JP2001290851A - Sphere dividing method and sphere system structure designed by the same - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a sphere dividing method which can secure the degree of freedom of the area ratio between adjacent divided areas. SOLUTION: When the interval L between a couple of small circular disks 10 which are arranged in parallel at an equal interval across each large disk, concretely, a spherical grating P is formed by crossing small circular disks 10, i.e., small circles 12 with each other by using six large circles of 5-time rotational symmetry that 5-4-3-time rotary symmetry (regular icosahedron) has, the spherical surface is divided by the small circles 12 into a nearly triangular area S3 and a thin and long nearly elliptic area S4 which is positioned around the nearly triangular area S3. The chords 32 of points where the vertexes of the nearly triangular area and the vertexes of the thin and long elliptic area gathers represent the sides of the regularly triangular surfaces S5 of the regular icosahedron by forming spherical gratings P and connecting adjacent spherical gratings P by the chords 32. This sphere dividing method using the small circles makes this polyhedron totally triangle when the polyghedron formed of chords 30 and 32 connecting adjacent grating points P is observed as well as a sphere dividing method using large circles.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method of dividing a spherical surface, and more particularly to a method of dividing a spherical surface suitable for designing a spherical dome, tent or golf ball.

[0002]

2. Description of the Related Art The design of dimple arrangement in a golf ball employs a method of dividing the surface of a golf ball into a plurality of regions and arranging a plurality of dimples in each of the divided regions. In the conventional design method, as a method of dividing the surface of a golf ball into a plurality of regions, a method using a regular polyhedron in order to secure its symmetry, a great circle (G
(Japanese Patent Laid-Open Nos. 1-223979 and 3-139).
371, JP-A-3-45275). Here, the great circle refers to a circle on a spherical surface that divides a sphere into two hemispheres. That is, the great circle refers to a circle on the sphere formed by a plane passing through the center of the sphere intersecting the sphere.

[0003] Also, R. Buckminster F., famous for geodetic dome,
(Uller) proposes a great circle dome as a prototype of a geodetic dome (“The World of the Dimaxion of Buckminster Fuller”, page 204, author: Co-authored by Buckminster Fuller / Robert W. Marks, publisher: Kashima Press). The great circle dome is designed based on a spherical division method in which a spherical surface is divided by a plurality of great circles included in a spherical polyhedron, and the skeleton of the dome is, for example, a product name NF sold by Gold Win −5150
And NV-1100, along a great circle.

[0004]

However, spherical division using a great circle does not have a high degree of freedom in the shape of a region divided by the great circle and / or the area ratio between adjacent regions, so that it has a practical viewpoint. In consideration of the above, there is a limit to the number of areas that can be divided by a great circle, that is, the number of divisions. For this reason, Fuller et al. Proposed Triacon Subdiv as a spherical division method that can be deployed on a huge dome.
Ition method and Alternate Subdivis
proposed two methods called the “ion method” (“GE
ODESICDOME ", page 12, Author: Borin
Van Loon, publisher: Tarquin Publ
ications). These two spherical division methods are intended to make the dome into a total triangle, that is, to configure the dome only with triangles. Therefore, ICOSAHEDRON (regular icosahedron) composed only of triangular faces,
OCTAHEDRON (regular octahedron), TETRAHED
RON (regular tetrahedron) is adopted as a basic model, and each surface of the basic model is divided into smaller triangles.

More specifically, these two spherical surface division methods subdivide a triangular surface of a basic model into small triangles, and project the vertices of each triangle onto a spherical surface inscribed by the basic model to form a spherical grid (Spherical G).
i), and the light source to be projected is set at the center of the basic model. A dome designed based on such a technique is called a geodetic dome (“Geodick Dome”).
esics "FIG. 103, Author: EDWARD P
OPKO, publisher: Industrializatio
n and Technology, for example, Montreal Dome (diameter: 84 meters), Radome (diameter: 15 meters), said "The World of Dimaxion" Fi
gs. 408-416), Dome Tent Nv-1107
(Sold by GoldWin).

SUMMARY OF THE INVENTION An object of the present invention is to provide a method of dividing a spherical surface which can secure the degree of freedom of the area ratio between adjacent divided regions in view of the practical limit of the conventional spherical division using a great circle. It is in. Another object of the present invention is to eliminate the enormous mathematical processing required for obtaining spherical grid points by the conventional projection method when performing highly segmented surface division such as a huge dome. It is an object of the present invention to provide a method for dividing a spherical surface.

Another object of the present invention is to provide a spherical surface dividing method suitable for visually performing spherical surface division using three-dimensional CAD software of a computer. Another object of the present invention is to provide a spherical division method which can freely determine an appropriate division number while visually confirming.

[0008]

SUMMARY OF THE INVENTION According to one aspect of the present invention, a plurality of small circles arranged in parallel with each great circle of a spherical polyhedron arranged concentrically with a spherical surface are provided. This is achieved by dividing the spherical surface by the intersection of these small circles. Here, a small circle (LesserCirc
Le or Small Circle) refers to a circle on a sphere that does not pass through the center of the sphere. Small circles are typically
It is preferable to arrange them on the left and right across the great circle. In this case,
Small circles may be set at equal intervals on the left and right sides of the great circle, or may be set at different intervals. In the design of a small dome or tent, it is sufficient to set 2 to 4 small circles parallel to each great circle, but if a large number of divided surfaces need to be set like a huge dome, more What is necessary is just to set a small circle.

The spherical polyhedron is typically selected from Plato Archimedes solids. The concept of the great circle included in the present invention will be described with reference to FIGS. Reference numeral 1 shown in FIG. 1 (A) indicates CUBOCTAHEDRON.
(A regular hexahedron). The equator 4 can be drawn on the regular hexahedron 1 when the rotation axis 3 passing through the centers of the mutually facing surfaces 2 of the regular hexahedron 1 is set. And this rotating shaft 3
Is applied to the other two sets of faces 2 facing each other,
As shown in FIG. 1B, three equators 4 that intersect each other at the center of each surface 2 can be drawn.

FIG. 2 shows a three-dimensional model in which three circular disks 6 each including the three equators 4 are installed concentrically with the regular hexahedron 1. In this three-dimensional model, the outer peripheral edges 8 of the three circular disks 6 (hereinafter, referred to as great circle disks) represent three great circles on a spherical surface concentric with a regular hexahedron. The three-dimensional model shown in FIG. 2 also has a pair of small circular disks 10 arranged in parallel with each of the great circular disks 6 with the great circular disk 6 interposed therebetween.
Are arranged at equal intervals from the great circle disk 6. The outer peripheral edge 12 of the small circle disk 10 represents a small circle on the same spherical surface as the spherical surface through which the great circle 8 passes. FIG. 3 shows FIG.
3 shows a three-dimensional model obtained by removing the great circle disk 6 from the three-dimensional model shown in FIG.

From the three-dimensional models shown in FIGS. 2 and 3, when six small circles 12 cross each other,
It can be seen that the sphere is symmetrically divided by the intersection of the large circle 8 with the large circle 8. This means
The great circle included in the spherical polyhedron itself has symmetry, and it is theoretically clear that the small circle set based on this great circle has symmetry.

FIGS. 4 and 5 show models corresponding to the three-dimensional model of FIG. 3, respectively. FIG. 4 shows the distance L2 between the large circle disk 6 and the small circle disk 10 in the case of FIG. An example of a case where the interval is set to be larger than the interval L1 (L2> L1) will be described. FIG. 5 shows an example in which the distance L3 between the small disk 10 and the large disk 6 is set smaller than the distance L1 in FIG. 3 (L3 <L1).

By comparing the three-dimensional models of FIGS. 3, 4 and 5, the following can be easily understood. That is, according to the spherical dividing method of the present invention, by changing the installation position of the small circular disk 10, that is, the distance L from the large circular disk 6, the area of each divided region sandwiched by the small circular disks 10 intersecting each other. The ratio can be changed. Therefore, by appropriately setting the installation position of the small circular disk 10, the area ratio of each divided area of the spherical division can be arbitrarily adjusted. In addition, as will be apparent from the following description, by changing the installation position of the small-circle disk 10, the number of divided spherical surfaces, that is, the number of divided regions can be increased or decreased. Also,
By adding an arbitrary number of additional small discs parallel to the great disc 6 to these three-dimensional models, the number of divided areas can be increased. Theoretically, there is no limit on the number of small circle desks that can be added.

According to another aspect of the present invention, ICOSAH
EDRON (regular icosahedron) or CUBOCTAHEDR
Using a great circle selected from the first and second great circles of ON (quasi-regular tetrahedron), setting one or more small circles in parallel with the selected great circle, and intersection of these first small circles The object of the present invention is achieved by dividing the spherical surface by. The small circles to be set are typically arranged at equal intervals on the left and right sides of each great circle, but small circles may be arranged at different intervals on the left and right sides of each great circle. The first and second great circles referred to here will be described below with reference to FIGS.

Fuller's study has already shown that the following relationship exists between each model of the Plato Archimedes solid and the great circle (R. Buckmin):
"Synergetics" by Ster Fuller
Explorations in the Geome
try of Thinkimg ", p. 456, publisher:
MACMILLAN PUBLISHING Co I
nc. ). First, FIGS. 6 to 9 show a first great circle (primary great-circle) of the icosahedron.
15 is a drawing for explaining the concept of le). First, FIG.
Shows the case where the rotation axis 3 passing through the midpoint of the side 18 forming the triangular surface 16 of the icosahedron 14 is set (2
Rotational symmetry). In this case, 15 equators 4 are formed. FIG. 7 shows a case where the rotation axis 3 passing through the opposing vertices of the regular icosahedron 14 is set (five-fold rotational symmetry). In this case, six equators 4 are formed. FIG.
Shows a case where the rotation axis 3 passing through the center of the triangular surface 16 of the icosahedron 14 is set (three-fold rotational symmetry). In this case, ten equators 4 are formed.

6 to 8 add up to 31 in total, and the 31 equators 4 are represented on a spherical surface by 31 great circles 8 shown in FIG. It is a first great circle of the facepiece. These 31 great circles are 5
It represents the maximum number of great circles included in each spherical polyhedron included in the 3-2 times rotationally symmetric system. As the spherical polyhedron included in the 5-3-2 times rotationally symmetric system, for example, DODECA
HEDRON, ICOSIDODECAHEDRON,
TRUNCATED-DODECAHEDRON, SN
UB-DODECAHEDRON, TRUNCATED
-ICOSAHEDRON, TRUNCATED-IC
OSIDEDECAHEDRON, RHOMBICOS
IDEDECAHEDRON, RHOMBIC TRI
There is ACONTAHEDRON (diamond 30 face).

FIGS. 10 to 14 are diagrams for explaining the concept of the first great circle included in the quasi-tetrahedron 20. FIG.
FIG. 10 shows a case where the rotation axis 3 passing through the center of the triangular surface 22 of the quasi-tetrahedron 20 is set (three-fold rotational symmetry). In this case, four equators 4 are formed. FIG. 11 shows a case where the rotation axis 3 passing through the center of the quadrangular surface 24 of the quasi-tetrahedron 20 is set (four-fold rotational symmetry). In this case, three equators 4 are formed. FIG.
Reference numeral 2 denotes a case where the rotation axis 3 passing through the vertex of the quasi-regular tetrahedron 20 is set (twice rotational symmetry). In this case, six equators 4 are formed. FIG. 13 shows a case where a rotation axis passing through the midpoint of the opposite side 26 of the quasi-tetrahedron 20 is set (another two-fold rotational symmetry). In this case, twelve equators 4 are formed.

The total number of the equators 4 in FIGS. 10 to 13 is 25, and the 25 equators 4 are represented on a spherical surface by 25 great circles 8 shown in FIG.
It is a first great circle of a facepiece. These 25 great circles are 4-3
It represents the maximum number of great circles included in each spherical polyhedron included in the -two-fold rotationally symmetric system. Examples of the spherical polyhedron included in the 4-3-2 times rotationally symmetric system include, for example, TETRAH
EDRON (regular tetrahedron), TRUNCATED-TET
RAHEDRON, HEXAHEDRON (regular hexahedron), OCTAHEDRON (regular octahedron), TRUNC
ATED-OCTAHEDRON, TRUNCATED
-CUBE, SNUB-CUBE, RHOMBIC
UBOCTAHEDRON, TRUNCATED-CU
BOCTAHEDRON, RHOMBIC DODEC
There is AHEDRON (diamond dodecahedron).

Fuller calls the above-mentioned great circle a great circle based on the first symmetry, and additionally, a great circle based on the second symmetry (Secondary Great-circle).
("Synergy" above)
tics Explorations in the
"Geometry of Thinking" No. 325
Page and FIG. 11320.01B). According to Fuller, the sum of the first and second great circles is 12 in a regular icosahedron.
It is clarified that the number is one, and that the quasi-tetrahedron has 72 second great circles. The concept of great circles included in the present invention includes great circles revealed by studies of these fullers.

The difference between the method of dividing a spherical surface by a large circle proposed by Fuller and the method of dividing a spherical surface by a small circle according to the present invention becomes more clear with reference to FIGS. Positive 2
The octahedron has all its faces composed of equilateral triangles. However, according to the spherical division method using the fuller great circle,
As shown in FIG. 15, a spherical regular icosahedron can be formed by using a fan-shaped great disk 6a having a central angle of 63.435 °. In other words, the fan-shaped great circle disc 6a
When three triangular pyramids are formed and the vertices of all triangular pyramids are gathered at one point O1, the spherical surface can be divided by the same substantially triangular region S1. The adjacent substantially triangular area S1 is
When the spherical lattice points P are formed at the intersections and the adjacent spherical lattice points P are connected by chords 30, the chords 30 represent the respective sides of the regular triangular surface S2 of the icosahedron.

FIG. 16 shows the rotational symmetry of 5-4-3 times (positive 20 times).
The three-dimensional model is a three-dimensional model created by using six five-rotationally symmetric large circles of the cuboid) and adjusting the distance between a pair of small-circle disks 10 installed at equal intervals and in parallel with each large-circle disk interposed therebetween. . Specifically, when the small circular disks 10, that is, the small circles 12 intersect each other to form a spherical lattice P, the small circles 12
As a result, the spherical surface is divided into a substantially triangular region S3 and an elongated substantially elliptical region S4 located around the substantially triangular region S3. The point at which the vertices of these substantially triangular areas and the vertices of the elongated elliptical area gather is the spherical grid P
Is formed, and the adjacent spherical lattices P are connected by a chord 32, and the chord 32 represents each side of a face S5 of an equilateral triangle having a regular icosahedron.

The following becomes clear when looking at FIG. 15 and FIG. (1) The method of dividing a spherical surface according to the present invention, that is, the method of dividing a spherical surface by a small circle is limited to a specific form, but the chord 30 connecting adjacent lattice points P,
By observing the polyhedron made at 32, it is possible to make this polyhedron a total triangulation. (2) In the spherical division method using the great circle, as can be understood from FIG. 15, the great circle is discontinuous at the vertex P of the regular icosahedron. That is, the icosahedron of FIG. 15 is formed by aggregating a part of different great circles 8 at the vertices P. On the other hand, the regular icosahedron shown in FIG. 16 is inevitably formed by the small circular disks 10 extending continuously over the entire circumference crossing each other. (3) In the three-dimensional model of FIG. 15 showing a method of dividing a spherical surface by a great circle, all the fan-shaped great circle disks 6a converge on a sphere, that is, the center (O1) of a spherical regular icosahedron. On the other hand, in the three-dimensional model in FIG. 16, the convergence points (O2) of the three small circular disks 10 are offset from the center of the sphere. The convergence point (O2) coincides with each vertex of a regular dodecahedron which is concentric with the center of the sphere and is constituted by a total of twelve small circular disc 10 surface areas (not shown).

According to another aspect of the present invention, a pair of small circles which are parallel to each other are set by using the faces, sides, and vertices of a core model composed of a spherical polyhedron, and are set concentric with the core model. The sphere can be divided. As a method of setting a small circle using the core model, when the core model has planes parallel to each other, a small circle disk may be set along the parallel plane. A small circle may be set by truncating the surface. Known types of this truncation are face truncation, edge truncation, and vertex truncation.
A small disk may be set along the surface formed by these truncations. Further, one small circle along the surface of the core model and a small circle passing through the vertex or side of the core model and parallel to the one small circle may be set. By setting small circles parallel to each other using the core model in this manner, a spherical surface concentric with the core model can be symmetrically divided by the small circle.

Referring to the drawings, specific examples will be described. FIG. 17 shows an example of spherical division when a regular octahedron is adopted as the core model 34. An octahedron has four pairs of opposing faces, which are in a parallel relationship. Therefore, as shown in FIG. 17, the small circular disc 10 is formed along each face 34a of the regular octahedron.
18, four pairs of parallel small circles 10 can be set as shown in FIG. 18, whereby the spherical surface concentric with the regular octahedron can be formed into four pairs of small circles 1 parallel to each other.
It can be divided by two. If a large circle is added to the surface division method using small circles in FIGS.
As shown in FIG. 9, the large circle disk 6 may be added to the middle of each pair of small circle disks 10.

The spherical division according to the present invention may be performed using a plurality of mixed core models. 20 to 24 show two different models concentrically mixed. 20 and 21 illustrate a case where the vertex of the inner core model 38 is set at the center of the surface of the outer core model 36. FIG. 22 illustrates a case where the vertices of the inner core model 38 are set so as to divide the sides of the outer core model 36 at the golden ratio. FIG. 23 illustrates a case where the inner and outer core models 36 and 38 are set such that the vertices of the inner core model 38 divide the sides of the outer core model 36 into a golden ratio. FIG. 24 illustrates a case where a vertex of the inner core model 38 is set to a vertex of the outer core model 36.

Inner core model 38 and outer core model 36
Can be varied in various ways.
For example, in FIG. 20, the inner core model 38 is a regular octahedron, and the outer core model 36 is a regular hexahedron. On the other hand, in FIG. 21, the inner core model 38 is a regular hexahedron, and the outer core model 36 is a regular octahedron. That is, since FIG. 20 and FIG. 21 each have a dual relationship, 2 in FIG.
The relationship of FIG. 21 can be created by enlarging the inner core model 38 (regular octahedron) among the two core models.

On the way from FIG. 20 to FIG. 21, the relationship shown in FIG. 25 is obtained. As can be seen from this, when a plurality of core models are mixed, the relative position of the small circle can be changed by adjusting the relative size of the core models. Two core models 3 of the embodiment shown in FIG.
FIG. 26 shows the result of spherical division using the elements 6 and 38. In the spherical division using the core model, the aspect of the spherical division can be changed by changing the relative size between the core model and the spherical surface.

[0028]

DESCRIPTION OF THE PREFERRED EMBODIMENTS FIGS.
It is a three-dimensional model created according to the present invention based on a great circle included in two-fold rotational symmetry (quasi-tetrahedron). FIG.
FIG. 18 shows a three-dimensional model 40 composed of a total of eight small circular disks 10 set with respect to four great circular disks three-fold rotationally symmetric, and is a view of the same model as FIG. 18 from a different viewpoint. FIG. 2 shows a large circle disk 6 and a small circle disk 10 on a spherical surface based on the three-dimensional model 40.
8 As can be understood from FIG. 27, according to this dividing method, the spherical surface can be divided into a triangular region 44, a quadrangular region 46, and a hexagonal region 48.

The three-dimensional model 50 shown in FIG. 28 is obtained by adding the great circle disk 6 to the method of dividing the plane by the small circle. That is,
The three-dimensional model 50 in FIG. 28 includes four large circle disks 6 and eight small circle disks 10 that are rotationally symmetric three times. According to the spherical surface division method of the present invention from the three-dimensional model 50, the spherical surface can be divided into a triangular region 54, a quadrangular region 56, and a rhombic region 58. Further, the quadrangular region 56 and the rhombic region 58 can be triangulated by adding three great circles of a spherical regular octahedron to a spherical surface, whereby the entire spherical surface can be divided into triangles. In the three-dimensional model 50, it can be seen that one spherical triangular region 52 of a spherical regular octahedron surrounded by three virtual lines 51, which is a part of a great circle, is completely triangulated.

In the three-dimensional model 50 shown in FIG. 28, when the radius of the three-dimensional model 50 sphere is fixed and the distance between the large-circle disk 6 and the small-circle disk 10 is relatively narrowed, a three-dimensional model 55 shown in FIG. As can be understood by comparing the two three-dimensional models 50 and 55, the three-dimensional model 55 in FIG. 29 has a larger number of divided regions on the spherical surface than the three-dimensional model 50 in FIG. As is apparent from this, according to the spherical surface dividing method according to the present invention, the form of the spherical surface division can be changed by changing the interval between the small circles located across the great circle.

In the three-dimensional model 60 shown in FIG. 30, four great circle disks three times rotationally symmetric are indicated by reference numeral 6 (3).
Reference numeral 6 (4) refers to three great circle disks having four times rotational symmetry.
It is indicated by. For each great circle on the sphere formed by these great circle disks, a pair of small circle disks arranged in parallel with each other are set. In the three-dimensional model 60, a form of spherical division in which the great circle disk and the small circle disk intersect with each other is shown. These small circle desks are designated by reference numerals 10 (3) and 10 (4). The three-dimensional model 65 shown in Fig. 31 is a three-dimensional model 65 in which the small circular disk 10 (4 with respect to the four-fold rotational symmetry is omitted from the three-dimensional model 60.
Spherical surface when using three great circular disks 6 (3) and eight small circular disks 10 (3) and three rotationally symmetric three large circular disks 6 (4). This shows one mode of division.

FIGS. 32 to 34 show three-dimensional models created according to the present invention based on the great circle included in the 5-4-3 rotational symmetry. FIG. 32 shows a three-dimensional model 70 composed of small circular disks 10 arranged at equal intervals in pairs with respect to each of the six large circular disks symmetrical five times. According to the three-dimensional model 70, a pentagonal area 76 and a hexagonal area 78
Can be divided so as to surround the area with a triangular region 74. This is of great interest. In other words, in the conventional dome design of Fuller et al., A method of forming a pentagonal area and a hexagonal area by adopting a method of forming a pentagonal area and a hexagonal area by omitting an appropriate triangular area and then omitting an appropriate triangular area was adopted according to the present invention. As is apparent from the three-dimensional model 75, the pentagonal region and the hexagonal region can be directly created without any troublesome work as in the related art.

The three-dimensional model 70 has a five-fold rotationally symmetric
When three great circle disks 6 are added, the three-dimensional model 7 shown in FIG.
It becomes 5. The hexagonal area 78 shown in FIG. 32 can be subdivided by the great circle disk 6 into a triangular area 74 and a rhombic area 76 as shown in FIG. This three-dimensional model 75 shows a case where all the great circles 8 are exactly equally divided into 20 by the intersection of the small circle disk 10 and the great circle disk 6. In the three-dimensional model 70, one spherical triangular area 72 of a spherical regular icosahedron surrounded by three virtual lines 71 which are a part of a great circle is shown. It can be seen that the addition makes the entire sphere completely triangulated. When the radius of the sphere of the three-dimensional model 75 in FIG. 33 is made constant and the distance between the small circular disks 10 sandwiching the great circular disk 6 is relatively widened so as to always maintain the same distance, the three-dimensional model in FIG. It will be 80. As described above, the division number can be increased only by adjusting the interval between the small circular disks. Due to the intersection of the small circle disk 10 and the great circle disk 6, the spherical surface is divided into a triangular area 84, a pentagonal area 86, and a hexagonal area 8
8 can be symmetrically divided. As a result, in the spherical triangular region 82 of the spherical regular icosahedron surrounded by the imaginary line 81, it can be seen that the spherical region divided by the great circle and the small circle has three-fold rotational symmetry. This three-dimensional model 80
FIG. 3 shows a case where the circumference of the great circle 8 is equally divided into 30 by the small circle disk 10 and the great circle disk 6. In the three-dimensional model 90 of FIG. 35, a small circle closer to the great circle with the great circle in between is indicated by reference numeral 12 (No. 1), and a small circle farther from the great circle is indicated by reference numeral 12 (No. 2). It is. The three-dimensional model 90 can divide the sphere into a triangular region 94, a pentagonal region 96, and a hexagonal region 98. If the pentagonal region 96 and the hexagonal region 98 are subdivided into triangles, the sphere can be divided into three. All can be divided into triangular regions.
In the three-dimensional model 90, three adjacent pentagonal regions 9
An imaginary line 91 connecting the centers of 6 is a spherical positive 20 on the spherical surface.
One spherical triangular area 92 of the face is represented.

When the spherical surface is totally triangulated using the three-dimensional model 90, one side 91 of the spherical regular triangle 92 is formed.
Is divided into ten, and according to the definition of Fuller, 1
This means a spherical division of 0 division (10 Frequency). In other words, it is clear that the three-dimensional model 90 can perform spherical division in substantially the same form as ten divisions (10F) according to Fuller's geodetic theory. The three-dimensional model 100 of FIG. 36 shows a case where a pair of small circles 12 is newly added between the great circle 8 and the small circle 12 of the three-dimensional model 80 shown in FIG. In this case, all hexagonal regions shown in the three-dimensional model 80 are triangulated. Further, the three-dimensional model 90 of FIG.
From the comparison with the three-dimensional model 100 shown in the figure, it can be seen that the number of divisions increases only by moving the two pairs of small circles 12 away from the great circle 8 in parallel, as in the three-dimensional model 90. In particular, three-dimensional model 1 composed of the same number of great circle disks and small circle disks
When the three-dimensional model 100 is compared with the three-dimensional model 90 in terms of the number of faces when the three-dimensional model 90 is totally triangulated, the three-dimensional (6F)
Is a 720-hedron, but is divided into ten (10
In the three-dimensional model 90 of F), it can be seen that the number of spherical divisions into 2000 planes increases exponentially due to the movement of the interval between the small circular disk and the large circular disk.

The various spherical divisions described above can be used, for example, in designing domes and tents or in designing the arrangement of dimples on a golf ball. In designing the tent, when a great circle is used, an aggregate such as a rod may be arranged along the great circle and the small circle. In designing a dome, a distance between adjacent spherical lattice points is obtained from a three-dimensional model, and a straight or arc-shaped strut (an arc-shaped strut is, for example, a Gold strut presented as a conventional technique) between the lattice points. A substantially arc-shaped strut may be formed by bending a rod (frame) like a Win tent), or a surface material suitable for each divided area may be provided. The dome may be made by connecting these face materials to each other. When face materials or straight struts are used (particularly in the case of flat face materials), the grid points may be displaced outward to set the connection points of adjacent face materials or struts.

When a straight strut is used, it becomes a polyhedral dome. Further, Fuller et al. Proposed a structure composed of a compression material and a tension material, and named this structure a tensegrity. However, the spherical division according to the present invention may be used for the basic design of the tensegrity. If necessary, other polygonal regions except the triangular region may be subdivided by the reinforcing member to form a total triangle.

As is apparent from the above, according to the spherical surface dividing method of the present invention, the area ratio and the number of divisions of the divided region can be changed depending on the set position of the small circle.
Such processing can be easily performed by drawing on a computer using CAD software and 3D software.

[Brief description of the drawings]

FIG. 1 is a diagram showing a relationship between a rotation axis set at the center of a surface of a regular hexahedron and a great circle;

FIG. 2 is a diagram showing an embodiment of the present invention showing the intersection of a great circle disk and a small circle disk on the same spherical surface when a great circle is set concentrically with a regular hexahedron.

FIG. 3 is a view showing another embodiment in which a great circle disk is removed from the three-dimensional model of FIG. 2;

FIG. 4 is a diagram showing another embodiment in which the distance from the large circle disk to the small circle disk is set larger than that in FIG. 3;

FIG. 5 is a diagram showing another embodiment in which the distance between the large circle disk and the small circle disk is set smaller than that in FIG. 3;

FIG. 6 shows a rotation axis set at the center of a side of a regular icosahedron and 1
Diagram showing the relationship with five great circles

FIG. 7 is a diagram showing a relationship between a rotation axis set at a vertex of a regular icosahedron and six great circles;

FIG. 8 shows a rotation axis set at the center of the surface of a regular icosahedron and 1
Diagram showing the relationship with zero great circles

FIG. 9 shows a first icosahedron formed based on symmetry.
Diagram showing 31 great circles

FIG. 10 shows a quasi-regular tetrahedron (CUBOCTAHEDRO)
N) Diagram showing four great circles set at the center of the triangle

FIG. 11 shows a quasi-regular tetrahedron (CUBOCTAHEDRO)
N) Diagram showing three great circles set at the center of the square

FIG. 12 shows a quasi-regular tetrahedron (CUBOCTAHEDRO)
The figure which shows six great circles set in the center of the vertex of N)

FIG. 13 shows a quasi-regular tetrahedron (CUBOCTAHEDRO)
The figure which shows 12 great circles set in the center of the side of N)

FIG. 14 shows a quasi regular tetrahedron (CUBOCTAHEDRO).
FIG. 25 shows 25 first great circles formed based on N).

FIG. B. Diagram showing a regular icosahedron formed by the geodesic spherical segmentation method using Fuller's great circle

FIG. 16 is a diagram showing an embodiment of a regular icosahedron formed by the method of dividing a spherical surface using a small circular disk according to the present invention;

FIG. 17 is a layout diagram of a small circular disk when a regular octahedron is set as a core model.

FIG. 18 is a view showing another embodiment of the present invention in which a spherical surface concentric with the core model of an octahedron is divided by four pairs of small circles parallel to each other;

FIG. 19 is a layout diagram of a large circle disk and a small circle disk when a large circle disk is added to the middle of a small circle disk.

FIG. 20 is a diagram showing a relationship between a regular hexahedron of an outer core model and a regular octahedron of a dual inner core model;

FIG. 21 is a diagram showing the relationship between an outer core model regular octahedron and a dual inner core model regular hexahedron;

FIG. 22 is a diagram illustrating a relationship between vertices of an inner core model regular octahedron that divides an edge of the regular icosahedron of the outer core model by a golden division ratio.

FIG. 23 is a diagram illustrating a relationship between vertices of an inner core model icosahedron that divides an edge of an octahedron of an outer core model by a golden division ratio.

FIG. 24 is a diagram illustrating a relationship between vertices of a regular dodecahedron of an outer core model and vertices of a regular hexahedron of an inner core model.

FIG. 25 is a diagram showing a relationship in which the midpoint of the side of the inner core model regular hexahedron coincides with the midpoint of the side of the regular octahedron of the outer core model.

26 is a layout diagram of a small circle and a great circle when the two types of core models of FIG. 25 are used.

FIG. 27 is a diagram showing an embodiment according to the present invention in which the model of FIG. 18 is viewed from different viewpoints;

FIG. 28 is a view showing another embodiment in which a large circle disk and a small circle disk intersect.

FIG. 29 is a diagram showing another embodiment according to the present invention for increasing the number of spherical divisions by reducing the interval between small circular disks.

FIG. 30 is a view showing another embodiment according to the present invention in which a pair of small circular disks arranged on two types of great circular disks having different symmetries intersect with each other.

FIG. 31 shows another embodiment according to the present invention, omitting the small circular disc for four-fold rotational symmetry.

FIG. 32 shows another embodiment according to the present invention intersecting small circular disks arranged in pairs at equal intervals on each of six great circular disks symmetrical five times.

FIG. 33 is a diagram showing another embodiment according to the present invention in which six great circle disks are added to a three-dimensional model 75 to intersect with each other.

FIG. 34 is a view showing another embodiment according to the present invention for increasing the number of spherical divisions by enlarging the interval between a pair of small circular disks sandwiching a five-fold symmetric great circular disk.

FIG. 35 is a view showing another embodiment according to the present invention in which the distance between two pairs of small circle disks sandwiching a five-fold symmetric large circle disk is increased to increase the number of spherical divisions.

FIG. 36 is a view showing another embodiment according to the present invention in which two pairs of small circle disks sandwiching a five-fold symmetric large circle disk are arranged and a hexagonal area is triangulated;

[Explanation of symbols]

 6 Large circle disk 8 Large circle 10 Small circle disk 12 Small circle P Spherical lattice

Claims (20)

[Claims] 1. A spherical surface concentric with a spherical polyhedron is set, and a plurality of small circles arranged in parallel with each great circle of the spherical polyhedron on this spherical surface and arranged on both sides of each great circle are defined by A method for dividing a spherical surface, wherein the method is set for each great circle and the spherical surface is divided by the intersection of these small circles. 2. The method according to claim 1, wherein the small circle comprises one or a plurality of pairs of small circles arranged at equal intervals with the great circle interposed therebetween. 3. The spherical surface dividing method according to claim 1, wherein a spherical surface is divided by adding a great circle of the spherical opposing body. 4. The method according to claim 1, wherein the spherical polyhedron is a Plato Archimedes solid. 5. A plurality of small circles arranged in parallel with each of the great circles selected from the first and second great circles of the regular icosahedron and arranged on the left and right of each great circle, A spherical surface dividing method characterized by dividing a spherical surface by the intersection of the small circles. 6. The spherical division method according to claim 5, wherein the small circle comprises one or a plurality of pairs of small circles arranged at equal intervals with the great circle interposed therebetween. 7. The method for dividing a spherical surface according to claim 5, wherein the spherical surface is divided by adding the selected great circle. 8. A plurality of small circles arranged parallel to each of the great circles selected from the first and second great circles of the quasi-regular tetrahedron and arranged on the left and right of each great circle are set. A spherical surface is divided by the intersection of the small circles. 9. The method of claim 8, wherein the small circle comprises one or a plurality of pairs of small circles arranged at equal intervals with the great circle interposed therebetween. 10. The method for dividing a spherical surface according to claim 8, wherein the spherical surface is divided by adding the selected great circle. 11. A method of setting a spherical surface concentric with a core model and dividing the spherical surface by setting parallel small circular disks extending along a surface, a vertex, a side, or a surface obtained by truncating the surface of the core model. A spherical surface division method characterized by the above-mentioned. 12. The method according to claim 11, wherein the core model comprises a plurality of concentrically set spherical polyhedrons. 13. The spherical surface dividing method according to claim 11, wherein the spherical surface is divided by setting a great circle between the small circles in addition to the pair of small circles parallel to each other. . 14. A spherical structure divided into spherical surfaces by arranging a plurality of small circle groups parallel to each other in a rotationally symmetric manner. 15. The spherical structure according to claim 14, wherein the small circle group has a great circle parallel to the small circles. 16. The spherical structure has a polyhedron shape, and the vertices of the polyhedron are spherical lattice points formed by the small circles intersecting each other or the small circle and the great circle.
A spherical structure according to claim 14. 17. A spherical structure designed based on the method of dividing a spherical surface according to any one of claims 1 to 13. 18. The spherical structure according to claim 17, wherein the spherical structure has a polyhedral shape, and vertices of the polyhedron are spherical lattice points formed by the spherical surface division method. 19. The spherical structure according to claim 17, wherein the spherical structure is designed based on the remaining spherical lattice points excluding some of the spherical lattice points. 20. The spherical structure according to claim 17, wherein a part of the sides dividing the divided area formed by the spherical surface division method is omitted.