] [5] [6] [7]. Some researchers have noticed that parametric continuities were usually too stiff to meet for a scheme of surface construction. For this reason, Barsky and DeRose [3] [4] devised the Beta-spline surfaces that meet the geometric continuities rather than parametric continuities. According to the studies [3] [4] [5] [6] [7], parametric continuities can shrink the set of parameterizations by excluding ones that can generate geometrically smooth curves or surfaces. The reason for this is that the condition of parametric continuity of a given order is stricter than one of the geometric continuity of the same order, which will be discussed in this paper.
Researchers in applications of surface construction and reconstruction usually On the other hand, the existent theory of algebraic topology [8] has established the mathematics foundation for the study of this paper, which can be used to present the definition of a system of surface construction in a mathematical way, which will be introduced in Section 3. In this system, our further discussion will progress to the geometric properties of these two classes of continuities. To support the further discussion in this paper, the theories of modern geometry [9] and differential geometry [10] [11] must be introduced. In this way, the study of this paper can bridge the gap between the surface construction and the related mathematics theories.
With the above purpose and theories, this paper uncovers the geometric properties of these two classes of continuities of less than and equal to second order and presents the proofs for the corresponding propositions. These propositions can be applied to the existent or promising schemes of surface construction or reconstruction, which can provide a convincing theory for researchers to establish their schemes in surface construction and reconstruction.

System of Surface Construction
In order to follow the researches of predecessors mentioned above and save the extra effort in the existent proofs, we put our discussion in the context of the related theory that has been established. A system of surface construction or reconstruction is defined in algebraic topology firstly here. From the point of view of constructing a surface from a network of sampled vertices with the computer technology, surface reconstruction can be included in surface construction.
Thus, we will substitute surface construction or reconstruction with surface construction in the rest of this paper. This paper will develop its deduction on the basis of the existent theory of algebraic topology, especially on the treatise of Massey [8]. Therefore, the definitions here will start from this basis. It is also required to mention that this paper will focus on surfaces that are orientable and locally connected since these types of surfaces are ones usually studied in CG, CAD, CAGD, and their related ap- Two points, { } , a b , the images of the end points of the interval I denote the end points of the segment, and the segment f is called to join its end points. The segment f is orientable so that a , the point from which the segment starts, is called initial point and b, the point which the segment reaches, is called terminal point. Definition 2. A surface S is called segment-wise connected if any two points of S can be joined by a segment. A segment-wise connected surface is connected. The segment components of S are the maximal segment-wise connected subsets of S. A surface is locally segment-wise connected if each point has a basic family of segment-wise connected neighborhoods. A surface that is connected and locally segment-wise connected is segment-wise connected. Definition 3. Let S be a surface. S is also a topological space. Let M be a system of surface construction. M is a pair { } , S ϕ , where Ŝ is a constructed surface and : S S ϕ → a continuous map such that the following condition holds: Each point x S ∈ has a segment-wise-connected open neighborhood K such that each segment component of ( ) is nonempty. The map ϕ is also called a projection. Any open neighborhood K that meets the above condition is called a prime neighborhood. Definition 3 is analogous to the definition of the covering space in algebraic topology [8]. The purpose of doing so is to channel surface construction of CG, CAD, and CAGD into algebraic topology of which the established theory can be applied directly to this paper. It is also natural and proper because a surface is one class of topological spaces and a surface constructed by a method in CG, CAD, and CAGD can be corresponding to a covering space of a target surface that is studied in CG, CAD, and CAGD. That is, the covering space in algebraic topology is transferred into the system of surface construction in CG, CAD, and CAGD here. As a constructed surface is a composite surface, they are interchangeable in this paper.

Parametric Continuities and Geometric Continuities
In order to explore the geometric meanings and properties of smooth conditions between adjacent patches of a composite surface, it is necessary to define two different types of continuities, parametric continuities and geometric continuities, which have been presented for years in CG, CAD, and CAGD [3] [7], and are still used recently [12] [13]. Here, they are defined in the context of the system of surface construction.
The local continuity of a patch is obvious if a set of algebraic formulae with definite continuities are used to express the patch. In CG, CAD and CAGD, researchers usually discuss continuities on the boundaries between pairs of patches when they present their schemes of surface construction. The focus of this paper is put primarily on the boundaries between pairs of patches that are sewed together to form a composite surface.
Let us move onto the definitions of parametric and geometric continuities. Definition 5 We say that the above two curve segments, ci l and cj l , meet with the n th -order geometric continuity ( n G ) if there exists an algebraic curve a l which meets both curves ci l and cj l at p with contact of order n in the sense that the first n terms in the Taylor series expansions about the point p of the two given segments, ci l and cj l , and the algebraic curve a l all agree at p.
If these two segments, ci l and cj l , join with the parametric continuities with respect to the natural arc length parameterization and with the assistance of the chain rule, ci l and cj l meet the conditions of geometric continuities. That is, the condition of 0 G is written as, (2) The condition of 1 G is written as The condition of 2 G is written as In CG, CAD, and CAGD, parametric continuities k C are usually too stiff for researchers to find a set of parametric expressions that satisfy them in practical Since patches of a composite surface are usually represented by bi-parametric algebraic expressions in CG, CAGD, and CAD, it is convenient to represent the geometric continuities with bi-parametric formulae. Bi-parametric formulae of geometric continuities will be presented in the next section.

Bi-Parametric Formulae of Geometric Continuities
Give two bi-parametric patches, { } , j c w t are expressed by radius vectors in differential geometry. These two patches meet at a point,  . According to Definition 5 and the study of Hoschek and Lasser [7], the condition of zeroth-order geometric continuity 0 G is as follows, The condition of first-order geometric continuity 1 G is written as follows, where ( ) The condition of second-order geometric continuity 2 G is written as follows, where ( ) Before discussing the properties of continuities, let us see how to represent the first and second derivatives in Equations ((3), (4), (6) and (7)) in the discrete way in CG, CAGD, and CAD in order to see which points in the neighborhood of an assessed point should be involved in computing with computer technology.
In practice, these points are influenced by a concrete scheme of surface con- It is necessary to notice that the first derivatives in Equations ( (3), (4), (6) and  In an analogous way, the second derivatives in Equations (4) and (7) can be represented approximately with the ratio of the difference of the corresponding first derivatives to the parametric variation. Thus, the first and second columns of points on both sides in the neighborhood of the assessed point are involved in computing the second derivatives. They are 1 p , 2 p , 3 p and 4 p . Figure 1 shows the first and second columns of points on both sides in the neighborhood of an assessed point p.
We will explore the properties of first order continuities in Section 6 and the properties of second order continuities in Section 7, respectively.

Properties of First-Order Continuities
Remark 1. Let us see Equations (1) and (3)   There exists a patch ĵ c S ⊂ , where j c is locally 1 C and Ŝ is globally 1 G .
In the patch j c , let us take a bi-parametric curve segment, Remark 2. If we choose two different natural (arc) parameters along two different directions as two parameters of w and t, respectively, two families of curves are corresponding to the network of two coordinates, w and t. Since each of parameters, w and t, is a natural (arc) parameter, and We should notice that i c also has the coordinate network of two corresponding coordinates, u and v, and In other words, the coordinate network of j c can be extended to its neighboring patches. That is, for any of .  In the analysis of Remark 2, we know that in j c the coordinate network of t and w is composed of ( ) and Let us examine the geometric meanings of Equations (6), (10) and (11).  (9), at the common point p.
In other words, with globally 1 G , the pair of partial derivative vectors of each coordinate network hold the linear relationships with the corresponding partial derivative vectors of its neighboring patches.
Let us see carefully the matrix 11 T of Equation (6). With the theories of modern geometry and transformation group [9], if the square matrix 11 T is full rank, 11 T can perform twisting or scaling on a vector by left multiplying the vector.
Therefore, unlike a surface of the globally 1 C , when a coordinate curve segment in a patch of Ŝ is extended to the corresponding coordinate curve segment in its neighboring patch, it may twist or scale. These observations are just what is said in Proposition 1. That is, parameters, where cj l is the radius vector and What about only holding the orientation of the differential arc when crossing the boundary between two adjoining patches? The answer to this question is in the following Remark 5.
Remark 5. If only the differential arc orientation is held, one of two adjoining patches, i c and j c , has to be changed its own dimensions by being mapped or transformed into the other's. Or, they keep the difference of their dimensions and each of them holds its own distinct geometric details. With the theory of differential geometry [10], the condition of holding only differential arc orientation also means that at the point In this situation, we will give the Proposition 2 and prove that this condition is equivalent to two conditions of Remark 3, i.e. locally 1 C and globally 1 G .
This case is also referred to as angle-preserving.
Proposition 2. The condition of angle-preserving is equivalent to the conditions that patches of a composite surface Ŝ are locally 1 C and the surface Ŝ G .
The angle-preserving case may not be as obvious as the distance-preserving one in Remark 4. The simplified proof is given as follows.
Proof. From the definition of angle-preserving in the theory of differential geometry, it results in the fact that the sufficient and necessary conditions of angle-preserving are to hold the same ratio of the first fundamental forms. That is, given two first fundamental forms of the two joined patches, i c and j c , which have been defined in Equation (13), they can be written as follows,   (6), the 1 G condition, replace u and v with w and t, respectively, in Equation (15). And using Equation (14), the following equation can be led.
Equation (17) is just the sufficient and necessary conditions for angle-pre-

serving. ☐
We have discussed the case that the patches of a composite surface are locally 1 C and the composite surface is globally 1 G . Now let us take that idea a step further and think about the situations of 2 C and 2 G in the next section.

Properties of Second-Order Continuities
According to the theory of differential geometry [10] [11], if Ŝ is globally 2 C , the second fundamental form of any curve ĉ j j l c S ⊂ ⊂ is written as follows, (19) In Equation (19), ci p II is the second fundamental form at the point of ( ) With the condition of Equation (19), both the orientation and size of the bending rate are held. In other words, if we expect to join two patches together with the bending rate invariant, these two patches must hold the orientation and size of the bending rate when crossing the boundary between them.
This case is called bending-rate-preserving in this paper. That is, the bending rate is kept when a point flowing crossing the boundary between two adjoining patches. In other words, the bending rate is invariant at a common point of two It may be too stiff for researchers to construct a composite surface to hold the Alternatively, what about only holding the bending orientation of the natural arc when crossing the boundary between two adjoining patches? The answer to this question is in the following Remark 7. We will check a composite surface that is locally 2 C and globally 2 G .
Remark 7. If only the bending orientation is kept, one of two patches, i c and j c , has to change the dimensions of its bending rate by mapping or transforming it into the other's. Or, they keep the difference of these dimensions and each of them holds its own changing rate.
In this paper, this case is called normal-section-curvature-preserving (in short NSC-preserving). Later in this section, we will give the reason for naming this term in this way. Let us study in details the condition of the NSC-preserving.
Proposition 3. Let i c and j c be a pair of adjoining patches of the composite surface Ŝ . The condition of which Ŝ is NSC-preserving is equivalent to the conditions that patches of the composite surface Ŝ are locally 2 C and the composite surface Ŝ is globally 2 G . That is, Equations (5), (6), and (7) are satisfied for any pair of i c and j c in Ŝ . Proof. It does not lose the generality to form ci j l − by connecting two seg-Advances in Pure Mathematics where cj k is the curvature of the curve cj l at the point p, and cj θ is the in- where ci k (or cj k ) is the curvature of ci l (or cj l ), and ci θ (or cj θ ) is the angle between the normal vector n and ci β (or cj β ). ci β (or cj β ) is principal normal vector of ci l (or cj l ) at the point p, respectively.
Equation (25) gives the reason why we have the term, NSC-preserving (normal-section-curvature-preserving) for the case that the bending orientation is kept.
In the next section, we will give some examples of composite surfaces with definite continuities.

Application Examples of Surface Continuities
We use three examples to show how to apply the theory in Sections 6 and 7.
Example 1. The study of Lávička et al. [12] presents a scheme of smooth surface interpolation using patches with rational offsets. Their scheme is the Hermit interpolation with polynomial Pythagorean normal vectors. The surfaces constructed with their method have rational offsets at given points in a rectangular grid, which can give local geometrical details. Their method can produce local patches with piecewise rational 1 C and the composite surface with global 1 G .
Their method is a simple functional algorithm for computing composite surfaces without the need for any subsequent re-parameterizing or trimming of the parameter domain. In their paper [12], we can see that patches with Pythagorean-normal-vector method interpolating the same boundary data may bring out a smooth composite surface by a suitable choice of the tangent vectors or produce a ridge look with the specific choice (shown in Figure 8 in their paper [12]). For the situation of locally 1 C in patches and globally 1 G in the composite surface, Remark 3 and Remark 5 have given a detailed explanation.
Example 2. In the study of Shen et al. [13], they address a method that converts multiple NURBS (non-uniform rational B-splines) patches to a single untrimmed NURBS-compatible subdivision surface in order to avoid a gap between a pair of adjacent NURBS patches, which are usually trimmed and stitched First of all, a gap between a pair of adjacent NURBS patches is resulted from breaking the 0 C continuity condition when crossing the boundary between these two patches. In other words, the NURBS patches are trimmed without globally holding the 0 C continuity.
In the procedure of Shen et al. [13], a coarse quadrilateral mesh is created first from the trimmed bi-cubic NURBS patches. Then, this coarse mesh is refined to form a non-uniform subdivision mesh that is maintained to retain the boundary geometry property of each NURBS patch through boundary curve spacing and surface fitting. In mathematics sense, the method maps a collection of trimmed bi-cubic NURBS patches into a single subdivision surface of the non-uniform subdivision mesh.
A single subdivision surface with the global 1 C can be produced by merging the subdivision patches (see Figure 12 in their study [13]). Even better, some of their converted model can maintain inter-patch continuity up to the 2 C continuity (see Figures 1 and 17 in their study [13]).
As mentioned in their paper, in some cases, pre-processing and inter-fitting are necessary to satisfy the smoothness requirements. That is, they pre-process the coarse mesh of NURBS and trim it into the subdivision mesh; and then fit the subdivision surface with tight tolerance and fairness functions to approximate the collection of NURBS patches. The former brings out a new knot mesh that replaces the original NURBS one. The latter produces a new subdivision surface that substitutes the collection of NURBS patches to represent a target surface in a visibly smooth way.
The authors indicated that since the process of trimming for 1 C and 2 C might change the model edge curve, they regarded the maintenance of smoothness as more important than exactly fitting the edge curve, and their models of 1 C (see Figure 12 in their study [13]) and 2 C (see Figures 1 and 17 in their study [13]) were obtained in this way.
Example 3. The study of Liu [14] presented the Progressive and Mixing Algorithm (PAMA) which can give the inter-patch 0 C continuity for a ridge looking, or the inter-patch 1 G continuity for a smooth look in different cases.
PAMA adopts bi-parameterization and quadrangle patches. With the Betasplines [3] [4] [15] which have applied geometric continuities to curve and surface construction, PAMA was created for surface construction and edition with human-computer interaction. Without the cost of trimming, PAMA has been applied to the real-time FPGA (field programmable gate array) implementation.

Results and Conclusions
In this paper, we have explored two classes of continuities, parametric continuities and geometric continuities less than and equal to second order, and their geometric properties. The presented propositions can be applied to verify the continuities of surfaces constructed in computer. We can summarize the geometric properties presented in this paper as follows. 0 C and 0 G continuities can be satisfied by the connection of two adjoining patches with a sharp look such as a ridge, along their common boundary. 1 C continuity can be satisfied by identifying the first parametric derivatives on both sides along the common boundary, which restricts the parametric representations of the first column points on both sides around a boundary point of the adjoining patches. The first column points have been shown in Figure 1 in Section 5.

1
G continuity can be satisfied by aligning the first parametric derivatives on both sides along the boundary with the flexibility given by four arguments of the first inter-parametric partial derivatives (see Equation (6)), which relaxes the restriction carried by 1 C continuity but maintains appropriately a smooth look similar to 1 C continuity.

2
C continuity can be achieved by identifying the second parametric partial derivatives on both sides along the common boundary, which heavily restricts the parametric representations of the first two columns of points on both sides around a boundary point of the adjoining patches. This stiff condition can be met by constructing a surface as a whole set of bi-parametric cubic representations. The first and second column points are shown in Figure 1 in Section 5.

2
G continuity can be attained by aligning the first and second parametric partial derivatives on both sides along the common boundary with the flexibility given by two matrices of parameters involved with the first and second inter-parametric partial derivatives, that is, parameters of the parametric partial derivatives (see Equation (7)), which relaxes the stiff restriction of 2 C continuity but holds a smooth look similar to 2 C continuity.
The study in this paper is limited to the geometric properties of continuities less than and equal to second order. The properties of continuities of higher order have not been found in this paper.
In future, we will keep doing research on bridging the gap between the existent mathematics theories and the surface construction in computer, and extend the study on a composite surface into its algebraic topology. Its purpose is to find a mathematical basis to make a comparison between a composite surface and its target surface that would be represented by the composite surface.