1. Introduction
Nowadays, virtual modeling and reverse engineering are extensively used in preserving and studying cultural-heritage-relevant artifacts [
1]. In particular, such issues as shape perception enhancement, restoration and preservation, monitoring, and object interpretation and collection analysis require the support of solid CAD modeling and reverse engineering at micro, meso, and macro levels [
1]. Although the size of the digitally searchable collection of virtual cultural artifices is growing very fast, there has been less progress in shape perception and analysis [
1]. In order to enhance shape perception and analysis, the concept of symmetry can play an important role, because of the following reason: Hann [
2] reported a large number of patterns collected from numerous sources (ancient civilizations of stupas and mandalas, Islamic Spain, Safavid Persia, Ottoman Turkey, Indian subcontinent, Indonesia, dynastic China, Korea, and Japan). The reported patterns exhibit symmetry that can be analyzed using some simple geometric entities, namely, segments of intersecting circles (“vesical piscis”), construction based on equilateral triangles and vesical piscis, construction based on hexagons, construction based on “four circles over one,” equilateral triangles, isosceles triangles, equilateral triangles, right angle triangles, regular hexagons, regular pentagons, “5-4-3 triangles,” construction based on 36 degree isosceles triangles, equilateral triangle grids, regular hexagon grids, squire grids, square and root triangles, construction based on the golden section triangle and diagonal of a rectangle, whirling square rectangles, squares with Brune’s star-type divisions, and Brune’s stars with intersections [
2] (pp. 16–24,34). In addition, such simple operations as translation, two-fold rotation, reflection, and glide-reflection applied to the abovementioned geometric entities can create patterns that are often seen in the cultural artifacts of the abovementioned civilizations [
2]. Moreover, analyzing symmetry can facilitate the geometric modeling performed on the digital data of the artifacts with cultural significance [
3,
4].
The cultural heritage of indigenous people (e.g., Nahuas living in Mexico and El Salvador, Ainu living in the northern hemisphere of the world, Assyrians living in Western Asia, and the like) is full of aesthetic patterns, which can also be studied using the concept of symmetry. The focus of this article is the patterns of the indigenous people living in the northern part of Japan, known as Ainu [
5,
6]. Ainu people decorate their houses, clothing, ornaments, utensils, and spiritual goods using some unique patterns [
7,
8,
9,
10,
11,
12]. The patterns exhibit symmetry, carry their identity, and a sense of aesthetics. Nowadays, different kinds of souvenirs and cultural artifacts are crafted with Ainu patterns, which are cherished by many individuals in Japan and abroad [
13]. Thus, the Ainu patterns carry both cultural and commercial significance. Many shops all over Hokkaido sell souvenirs and artifacts crafted with Ainu patterns. There are even shopping streets (e.g., the shopping street at Akan Sap [
14]) that are specialized in products crafted with Ainu patterns.
Figure 1 shows some pictures of such products taken recently at the shopping street located in Akan Spa. A great deal of craftsmanship is needed to produce the Ainu patterns precisely. The remarkable point is that there is a lack of human resources who can produce the patterns. This trend will remain the same in the foreseeable future, creating a pressing need to preserve the Ainu pattern-making craftsmanship. To preserve the craftsmanship, one of the pragmatic options is to use digital manufacturing technology. From this perspective, this article presents a methodology to create both virtual and physical prototypes of some Ainu patterns, and also demonstrates its efficacy through some case studies. Particularly, a point cloud-based approach was adopted to model the patterns. The point cloud representing a pattern was then used to create a virtual model of the pattern in the form of a solid CAD model. Finally, the triangulation data of each solid CAD model were used to run a 3D printer for producing a physical prototype (replica of the pattern). The virtual and physical prototypes of both basic (Hokkaido) Ainu patterns and some complex patterns were produced by applying the proposed methodology.
Therefore, the rest of this article is organized as follows.
Section 2 presents the basic motifs that are used to create all kinds of patterns found in the artifacts used by the Ainu community living in Hokkaido.
Section 3 describes a methodology to model a pattern, and, thereby, to produce its virtual and physical prototypes.
Section 4 presents some results obtained by applying the methodology.
Section 5 concludes this article.
2. Basic Motifs
Before presenting the methodology intended to mimic the Ainu pattern-making craftsmanship using digital manufacturing technology, it is important to know about the origins of the patterns as reported in the literature. As in other cases, Ainu patterns can also be classified into two broad categories, namely, primary patterns and synthesized patterns. The primary patterns are hereinafter referred to as basic motifs. As a result, any combinations of the basic motifs yields a synthesized pattern. This section describes only the basic motifs of the Ainu community living in Hokkaido. The Ainu communities living in other regions have their own basic motifs, because the motifs carry the identity and a sense of aesthetics of the respective community.
Numerous authors have reported the basic motifs (e.g., see the work [
7,
8,
9,
11]) used by the Ainu community living in Hokkaido [
10,
12]. The Sapporo city authority has summarized the basic motifs into fourteen types, which are listed in
Table 1.
As seen in
Table 1, the first twelve motifs have names, whereas the other two do not have any name. In particular, Motif No. 1 (called Ayus in the Ainu language) takes the shape of a thorn. Motif No. 2 (Morew in the Ainu language) takes the shape of a spiral. Motif No. 3 (called Arus-Morew in Ainu language) takes the shape of a spiral with small thorns. Motif No. 4 (called Sikike-nu-Morew in the Ainu language) takes the shape of a spiral with corners. Motif No. 5 (called Sik in the Ainu language) takes the shape of an eye. Motif No. 6 (called Utasa in the Ainu language) takes the shape of an intersection. Motif No. 7 (Uren-Morew in the Ainu language) takes the shape of two spirals. Motif No. 8 (Ski-uren-Morew in the Ainu language) takes the shape of two spirals with an eye. Motif No. 9 (called Morew-etok in the Ainu language) takes the shape of a spiral plant. Motif No. 10 (called Punkar in the Ainu language) takes the shape of a vane. Motif No. 11 (called Apapo-piras (u) ke in the Ainu language) takes the shape of a flower. Motif No. 12 (called Apapo-epuy in the Ainu language) takes the shape of a flower bud. Motifs No. 13 and 14 do not have Ainu names, but they look like a heart type shape and a fishing bell shape, respectively. Although some authors have studied the significances of these motifs [
7,
8,
9,
11], a more comprehensive study lies ahead. As far as modeling is concerned, the motifs called Morew, Sik, Utasa, and Ayus have been frequently used, and thus can be defined as the main motifs. Arus-Morew, Sikike-nu-Morew, Sikike-nu-Morew, and Uren-Morew are the motifs created by modifying the main motifs, and thus can be defined as the first-order synthetic motifs. The motifs called Morew-etok, Punkar, Apapo-piras (u) ke, and Apapo-epuy represent plants, and thus can be defined as the plant motifs. The other motifs (i.e., Motifs No. 13 and No. 14) can be defined as the other motifs.
3. Pattern Digitization Methodology
This section describes a general methodology that can be used to digitize a 2D pattern for the sake of virtual and physical prototyping. Before proposing the methodology, some of the salient points must be described first, as follows.
The digital technology called reverse engineering [
16] has been extensively used in cultural artifact preservation and prototyping [
17,
18,
19,
20,
21,
22,
23,
24,
25] because it (reverse engineering) helps create digital or geometric models of existing physical objects when these models (digital or geometric models) are unavailable. In most cases, reverse engineering uses scanning (e.g., [
19,
20] or image processing [
18] techniques to extract the shape, topological, and texture information of an existing physical object. The extracted information must be processed to create the (digital) virtual model of the object using CAD systems. Once the digital data of the virtual models are available, physical models (often referred to as prototypes or replicas) can be produced using a manufacturing process (additive manufacturing (e.g., 3D printing) [
21], subtractive manufacturing (CNC machining), or formative manufacturing (casting)). However, depending on the level of automation and accuracy, the object information obtained by the scanning or image processing techniques must undergo a set of sophisticated and complicated transformations. The transformations include surface reconstruction from the point clouds obtained by 3D scanning or imaging (photogrammetric technique), reduction of redundant information (e.g., reduction of redundant points from a point cloud), and hierarchical decomposition of a point cloud or image data based on depth, color, and density, and other factors. Among others, Geng and Bidanda [
16] and Tashi et al. [
25] provided a relatively comprehensive account of the computational complexity associated with the transformations mentioned above. However, studying the abovementioned transformations is an active field of research [
26], as there are many unsolved issues.
In order to avoid the abovementioned computational complexity, other alternative approaches can be considered. In this respect, the following three studies can be noted. The first alternative approach described here is taken from the work of Tamaki et al. [
27], where it was shown that a special type of photo-curing resin can be used to extract the shape information of an object (in their case, a snow crystal) for the sake of physical prototyping. This technique works well when the object is tiny. The parameters that affect the curing process of the resin play an important role in the shape information extraction process. Another alternative approach was developed by Rojas-Sola et al. [
28], where the virtual model of a historical artifact (in this case, an engine) was constructed using a commercially available CAD system. The 2D sketch of the artifact was used as the reference model while building the virtual model. This technique relies on the perception of the model builder, and worked well when the object consisted of some simple geometrical shapes (e.g., plate, line, circle, cylinder, and similar). The authors did not report any results regarding the physical prototyping of the artifact (the engine). The other alternative was developed by Tashi et al. [
25], who used analytical point clouds to represent the relevant segment of a given artifact for virtual and physical prototyping. In this approach, both a set of equations and an algorithm were needed to create the desired point cloud. The point cloud was used to create a solid CAD model (virtual prototype) of the object using commercially available CAD systems. The solid CAD model was used to create a physical prototype using a commercially available 3D printer. In this article, this approach was adopted. For the sake of systemization of the whole modeling process, three domains were considered, namely, the modeling domain, virtual prototyping domain, and physical prototyping domain.
Figure 2 schematically illustrates these three domains. As seen in
Figure 2, in the modeling domain, a model builder first studies the object and determines the pattern to be modeled. Subsequently, the model builder fixes the values of the parameters of the point cloud creation algorithm (shown below), namely, center point (
Pc), initial length (
d), initial angle (
ϕ), instantaneous distance (
ri|
i = 0,1,…,
n), and instantaneous rotational angle (
θi|
i = 0,1,…,
n). If the model builder is satisfied, then the output of the algorithm is collected as the point cloud (
PC) that is the model of the desired pattern. In the virtual prototyping domain, the model builder inputs the point cloud (
PC) to an appropriate CAD system and performs solid modeling using the functions offered by the system. This results in a virtual prototype (solid CAD model). Finally, the physical prototyping domain acknowledges the triangulation data (often referred to as STL data) of the solid CAD model and uses a commercially available 3D printer to produce the physical prototype of the pattern. Other manufacturing means can be used if preferred.
The mathematical settings of the point cloud creation algorithm (Algorithm 1) that were used to model the Ainu patterns shown in this article were adopted from Tashi et al. [
25]. The algorithm consists of four steps, as shown below. The first step is the input step, the second step is the calculation step, the third step is the iteration step, and the last step is the output step. In the input step, the center point
Pc = (
Pcx,
Pcy) ∈ ℜ
2, initial distance
d > 0, initial angle
ϕ ∈ ℜ, instantaneous distances (
ri ∈ ℜ|
i = 0,1,…,
n), and instantaneous rotational angles (
θi ∈ ℜ|
i = 0,1,…,
n) are defined. In the calculation step, the initial point
P0 = (
P0x,
P0y) is calculated, which is a point at a distance
d from
Pc on the line
PcP0, making an angle
ϕ with the
x-axis in the counter-clockwise direction. In the iteration step,
P0 is first rotated in the counter-clockwise direction of the
x-axis using
θi to create the points
Pi = (
Pix,
Piy),
i = 0,1,…,
n. Afterward, the points denoted as
Pi = (
Pix,
Piy),
i = 0,1,…,
n, are placed at the distances given by
ri from
Pc, resulting in the points
Pei = (
Pix,
Piy),
i = 0,1,…,
n. In the output step, the points
Pei = (
Pix,
Piy),
i = 0,1,…,
n, are collected as the point cloud denoted as
PC = (
Pei|
i = 0,1,…,
n).
The working principle of the algorithm is schematically illustrated in
Figure 3, using the first four iterations for some arbitrary values of the instantaneous distance and rational angle. The algorithm can be used for creating different kinds of planar shapes not limited to the patterns shown in this article. Since most of the cultural-heritage-relevant shapes reported here and elsewhere [
2] are symmetrical, the point cloud (
PC) that models a segment of a given symmetrical shape can be rotated and translated several times to create a set of symmetrical
PCs that model the other segments of the shape. In this case, the parameters called initial angle and distance become instrumental. The reason is as follows: If
PC1 and
PC2 are created by setting
ϕ =
a,
ϕ =
b, respectively, then both
PC1 and
PC2 are symmetrical point clouds but separated by an angular difference of
b –
a. Similarly, if
PC1 and
PC2 are created by setting
d =
a,
d =
b, respectively, then both
PC1 and
PC2 are symmetrical point clouds but placed at a distance (
b–
a)cos
ϕ in the
x-direction and (
b–
a)sin
ϕ in the
y-direction. As a result, from the viewpoint of symmetrical patterns, both initial angle and distance bear a great deal of significance.
Algorithm 1 Point Cloud Creation Algorithm |
1 Define: | Center Point Pc = (Pcx,Pcy) ∈ ℜ2, Initial Length d > 0, Initial Angle ϕ ∈ ℜ Instantaneous Distance (ri ∈ ℜ|i = 0,1,…,n) Instantaneous Rotational Angle (θi ∈ ℜ|i = 0,1,…,n) |
2 Calculate: | P0 = (P0x, P0y) so that and |
3 Iterate: | For i = 0,1,…,n Rotate P0 by an angle θi around Pc in the counter-clockwise direction to create Pi = (Pix,Piy) so that
Extend Pi to Pei (a point on the line PcPi at a distance ri from Pc)
End For |
4 Output: | PC = (Pei|i = 0,1,…,n) |
When a representative segment of a symmetrical pattern is modeled by
PC, the user must be aware of how to fix the values of instantaneous distance and rotational angle. In most cases, a simple monotonic function (a straight line with positive or negative slope) or tent-like function can be used to fix the values of instantaneous distances and rotational angles.
Table 2 shows an example of how a point cloud
PC evolves where a straight line with a positive slope and a tent-function are used to fix the values of instantaneous rotational angles and distances, respectively. This example reveals the fact that at the point of inflection (see the plot of
ri), the
PC changes its direction while the other parameter keeps its monotonicity (compare the plot of
θi with that of
ri). The nature of these two parameters can be understood more clearly from the models of the Ainu patterns presented in the next section.