Table tennis ball with markings enabling detection of ball rotation

Abstract

The invention describes a table tennis ball (2) on the spherical ball surface (6) of which a marking (8) is applied which makes the ball spin detectable with measuring technology. The marking (8) comprises a number of marking points (P i ) which are distributed on the ball surface (6) in such a way that the standard deviation of the lengths (Z i,j ) of the great circle lines (20) between each marking point (P i ) and its three nearest neighbours (14, 16, 18) is at least 12% of the average value (μ) of these lengths (Z i,j ) and the minimum length of the great circle lines (20) between each marking point (P i ) and its three nearest neighbours (14, 16, 18) is at least 40% of the average value (μ) of these lengths (Z i,j ).

Description

Technical Field

[0001] This invention relates to a ping-pong ball (hereinafter referred to as "the ball") having a spherical surface and markings applied to the surface such that the ball's spin can be detected by measurement techniques, wherein the markings comprise a number of marked points. Such a ping-pong ball is known, for example, from WO 2020 096120 A1. Background Technology

[0002] In table tennis, besides ball speed, ball spin (also known as "self-spin") is crucial because it decisively influences the ball's trajectory and its rebound behavior on the table and the opponent's racket. Especially in competitive table tennis, it is essential to determine the spin as accurately as possible during flight to allow coaches to objectively monitor and specifically improve players' techniques. Furthermore, in tournaments, and within the scope of competition review and sports reporting, it is necessary to measure and analyze the ball's motion (as much as possible in real time).

[0003] To make the spin of a ping-pong ball visible in flight, it is known in principle to apply markings to the surface of the ball. For example, the German Wikipedia entry on "ping-pong" (https: / / de.wikipedia.org / wiki / Tischtennisball; version as of April 17, 2020, 20:51) describes a ping-pong training ball with a colored pattern consisting of large areas of color printed on its surface as a mark for spin identification.

[0004] In a table tennis ball known from WO 2020 096120 A1, the markings applied for spin identification consist of six marking points evenly distributed on the surface of the ball. Each marking point has a diameter between 5 mm and 13 mm.

[0005] Similar to table tennis, golf also requires the measurement of ball spin during flight. Here, markings, particularly in the form of dotted patterns, applied to the ball's surface are used to allow for the measurement of spin at the tee shot. Relevant golf balls are known, for example, from CN 107 543 530 A, KR 102 101 512 B1, US2018 0353828 A1, and US 7 062 082 B2. The markings disclosed herein each have one or more groups comprising a plurality of closely adjacent marked dots.

[0006] Alternatively, according to CN 106 643 662 A, markings with one-dimensional or two-dimensional structures (e.g., lines) are applied to golf balls to identify spin. Again, as an alternative, JP 2016 218014 A suggests using manufacturer markings applied to the golf ball as markings to make spin detectable.

[0007] While automated methods for measuring spin in golf balls have been successfully used, a similar method has not yet been established for table tennis. This is primarily because traditional measurement methods are too slow for meaningful application in table tennis. Compared to golf, table tennis is a very fast-paced sport characterized by a high frequency of shots; during ball exchanges, the ball typically crosses the net more than once per second. Worse still, unlike golf, the ball's position and orientation are unknown at the start of the measurement. In golf, spin measurement is typically performed at impact, before which the ball is stationary (thus accurately defining its position and orientation), while in table tennis, spin is measured in flight (mostly upon crossing the net). However, because the trajectory of a table tennis ball can only be predicted in advance with very high inaccuracies, the measuring device used for calculating table tennis spin must first locate the ball before it can measure the spin, which takes considerable time. Furthermore, the orientation of the table tennis ball at the point of detection by the measuring device is unknown beforehand and must be determined first. For these reasons, until now, spin measurement of a ping-pong ball in flight has only been possible under favorable conditions, and only with a considerable time delay depending on the progress of the match. Reliable spin information is currently unavailable in every match situation (especially near real-time, i.e., close to the time interval between two shots). Summary of the Invention

[0008] The objective of this invention is to enable the automatic detection of the spin (rotation) of a ping-pong ball in flight in a particularly fast and reliable manner.

[0009] This task is solved according to the invention by means of a ping-pong ball having the features described below. Accordingly, a ping-pong ball is described, having a spherical surface and markings applied to the surface such that the ball's rotation (spin) can be detected by measurement techniques. As is known from WO 2020 096120 A1 itself, the markings comprise a plurality of marked points. The number of marked points is described below by the variable N. The radius of the ping-pong ball is described below by the variable R.

[0010] Preferably, the markings associated with making spin detectable consist solely of dot marks. In other words, in this case, the markings are pure dot patterns with no other structure besides the dot marks. That is, in this variant of the invention, the ping-pong ball may also have other structures or patterns on its surface. However, such other structures are not part of the markings associated with knowing the spin and are not evaluated when knowing the spin.

[0011] According to the present invention, the distribution of marked points on the surface of a sphere is characterized by a certain number of criteria:

[0012] According to the first criterion, the standard deviation of the length of the great circle between each marker point and its three nearest neighbors is at least 12% of the average of these lengths. Therefore, when calculating the standard deviation, the distance between all marker points and their respective three nearest neighbors is taken into account.

[0013] According to the second criterion, the minimum length of the great circle between each marker point and its three nearest neighbors must be at least 40% of the average length of these great circles. In other words, according to the second criterion, the length of the great circle between a marker point and its three nearest neighbors must be at least 40% of the average length of these great circles.

[0014] As an addition to or alternative to the second criterion, the distribution of points is characterized in that the minimum length of the great circle between each marked point and its three nearest neighbors is at least 120% of the quotient of the radius of the sphere and the square root of the number of points N. In other words, according to the third criterion, the length of the great circle between a marked point and its three nearest neighbors is no less than 120% of the quotient of the radius of the sphere and the square root of the number of points.

[0015] The second and third criteria also take into account the distance between each marked point and its three nearest neighbors.

[0016] This invention is based on the understanding that the feasibility of reliably and numerically simple and thus quickly determining the spin of a ping-pong ball depends primarily on the type of markings applied to the ping-pong ball.

[0017] In the first step, the present invention is based on the consideration that markers with one-dimensional or two-dimensional structures are not conducive to the automatic determination of spin, because such structures can only be detected by numerically complex and therefore relatively time-consuming image recognition measures (especially by segmentation). In contrast, point-like structures are known to be detectable in a way with very low numerical complexity. Therefore, the present invention focuses on developing a marker consisting of, or at least including, a certain number of marker points.

[0018] However, the uniform distribution of markers suggested in WO 2020 096120 A1 is considered unfavorable because it makes it impossible to clearly identify the ball's orientation from photographic images of the ping-pong ball. More specifically, the suggested orientation of the ball has six distinct orientations, which are marked to reflect the ball itself, thus making these orientations indistinguishable in photographic images of the ball. This redundancy in marking is known to cause markers that can be identified in different images to be incorrectly assigned to each other when comparing and evaluating time-series images of the ball, which in turn leads to errors in spin calculations.

[0019] However, the grouping of marker points suggested in CN 107 543 530 A, KR 102 101 512 B1, US 2018 0353828 A1, and US 7062 082 B2 has also proven to be meaningless for spin detection in table tennis. This is primarily because the ball's orientation (unlike in golf at the moment of impact) is unknown at the start of a table tennis match when measured using measurement techniques. Therefore, since the table tennis ball can enter the detection range of the measuring device in any orientation, at the time of detection, many clustered measurement points have a relatively high probability of not being on the ball side facing the measuring device, and thus being poorly imaged or not imaged at all. In these cases, conventional spin detection based on the evaluation of a temporal image of the ball often fails. Although, this problem is known to be solved by arranging the groups of points more densely on the ball's surface. However, this comes at the cost of high numerical complexity in identifying the markers and reduced accuracy in spin detection (due to the large number of markers and their denser stacking).

[0020] In view of the shortcomings of the prior art, the present invention adopts a compromise approach. It is based on the idea that while the markers are distributed as evenly as possible globally (i.e., involving the entirety of the markers), they are distributed as unevenly and irregularly (pseudo-randomly) locally (i.e. involving individual markers and their nearest neighbors).

[0021] The global uniformity of the point distribution enables the identification of particularly large subgroups of markers in every possible orientation of the sphere, even when the overall marking consists of only a relatively small number of markers. This ensures that the detected markers are identified numerically without complexity. However, the local non-uniformity of the point distribution greatly simplifies the identification of individual markers, as each marker can be easily, clearly, and reliably identified by considering its arrangement relative to its neighbors.

[0022] The aforementioned characteristics, namely, the global uniformity of the distribution of the marked points according to the invention on the one hand and the local non-uniformity on the other, can be described particularly simply and clearly by the length of the great circle line between each marked point and its three nearest neighbors. The term "great circle line" here refers to the shortest connection between the separately observed marked points on the surface of a sphere.

[0023] The "nearest neighbor" of the observed marker point ("center point") is always the marker point connected to the center point via a great circle of the shortest length. Correspondingly, the "second nearest neighbor" and "third nearest neighbor" of the center point are always those marker points connected to the center point via great circles of the second or third shortest length. In this definition, each marker point has one nearest neighbor, one second nearest neighbor, and one third nearest neighbor, and the center point can be represented according to this definition. The markers are preferably chosen such that, for each center point, the three nearest neighbors are paired with each other at different distances, thus explicitly assigning these neighbors to the center point as the nearest neighbor, second nearest neighbor, and third nearest neighbor. However, within the scope of this invention, one or more marker points may also have the same distance from two of their three nearest neighbors. In other words, in individual cases, the nearest neighbor and the second nearest neighbor and / or the second nearest neighbor and the third nearest neighbor may be equidistant from the center point.

[0024] A group of markers consisting of an observed marker (center point) and its three nearest neighbors is also referred to below as a "four-point network". A four-point network has a number of markers corresponding to the number of markers.

[0025] For the sake of linguistic simplicity, the length of the great circle will be referred to as the "point spacing" both above and below. The respective point spacing is measured along the surface of the sphere (depending on the orientation of the great circle to which it belongs).

[0026] The length of the great circle connecting the center point of each of the four points in the observed network to one of its three nearest neighbors is referred to below as the "center-to-center distance" and is denoted by the formula symbol Z. i,j Let's explain. The rotation variable *i* (where *i* = 1, 2, ..., N) refers to the marker point considered the center point in its respective relation. The rotation variable *j* (where *j* = 1, 2, 3) refers to the nearest, second nearest, or third nearest neighbor to the *i*-th center point. In this sense, the formula symbol Z... 5,2 For example, it refers to the center point distance between the fifth marker point, which is considered the center point, and its next nearest neighbor.

[0027] The aforementioned four-point network has proven particularly advantageous because the three nearest neighbors are known, especially when the markers are nearly uniformly distributed, to form a more or less distinct and closed first shell around their respective center points. Here, the three nearest neighbors are characterized by their similar point spacing from the center point, while the center point (depending on the number of points and the uniformity of the pattern) has a more or less significant point spacing from its more distant neighbors.

[0028] The aforementioned first criterion for the point distribution, according to which the standard deviation (hereinafter referred to as σ) of the length of the great circle between each marked point and its three nearest neighbors is at least 12% of the average of these lengths (hereinafter referred to as µ), can thus be written as:

[0029] Equation 1

[0030] in Equation 2

[0031] and Equation 3

[0032] Preferably, the aforementioned center point spacing Z i,j The standard deviation σ is at least 15% of the mean µ. As can be seen from equations 2 and 3, when calculating the standard deviation σ and the mean µ, all marked points P are respectively... i The distance Z between the three nearest (center) points i,j Add them together, that is, add Z together. 1,1 Z 1,2 Z 1,3 Z 2,1 Z 2,2 Z 2,3 ... Z N,1 Z N,2 Z N,3 Add them together.

[0033] A smaller standard deviation σ indicates that, in any case, for most marker points P i In this context, the three nearest neighbors are located at the marked point P, which is considered the center point. i Within the relatively thin ring surrounding it, or (in other words) in any case, most of the marked points P i The points are arranged so that their distances to their three nearest neighbors are very close. Therefore, the characteristic parameters σ and σ / µ represent measures of the uniformity of the point distribution. The minimum standard deviation given earlier ensures that the point distribution within the four-point network does not become overly uniform.

[0034] The second criterion for the distribution of the points mentioned above is that the minimum length of the great circle between each marked point and its three nearest neighbors is the minimum (center) distance d on the surface of the sphere. min It is at least 30% of the average of these lengths, and therefore can be written as:

[0035] Equation 4

[0036] Preferably, the minimum (i.e., minimal) point spacing d min It should be at least 50% of the average value µ, and especially at least 60%. As can be seen from Equation 4, the second criterion also applies to all marked points P. i The distance Z between the three nearest (center) points of each i,j That is, applicable to Z 1,1 Z 1,2 Z 1,3 Z 2,1 Z 2,2 Z 2,3 ... Z N,1 Z N,2 Z N,3 .

[0037] The second criterion for point distribution eliminates the possibility of excessively small spacing between marker points. This, in particular, prevents the formation of marker point clusters. Therefore, the second criterion ensures approximately global uniformity in the point distribution.

[0038] Another measure of the global uniformity of point distribution is the minimum (center) point spacing d. min The ratio of the marker point P to the sphere's radius R. Assuming N marker points are distributed as evenly as possible on the sphere's surface, the ratio of marker point P to the sphere's radius R is... i Generally, the points should be as far apart as possible. In other words, avoid pairs of points with minimal spacing (i.e., groups of closely adjacent markers). Similarly, it can be assumed that, in the case of the most uniform distribution possible, each marker is arranged in a different fraction of the surface of the sphere, that is, each marker is more or less arranged in a portion of the surface of the sphere that has no other markers attached, wherein each of these portions has an extension dimension A of the fraction of the surface of the sphere. p .

[0039] Equation 5

[0040] If we now assume these partial surfaces are approximately flat circular surfaces, then according to Equation 5 and the formula for the area of ​​a circle (A... p ≈0.25·π·d G 2 Estimate the diameter d of part of the surface. G :

[0041] Equation 6

[0042] Therefore, the diameter d G This represents a measure of the typical spacing between uniformly distributed markers and their respective neighbors on the surface of a sphere. As can be seen from Equation 6, this typical spacing decreases approximately with the square root of the number of points N. Because the dimensional extension of the sphere's surface is limited, the spacing between markers generally cannot be increased without decreasing the spacing at other points on the sphere's surface. Therefore, the minimum point spacing d... min With diameter d G The ratio is a measure of how strongly the point distribution deviates from global uniformity.

[0043] According to the aforementioned third criterion, the minimum point spacing d min At least the typical spacing d calculated according to Equation 6 G 30% or (equivalently) at least 120% (i.e., 1.2 times) of the quotient of the square root of the sphere radius R and the number of points N:

[0044] Equation 7

[0045] Preferably, the minimum point spacing d min At least the typical spacing d G 37.5%, especially at least 45%. Equivalently, the minimum point spacing d min Preferably, it is at least 150% (i.e., 1.5 times) of the quotient of the square root of the sphere radius R and the number of points N, and more particularly at least 180% (i.e., 1.8 times). In a pattern with 18 marker points (N=18), the minimum point spacing d min Therefore, it is at least 28% of the sphere radius R, preferably at least 35%, and especially at least 42%.

[0046] As mentioned above, the second and third criteria are alternatives to each other or considered in combination within the scope of the invention, but are always considered in combination with the first criterion for characterizing the sphere pattern according to the invention.

[0047] The length of the great circle connecting the three nearest neighbor points is different from the distance Z between the center points. i,j The area is referred to as "peripheral point spacing" Q. i,k Here, the rotation variable *i* refers to the marker point considered as the center point. Conversely, the rotation variable *k* (where *k* = 1, 2, 3) represents:

[0048] - For k=1, the distance between the nearest neighbor and the second nearest neighbor of the i-th marked point.

[0049] - Regarding k=2, the distance between the second nearest neighbor and the third nearest neighbor of the i-th marked point, and

[0050] - For k=3, the distance between the third nearest neighbor and the nearest neighbor of the i-th marker.

[0051] Therefore, in this sense, the formula symbol Q is chosen as an example. 5,2 This refers to the distance between the second nearest neighbor and the third nearest neighbor of the fifth marker point, which is considered the center point.

[0052] In a preferred embodiment of the invention, the marker points are distributed on the surface of the sphere according to a fourth criterion, such that the distance Q between the outer points of each four-point network is... i,k The range (hereinafter referred to as ΔQ) i In other words, the difference between the longest and shortest great circle between the three nearest neighbors of each marker point is greater than the center distance Z. i,j 30% of the aforementioned average value µ:

[0053] For each i = 1, 2, ..., N

[0054] Equation 8

[0055] This fourth characteristic of the point distribution according to Equation 8 can be considered not only in combination with one of the aforementioned characteristics of the marking, but also advantageously considered in isolation in the ping-pong ball described at the very beginning, for characterizing a particularly suitable point distribution, and is thus regarded as an independent invention. Range ΔQ i Preferably greater than 35% of the average value µ, and especially even greater than 40%.

[0056] For the range ΔQ i The optimal determination of the lower limit is based on the understanding that the range ΔQ i This illustrates the deviation between the triangle formed by the three nearest neighbors of a four-point network and an equilateral triangle, and thus from a locally uniform point distribution. Therefore, the range ΔQ i A sufficiently large value clearly represents a particularly convincing measure of the expected local non-uniformity of the point distribution.

[0057] In a preferred embodiment of the invention, the number of markers is between 13 and 25, preferably between 16 and 21, and especially 18 or 19. This is based on the experience that 18 or 19 markers distributed on the surface of a sphere can achieve spin identification optimized in terms of numerical complexity and reliability.

[0058] With fewer markers, the following situation occurs (more frequently as the number of markers decreases): there are not enough markers clearly visible in the sphere image to allow for definitive identification of identifiable markers. This can lead to inaccuracies or errors in spin calculations.

[0059] On the other hand, as the number of markers increases, the number of times the groups of markers detected by measurement techniques must be compared to identify their probabilities also increases disproportionately. Thus, markers with more than 19 markers lead to an increase in numerical complexity when determining spin.

[0060] However, the advantages achieved by the present invention are also obtained to a less obvious degree by using markers containing 16, 17, 20 or 21 marker points; even less obvious are markers with 13 to 15 or 22 to 25 marker points.

[0061] The diameter of each marker is preferably chosen such that it is between 10% and 24% of the sphere's radius, more preferably between 15% and 20%, and especially 17.5%. In absolute terms, the diameter of each marker is preferably between 2.0 mm and 4.8 mm, more preferably between 3.0 mm and 4.0 mm, and especially 3.5 mm. This determination of the marker size has proven particularly advantageous because these markers are, on the one hand, large enough to be reliably detected on the image of the sphere even under adverse conditions (e.g., when the sphere is relatively far from the measuring device). On the other hand, the markers, thus determined in this way, are small enough that they can be considered as usable approximate point-like (i.e., sizeless) structures, thereby enabling numerically simple image evaluation. In another alternative embodiment, simple image evaluation is also advantageous because all markers have the same shape and size.

[0062] In one advantageous embodiment, the marking point is characterized by having infrared absorption or re-emission characteristics different from the rest of the ball's surface, thus making the marking point stand out in contrast from the rest of the ball's surface in the infrared (IR) range of the electromagnetic radiation spectrum, particularly in infrared images of the ping-pong ball. Specifically, the marking point is applied to the ball's surface using an infrared-sensitive coating (wherein this infrared-sensitive coating is used only for the marking point and not for other markings or patterns that may be present on the ball's surface). This feature enables automatic determination of spin by photographing the ball using an infrared camera, which is particularly insensitive to interference in the visible light spectrum (e.g., glare from spotlights or flashes). Furthermore, the contrasting infrared absorption or re-emission characteristics allow for selective photographing of the markings for spin determination. Thus, in addition to the markings, the ball can be largely freely decorated with other patterns or prints without interfering with spin determination.

[0063] In another embodiment of the invention, the markings are designed such that, by applying a specific common color within the visible range of the electromagnetic radiation spectrum, the marking points stand out against the background of the rest of the ball's surface. This feature also allows for the selective photographing of the markings for spin detection purposes, for example, by taking a color-filtered photograph of the ping-pong ball or through subsequent image processing. Therefore, this feature also allows for the application of other patterns or prints (one or more other colors) to the ball without interfering with spin detection.

[0064] In a particularly advantageous embodiment, the ping-pong ball has a marking system comprising 18 points. These points are collectively referred to below as P. i (Where i = 1, 2, ..., 18). These 18 marked points, each measured on the radius R of the sphere, are arranged at a center-to-center distance Z, as detailed below, to their nearest neighbor. i,j Inside:

[0065]

[0066] Table 1: Optimal center-point spacing for markers with 18 marker points

[0067] According to the marked point P in Table 1 i The arrangement scheme can be used not only in combination with one of the aforementioned features of the mark, but also advantageously used in isolation in the ping-pong ball described at the very beginning, and is thus considered an independent invention.

[0068] Furthermore, preferably, the adjacent points of each four-point network are spaced apart by the following peripheral point spacing Q. i,k Arrangement:

[0069]

[0070] Table 2: Spacing between outer points of the preferred marker with 18 marker points Attached Figure Description

[0071] The embodiments of the present invention are further described below with reference to the accompanying drawings. Wherein:

[0072] Figures 1 to 6 The table tennis ball is shown in six top views from six mutually perpendicular viewing directions. It has a spherical surface and markings applied to it so that its spin in flight can be detected by measurement techniques. These markings consist of 18 marked points distributed across the surface of the ball.

[0073] Figure 7 According to Figure 1 The illustration shows a subgroup of markers (“four-point network”), which is formed by the first marker point, which is regarded as the center point, and its three nearest neighbors.

[0074] Corresponding parts, dimensions, and structures are always given the same reference numerals in all the accompanying drawings. Detailed Implementation

[0075] Figures 1 to 6 The (ping-pong) ball 2 is shown in six top views from six mutually perpendicular viewing directions. Figures 1 to 6 The spatial relationships between the views shown are indicated by arrows labeled with Roman numerals in these figures: the Roman numerals indicate the viewing direction based on the corresponding numerical values ​​in the figures. Therefore, Figure 1 The arrow marked with the Roman numeral II indicates that it was based on Figure 2 The view produces the viewing direction, etc.

[0076] Ball 2 typically has a hollow outer shell 4, especially made of plastic. Ball 2 particularly has a ball radius R of 20 mm or a ball diameter of 40 mm, which conforms to the rules of table tennis.

[0077] In order to enable the rotation (spin) of the ball 2 during its flight to be detected by measurement techniques, the spherical surface 6 of the outer shell 4 (and thus the entire ball 2) is provided with markings 8. In the example shown, the markings 8 consist of 18 marking points P distributed on the spherical surface 6. i (Where i = 1, 2, ..., 18) constitutes this. In Figures 1 to 6 In the diagram, for clarity, mark 8 is schematically shown. Marker point P is located at the edge of the visible portion of the sphere surface 6. i The distortion in perspective is not shown.

[0078] Each marker point P i From diameter d ( Figure 7 A circular surface with a diameter of 3.5 mm (corresponding to 17.5% of the sphere's radius R) is formed. Mark point P. i Here, a coating with strong infrared absorption and / or infrared re-emission, a printing pigment / ink with strong infrared absorption and / or infrared re-emission (see, for example, DE 10 2008 049595 A1), or other coating with strong infrared absorption and / or infrared re-emission is applied to the spherical surface 6. Marking point P is located within the visible spectrum of the electromagnetic radiation spectrum (i.e., within the visible light spectrum). i This can have a color that contrasts with the surrounding spherical surface 6. In this case, the marker point P... iIt also stands out visually from the surrounding spherical surface 6. However, the marker point P i Preferably, it is transparent to the human eye or has the same or similar color as the surrounding spherical surface 6. In the latter case, the marker point P i It is only clearly visible in the infrared image of sphere 2, but it is invisible to human observers and to the photograph of sphere 2, or at least not noticeable.

[0079] The infrared-sensitive implementation scheme described above for marker 8 achieves this by making marker point P i In the infrared image of sphere 2, it stands out sharply from the rest of the sphere surface 6. Optionally, sphere 2 may also be additionally provided with additional structures (e.g., printing or patterns) applied to the sphere surface 6 by means of at least one color that has no absorption or only slight absorption of infrared light. Unlike this possible additional structure, the mark 8 also stands out strongly in the infrared image, so that, conversely, the possible structures do not affect or impair the information about the spatial orientation of sphere 2 presented by the mark 8.

[0080] Marked point P on sphere 6 i The positions are shown in spherical coordinates in the table below, meaning they depend on the polar angle θ and the azimuth angle. To illustrate.

[0081]

[0082] Table 3: Marking point P in spherical coordinates i Position on the surface of a ping-pong ball

[0083] In this layout scheme, the marker point P i From a global perspective, the points are roughly uniformly distributed on the sphere 6. However, compared to an ideal uniform distribution, the marked points P... i They are still arranged in a pseudo-random manner in a recognizable way.

[0084] This characteristic of point distribution is particularly evident when considering local point environments (referred to as four-point networks10), each represented by a single labeled point P. i As the center point 12 and its three nearest neighbor points P i Formed based on marker point P. i The number of points can divide the 8 markers into 18 overlapping four-point networks 10.

[0085] The four-point network 10 of the first marker point P1 is exemplarily in Figure 7 As shown in the image. It can be seen here that...

[0086] - The nearest neighbor 14 of marker P1 is formed by marker P2.

[0087] - Point 16, the second nearest neighbor of marker P1, is formed by marker P5, and

[0088] - The third nearest neighbor point 18 of marker P1 is located at marker P. 17 form.

[0089] As defined above, the center point 12 of each four-point network 10 (in accordance with...) Figure 7 In the example, point P1 is marked with one of its neighboring points 14, 16, and 18 respectively (according to...). Figure 7 In the example, the marked points are P2, P5, or P. 17 The length of the 20-degree concentric circle connecting the two points is called the center-to-center distance Z. i,j Conversely, the length of the great circle 20 connecting the adjacent points 14, 16, and 18 of each four-point network 10 (as defined above) is called the outer point spacing Q. i,k For the four-point network 10 with the first marked point P1, in Figure 7 Draw the corresponding center point spacing Z. 1,1 Z 1,2 Z 1,3 Spacing Q with the outermost point 1,1 Q 1,2 Q 1,3 .

[0090] Regarding the 18 marked points P given in Table 3 i The layout scheme, with center point spacing Z i,j The values ​​are obtained from the right column of Table 1.

[0091] According to Equation 3, regarding the center point spacing Z... i,j The average value µ is obtained, which is equivalent to 0.80 times the sphere radius R (i.e., µ / R = 0.80). Therefore, when the sphere radius R is 20 mm, each marked point P i The average distance between its three nearest neighbors is 16.1 mm (i.e., µ = 16.1 mm). The standard deviation σ according to Equation 2, for the point distribution described in Table 3, yields a value of 0.126. The normalized standard deviation σ for the mean µ is 15.7% (i.e., σ / µ = 15.7%). Minimum (center) point distance d min At points P9 and P 14 The sphere is formed between these spheres and has a radius of 10.1 mm. This is equivalent to 63% of the average value µ (i.e., ).

[0092] Here, the minimum point spacing d min The typical spacing d is calculated based on Equation 6. G53.6%, or (equivalently) at least 214% of the quotient of the square root of the sphere radius R and the number of points N, or 50.5% of the sphere radius R.

[0093] Regarding the characteristic parameter σ / µ, the point distribution given in Table 3 is higher than that of the marked point P. i The most ideal uniform distribution; for 18 marked points P i For this uniform distribution, a comparative value of σ / µ = 10.1% has been found in the experiment. On the other hand, the point distribution given in Table 3 is also different from the average random point distribution, in which the standard deviation σ normalized to the mean µ is significantly higher; however, the minimum center distance often violates the condition according to Equation 4.

[0094] Regarding the point layout scheme in Table 3, the spacing Q of the outer points is... i,k The point arrangement scheme is derived from the right column of Table 2. Equation 8 is used to normalize the range ΔQ based on the average value µ. i For all four-point networks 10, greater than or equal to 42.8% (min{ΔQ) i / µ; i = 1, 2, ..., 18} = ΔQ 10 / µ = 42.8%.

[0095] The characteristic parameter ΔQ i The minimum value (which represents a measure of the local non-uniformity of the point distribution) is significantly higher in the case of the point distribution given in Table 3 than in the case of 18 marked points P. i The ideal uniform distribution or average random arrangement.

[0096] The point distribution characteristics in Table 3 indicate a clear uniformity in global observation, but exhibit particularly high irregularity in local observation. These two characteristics, when combined, result in the visible marker point P of sphere 2 in any view. i It is easier to identify clearly, and thus decisively facilitates the knowledge of spin.

[0097] To determine the spin of ball 2, a time-series image of ball 2 in flight is captured using a camera. An infrared camera is preferred to fully utilize the marker point P in the infrared spectrum. i The camera offers excellent contrast. It is preferably positioned laterally on the ping-pong table at net height, ensuring the camera is parallel to the table's horizontal orientation.

[0098] The images of the ball 2 captured are evaluated, for example, by applying the method known from US 7,062,082 B2, in order to determine the spin (in particular with respect to the axis of rotation, direction of rotation, and speed of rotation of the ball 2).

[0099] In particular, according to the invention, reference 8 in the embodiments shown in the accompanying drawings and detailed in Table 3, it is possible to obtain spin in a particularly simple numerical manner and, in particular, particularly quickly, thereby realizing for the first time the meaningful use of an automatic spin acquisition method in table tennis, especially in near real-time.

[0100] The present invention becomes particularly clear from the embodiments described above, but is not limited thereto. Rather, other embodiments of the invention can be derived from the above description.

[0101] List of reference numerals

[0102] 2 balls

[0103] 4. Outer shell

[0104] 6 spherical surfaces

[0105] 8 Marks

[0106] 10 Four-point network

[0107] 12 Center Points

[0108] 14 (Nearest) Neighboring Points

[0109] 16 (nearest) neighboring points

[0110] 18 (Third nearest) neighboring points

[0111] 20 large circles

[0112] d (diameter of the marker point)

[0113] P i Mark the points (i=1, 2, ..., 18)

[0114] Z i.j Center point spacing (i=1, 2, ..., 18; j=1, 2, 3)

[0115] Q i.k Spacing between outermost points (i=1, 2, ..., 18; k=1, 2, 3)

[0116] R sphere radius

Claims

1. A ping-pong ball (2), having a ball radius (R), a spherical surface (6), and a mark (8) applied to said surface (6) such that the ball's rotation can be detected by measurement techniques, wherein, The marker (8) includes a certain number N marker points. Its features are, The marking points are distributed on the surface of the sphere (6) in such a way that - The standard deviation of the length of the great circle (20) between each of the marked points and its three nearest neighbors (14, 16, 18) is at least 12% of the average (µ) of these lengths, and - The minimum length of the great circle line (20) between each of the marked points and its three nearest neighbors (14, 16, 18) is at least 40% of the average of these lengths (µ) and / or at least 120% of the quotient of the sphere radius (R) and the square root of the number of the marked points N.

2. The ping-pong ball (2) according to claim 1. in, The markers are distributed on the surface of the sphere (6) such that the standard deviation of the length of the great circle (20) between each marker and its three nearest neighbors (14, 16, 18) is at least 15% of the average (µ) of these lengths.

3. The ping-pong ball (2) according to claim 1 or 2. in, The minimum length of the great circle line (20) between each of the marked points and its three nearest neighbors (14, 16, 18) is at least 50% of the average of these lengths (µ).

4. The ping-pong ball (2) according to claim 1 or 2. in, The minimum length of the great circle line (20) between each of the marked points and its three nearest neighbors (14, 16, 18) is at least 150% of the quotient of the radius (R) of the sphere and the square root of the number of points N.

5. The ping-pong ball (2) according to claim 1 or 2, having a spherical surface (6) and a mark (8) applied to said surface (6) such that the ball's rotation can be detected by measurement techniques, wherein, The marker (8) includes a certain number of marker points. The markers are distributed on the surface of the sphere (6) such that the range of the length of the great circle (20) connecting the three nearest neighbors (14, 16, 18) of each marker is greater than 30% of the average length (µ) of the great circle (20) between each marker and its three nearest neighbors (14, 16, 18).

6. The ping-pong ball (2) according to claim 1 or 2. in, The number of markers is between 13 and 25.

7. The ping-pong ball (2) according to claim 1 or 2. in, The diameter (d) of each marker point is between 10% and 24% of the radius (R) of the sphere.

8. The ping-pong ball (2) according to claim 1 or 2. in, The diameter (d) of each marker point is between 2.0 mm and 4.8 mm.

9. The ping-pong ball (2) according to claim 1 or 2. in, All markers have the same shape and size.

10. The ping-pong ball (2) according to claim 1 or 2. in, The marker point has infrared absorption and / or infrared re-emission characteristics that are different from the rest of the spherical surface (6), thereby making the marker point stand out in contrast from the rest of the spherical surface (6) in the infrared range of the electromagnetic radiation spectrum.

11. The ping-pong ball (2) according to claim 1 or 2. in, The marker point has a different color from the rest of the spherical surface (6), so that the marker point stands out in contrast from the rest of the spherical surface (6) in the visible range of the electromagnetic radiation spectrum.

12. A ping-pong ball (2), having a ball radius (R), a spherical surface (6), and a mark (8) applied to said surface (6) such that the ball's rotation can be detected by measurement techniques, wherein, The marker (8) includes a certain number of marker points. The markers are distributed on the surface of the sphere (6) such that the range of the length of the great circle (20) connecting the three nearest neighbors (14, 16, 18) of each marker is greater than 30% of the average length (µ) of the great circle (20) between each marker and its three nearest neighbors (14, 16, 18).

13. The ping-pong ball (2) according to claim 12. in, The number of markers is between 13 and 25.

14. The ping-pong ball (2) according to claim 12 or 13. in, The diameter (d) of each marker point is between 10% and 24% of the radius (R) of the sphere.

15. The ping-pong ball (2) according to claim 12 or 13. in, The diameter (d) of each marker point is between 2.0 mm and 4.8 mm.

16. The ping-pong ball (2) according to claim 12 or 13. in, All markers have the same shape and size.

17. The ping-pong ball (2) according to claim 12 or 13. in, The marker point has infrared absorption and / or infrared re-emission characteristics that are different from the rest of the spherical surface (6), thereby making the marker point stand out in contrast from the rest of the spherical surface (6) in the infrared range of the electromagnetic radiation spectrum.

18. The ping-pong ball (2) according to claim 12 or 13. in, The marker point has a different color from the rest of the spherical surface (6), so that the marker point stands out in contrast from the rest of the spherical surface (6) in the visible range of the electromagnetic radiation spectrum.

19. A ping-pong ball (2), having a ball radius (R), a spherical surface (6), and a mark (8) applied to said surface (6) such that the ball's rotation can be detected by measurement techniques, wherein, The marker (8) includes a certain number of marker points. - Wherein, the mark (8) includes 18 mark points, - Wherein, starting from the first marker point P1 - The length Z of the great circle line (20) to its nearest neighbor P2 1,1 The ratio to the sphere radius (R) is 0.54 ± 20%. - The length Z of the great circle line (20) to the next nearest neighbor point P5 1,2 The ratio to the sphere radius (R) is 0.91 ± 20%, and - to its third nearest neighbor P 17 The length Z of the large circle line (20) 1,3 The ratio to the sphere radius (R) is 0.99 ± 20%. - Wherein, starting from the second marker point P2 - The length Z of the great circle line (20) to its nearest neighbor point P1 2,1 The ratio to the sphere radius (R) is 0.54 ± 20%. - The length Z of the great circle line (20) to the next nearest neighbor point P4 2,2 The ratio to the sphere radius (R) is 0.72 ± 20%, and - The length Z of the great circle line (20) to its third nearest neighbor point P5 2,3 The ratio to the sphere radius (R) is 0.83 ± 20%. - Wherein, starting from the third marker point P3 - To its nearest neighbor P 10 The length Z of the large circle line (20) 3,1 The ratio to the sphere radius (R) is 0.78 ± 20%. - To the next nearest neighbor P 15 The length Z of the large circle line (20) 3,2 The ratio to the sphere radius (R) is 0.84 ± 20%, and - The length Z of the great circle line (20) to its third nearest neighbor point P8 3,3 The ratio to the sphere radius (R) is 0.87 ± 20%. - Wherein, starting from the fourth marker point P4 - The length Z of the great circle line (20) to its nearest neighbor P2 4,1 The ratio to the sphere radius (R) is 0.72 ± 20%. - To the next nearest neighbor P 12 The length Z of the large circle line (20) 4,2 The ratio to the sphere radius (R) is 0.84 ± 20%, and - The length Z of the great circle line (20) to its third nearest neighbor point P5 4,3 The ratio to the sphere radius (R) is 0.85 ± 20%. - Wherein, starting from the fifth marker point P5 - To its nearest neighbor P 12 The length Z of the large circle line (20) 5,1 The ratio to the sphere radius (R) is 0.79 ± 20%. - The length Z of the great circle line (20) to the next nearest neighbor point P2 5,2 The ratio to the sphere radius (R) is 0.83 ± 20%, and - The length Z of the great circle line (20) to its third nearest neighbor point P4 5,3 The ratio to the sphere radius (R) is 0.85 ± 20%. - Wherein, starting from the sixth marker point P6 - To its nearest neighbor P 12 The length Z of the large circle line (20) 6,1 The ratio to the sphere radius (R) is 0.67 ± 20%. - To the next nearest neighbor P 18 The length Z of the large circle line (20) 6,2 The ratio to the sphere radius (R) is 0.73 ± 20%, and - to its third nearest neighbor P 15 The length Z of the large circle line (20) 6,3 The ratio to the sphere radius (R) is 0.94 ± 20%. - Wherein, starting from the seventh marker point P7 - To its nearest neighbor P 16 The length Z of the large circle line (20) 7,1 The ratio to the sphere radius (R) is 0.59 ± 20%. - To the next nearest neighbor P 17 The length Z of the large circle line (20) 7,2 The ratio to the sphere radius (R) is 0.76 ± 20%, and - to its third nearest neighbor P 15 The length Z of the large circle line (20) 7,3 The ratio to the sphere radius (R) is 0.90 ± 20%. - Wherein, starting from the eighth marker point P8 - To its nearest neighbor P 15 The length Z of the large circle line (20) 8,1 The ratio to the sphere radius (R) is 0.82 ± 20%. - The length Z of the great circle line (20) to the next nearest neighbor point P3 8,2 The ratio to the sphere radius (R) is 0.87 ± 20%, and - to its third nearest neighbor P 11 The length Z of the large circle line (20) 8,3 The ratio to the sphere radius (R) is 0.89 ± 20%. - Starting from the ninth marker point P9 - To its nearest neighbor P 14 The length Z of the large circle line (20) 9,1 The ratio to the sphere radius (R) is 0.51 ± 20%. - To the next nearest neighbor P 18 The length Z of the large circle line (20) 9,2 The ratio to the sphere radius (R) is 0.91 ± 20%, and - The length Z of the great circle line (20) to its third nearest neighbor point P8 9,3 The ratio to the sphere radius (R) is 0.95 ± 20%. - Wherein, from the tenth marker point P 10 Set off - To its nearest neighbor P 13 The length Z of the large circle line (20) 10,1 The ratio to the sphere radius (R) is 0.75 ± 20%. - To the next nearest neighbor P 17 The length Z of the large circle line (20) 10,2 The ratio to the sphere radius (R) is 0.75 ± 20%, and - The length Z of the great circle line (20) to its third nearest neighbor point P3 10,3 The ratio to the sphere radius (R) is 0.78 ± 20%. - Wherein, from the eleventh marker point P 11 Set off - To its nearest neighbor P 13 The length Z of the large circle line (20) 11,1 The ratio to the sphere radius (R) is 0.82 ± 20%. - To the next nearest neighbor P 10 The length Z of the large circle line (20) 11,2 The ratio to the sphere radius (R) is 0.88 ± 20%, and - The length Z of the great circle line (20) to its third nearest neighbor point P8 11,3 The ratio to the sphere radius (R) is 0.89 ± 20%. - Wherein, from the twelfth marker point P 12 Set off - The length Z of the great circle (20) to its nearest neighbor P6 12,1 The ratio to the sphere radius (R) is 0.67 ± 20%. - The length Z of the great circle line (20) to the next nearest neighbor point P5 12,2 The ratio to the sphere radius (R) is 0.79 ± 20%, and - The length Z of the great circle line (20) to its third nearest neighbor point P4 12,3 The ratio to the sphere radius (R) is 0.84 ± 20%. - Wherein, from the thirteenth marker point P 13 Set off - To its nearest neighbor P 10 The length Z of the large circle line (20) 13,1 The ratio to the sphere radius (R) is 0.75 ± 20%. - To the next nearest neighbor P 11 The length Z of the large circle line (20) 13,2 The ratio to the sphere radius (R) is 0.82 ± 20%, and - The length Z of the great circle line (20) to its third nearest neighbor point P1 13,3 The ratio to the sphere radius (R) is 1.02 ± 20%. - Wherein, from the fourteenth marker point P 14 Set off - The length Z of the great circle line (20) to its nearest neighbor P9 14,1 The ratio to the sphere radius (R) is 0.51 ± 20%. - The length Z of the great circle line (20) to the next nearest neighbor point P4 14,2 The ratio to the sphere radius (R) is 0.92 ± 20%, and - to its third nearest neighbor P 11 The length Z of the large circle line (20) 14,3 The ratio to the sphere radius (R) is 0.98 ± 20%. - Wherein, from the fifteenth marker point P 15 Set off - The length Z of the great circle line (20) to its nearest neighbor P8 15,1 The ratio to the sphere radius (R) is 0.82 ± 20%. - The length Z of the great circle line (20) to the next nearest neighbor point P3 15,2 The ratio to the sphere radius (R) is 0.84 ± 20%, and - The length Z of the great circle line (20) to its third nearest neighbor point P7 15,3 The ratio to the sphere radius (R) is 0.90 ± 20%. - Wherein, from the sixteenth marker point P 16 Set off - The length Z of the great circle line (20) to its nearest neighbor P7 16,1 The ratio to the sphere radius (R) is 0.59 ± 20%. - To the next nearest neighbor P 17 The length Z of the large circle line (20) 16,2 The ratio to the sphere radius (R) is 0.74 ± 20%, and - The length Z of the great circle line (20) to its third nearest neighbor point P5 16,3 The ratio to the sphere radius (R) is 1.10 ± 20%. - Wherein, from the seventeenth marker point P 17 Set off - To its nearest neighbor P 16 The length Z of the large circle line (20) 17,1 The ratio to the sphere radius (R) is 0.74 ± 20%. - To the next nearest neighbor P 10 The length Z of the large circle line (20) 17,2 The ratio to the sphere radius (R) is 0.75 ± 20%, and - The length Z of the great circle line (20) to its third nearest neighbor point P7 17,3 The ratio to the sphere radius (R) is 0.76 ± 20%, and - Wherein, from the eighteenth marker point P 18 Set off - The length Z of the great circle (20) to its nearest neighbor P6 18,1 The ratio to the sphere radius (R) is 0.73 ± 20%. - To the next nearest neighbor P 12 The length Z of the large circle line (20) 18,2 The ratio to the sphere radius (R) is 0.88 ± 20%, and - The length Z of the great circle line (20) to its third nearest neighbor point P9 18,3 The ratio to the sphere radius (R) is 0.91 ± 20%.

20. The ping-pong ball (2) according to claim 19. - in, Regarding the first marker point P1 - The length Q of the great circle (20) between the nearest neighbor P2 and the second nearest neighbor P5 1,1 The ratio to the sphere radius (R) is 0.83 ± 20%. - The second nearest neighbor P5 and its third nearest neighbor P 17 The length Q of the great circle line (20) between them 1,2 The ratio to the sphere radius (R) is 1.57 ± 20%, and - Third nearest neighbor point P 17 The length Q of the great circle line (20) between the nearest neighbor point P2 and its nearest neighbor point P2 1,3 The ratio to the sphere radius (R) is 1.53 ± 20%. - Wherein, regarding the second marker point P2 - The length Q of the great circle (20) between the nearest neighbor P1 and the second nearest neighbor P4 2,1 The ratio to the sphere radius (R) is 1.24 ± 20%. - The length Q of the great circle (20) between the second nearest neighbor P4 and its third nearest neighbor P5 2,2 The ratio to the sphere radius (R) is 0.85 ± 20%, and - The length Q of the great circle (20) between the third nearest neighbor P5 and its nearest neighbor P1 2,3 The ratio to the sphere radius (R) is 0.91 ± 20%. - Among them, regarding the third marker point P3 - nearest neighbor point P 10 and the next nearest neighbor P 15 The length Q of the great circle line (20) between them 3,1 The ratio to the sphere radius (R) is 1.61 ± 20%. - Next nearest neighbor P 15 The length Q of the great circle line (20) between the third nearest neighbor point P8 and its third nearest neighbor point P8 3,2 The ratio to the sphere radius (R) is 0.82 ± 20%, and - The third nearest neighbor P8 and its nearest neighbor P 10 The length Q of the great circle line (20) between them 3,3 The ratio to the sphere radius (R) is 1.45 ± 20%. - Among them, regarding the fourth marker point P4 - Nearest neighbor P2 and second nearest neighbor P 12 The length Q of the great circle line (20) between them 4,1 The ratio to the sphere radius (R) is 1.39 ± 20%. - Next nearest neighbor P 12 The length Q of the great circle line (20) between the third nearest neighbor point P5 and its third nearest neighbor point P5 4,2 The ratio to the sphere radius (R) is 0.79 ± 20%, and - The length Q of the great circle (20) between the third nearest neighbor P5 and its nearest neighbor P2 4,3 The ratio to the sphere radius (R) is 0.83 ± 20%. - Among them, regarding the fifth marker point P5 - nearest neighbor point P 12 The length Q of the great circle line (20) between the next nearest neighbor point P2 and P2 5,1 The ratio to the sphere radius (R) is 1.39 ± 20%. - The length Q of the great circle (20) between the second nearest neighbor P2 and its third nearest neighbor P4 5,2 The ratio to the sphere radius (R) is 0.72 ± 20%, and - The third nearest neighbor P4 and its nearest neighbor P 12 The length Q of the great circle line (20) between them 5,3 The ratio to the sphere radius (R) is 0.84 ± 20%. - Among them, regarding the sixth marker point P6 - nearest neighbor point P 12 and the next nearest neighbor P 18 The length Q of the great circle line (20) between them 6,1 The ratio to the sphere radius (R) is 0.88 ± 20%. - Next nearest neighbor P 18 and its third nearest neighbor P 15 The length Q of the great circle line (20) between them 6,2 The ratio to the sphere radius (R) is 0.97 ± 20%, and - Third nearest neighbor point P 15 and its nearest neighbor P 12 The length Q of the great circle line (20) between them 6,3 The ratio to the sphere radius (R) is 1.57 ± 20%. - Among them, regarding the seventh marker point P7 - nearest neighbor point P 16 and the next nearest neighbor P 17 The length Q of the great circle line (20) between them 7,1 The ratio to the sphere radius (R) is 0.74 ± 20%. - Next nearest neighbor P 17 and its third nearest neighbor P 15 The length Q of the great circle line (20) between them 7,2 The ratio to the sphere radius (R) is 1.60 ± 20%, and - Third nearest neighbor point P 15 and its nearest neighbor P 16 The length Q of the great circle line (20) between them 7,3 The ratio to the sphere radius (R) is 1.38 ± 20%. - Among them, regarding the eighth marker point P8 - nearest neighbor point P 15 The length Q of the great circle line (20) between the next nearest neighbor point P3 and the next nearest neighbor point P3 8,1 The ratio to the sphere radius (R) is 0.84 ± 20%. - The second nearest neighbor P3 and its third nearest neighbor P 11 The length Q of the great circle line (20) between them 8,2 The ratio to the sphere radius (R) is 1.00 ± 20%, and - Third nearest neighbor point P 11 and its nearest neighbor P 15 The length Q of the great circle line (20) between them 8,3 The ratio to the sphere radius (R) is 1.59 ± 20%. - Among them, regarding the ninth marker point P9 - nearest neighbor point P 14 and the next nearest neighbor P 18 The length Q of the great circle line (20) between them 9,1 The ratio to the sphere radius (R) is 1.39 ± 20%. - Next nearest neighbor P 18 The length Q of the great circle line (20) between the third nearest neighbor point P8 and its third nearest neighbor point P8 9,2 The ratio to the sphere radius (R) is 0.96 ± 20%, and - The third nearest neighbor P8 and its nearest neighbor P 14 The length Q of the great circle line (20) between them 9,3 The ratio to the sphere radius (R) is 1.33 ± 20%. - Among them, regarding the tenth marker point P 10 - nearest neighbor point P 13 and the next nearest neighbor P 17 The length Q of the great circle line (20) between them 10,1 The ratio to the sphere radius (R) is 1.22 ± 20%. - Next nearest neighbor P 17 The length Q of the great circle line (20) between the third nearest neighbor point P3 and its third nearest neighbor point P3 10,2 The ratio to the sphere radius (R) is 1.11 ± 20%, and - The third nearest neighbor P3 and its nearest neighbor P 13 The length Q of the great circle line (20) between them 10,3 The ratio to the sphere radius (R) is 1.45 ± 20%. - Among them, regarding the eleventh marker point P 11 - nearest neighbor point P 13 and the next nearest neighbor P 10 The length Q of the great circle line (20) between them 11,1 The ratio to the sphere radius (R) is 0.75 ± 20%. - Next nearest neighbor P 10 The length Q of the great circle line (20) between the third nearest neighbor point P8 and its third nearest neighbor point P8 11,2 The ratio to the sphere radius (R) is 1.45 ± 20%, and - The third nearest neighbor P8 and its nearest neighbor P 13 The length Q of the great circle line (20) between them 11,3 The ratio to the sphere radius (R) is 1.71 ± 20%. - Among them, regarding the twelfth marker point P 12 - The length Q of the great circle (20) between the nearest neighbor P6 and the second nearest neighbor P5 12,1 The ratio to the sphere radius (R) is 1.21 ± 20%. - The length Q of the great circle (20) between the second nearest neighbor P5 and its third nearest neighbor P4 12,2 The ratio to the sphere radius (R) is 0.85 ± 20%, and - The length Q of the great circle (20) between the third nearest neighbor P4 and its nearest neighbor P6 12,3 The ratio to the sphere radius (R) is 1.51 ± 20%. - Among them, regarding the thirteenth marker point P 13 - nearest neighbor point P 10 and the next nearest neighbor P 11 The length Q of the great circle line (20) between them 13,1 The ratio to the sphere radius (R) is 0.88 ± 20%. - Next nearest neighbor P 11 The length Q of the great circle line (20) between the third nearest neighbor point P1 and its third nearest neighbor point P1 13,2 The ratio to the sphere radius (R) is 1.84 ± 20%, and - The third nearest neighbor P1 and its nearest neighbor P 10 The length Q of the great circle line (20) between them 13,3 The ratio to the sphere radius (R) is 1.33 ± 20%. - Among them, regarding the fourteenth marker point P 14 - The length Q of the great circle (20) between the nearest neighbor P9 and the second nearest neighbor P4 14,1 The ratio to the sphere radius (R) is 1.12 ± 20%. - The second nearest neighbor P4 and its third nearest neighbor P 11 The length Q of the great circle line (20) between them 14,2 The ratio to the sphere radius (R) is 1.88 ± 20%, and - Third nearest neighbor point P 11 The length Q of the great circle line (20) between the nearest neighbor point P9 and its nearest neighbor point P9 14,3 The ratio to the sphere radius (R) is 1.05 ± 20%. - Among them, regarding the fifteenth marker point P 15 - The length Q of the great circle (20) between the nearest neighbor P8 and the second nearest neighbor P3 15,1 The ratio to the sphere radius (R) is 0.87 ± 20%. - The length Q of the great circle (20) between the second nearest neighbor P3 and its third nearest neighbor P7 15,2 The ratio to the sphere radius (R) is 0.90 ± 20%, and - The length Q of the great circle (20) between the third nearest neighbor P7 and its nearest neighbor P8 15,3 The ratio to the sphere radius (R) is 1.58 ± 20%. - Among them, regarding the sixteenth marker point P 16 - Nearest neighbor P7 and second nearest neighbor P 17 The length Q of the great circle line (20) between them 16,1 The ratio to the sphere radius (R) is 0.76 ± 20%. - Next nearest neighbor P 17 The length Q of the great circle line (20) between the third nearest neighbor point P5 and its third nearest neighbor point P5 16,2 The ratio to the sphere radius (R) is 1.57 ± 20%, and - The length Q of the great circle (20) between the third nearest neighbor P5 and its nearest neighbor P7 16,3 The ratio to the sphere radius (R) is 1.67 ± 20%. - Among them, regarding the seventeenth marker point P 17 - nearest neighbor point P 16 and the next nearest neighbor P 10 The length Q of the great circle line (20) between them 17,1 The ratio to the sphere radius (R) is 1.47 ± 20%. - Next nearest neighbor P 10 The length Q of the great circle line (20) between the third nearest neighbor point P7 and its third nearest neighbor point P7 17,2 The ratio to the sphere radius (R) is 1.21 ± 20%, and - The third nearest neighbor P7 and its nearest neighbor P 16 The length Q of the great circle line (20) between them 17,3 The ratio to the sphere radius (R) is 0.59 ± 20%. - Among them, regarding the eighteenth marker point P 18 - Nearest neighbor P6 and second nearest neighbor P 12 The length Q of the great circle line (20) between them 18,1 The ratio to the sphere radius (R) is 0.67 ± 20%. - Next nearest neighbor P 12 The length Q of the great circle line (20) between the third nearest neighbor point P9 and its third nearest neighbor point P9 18,2 The ratio to the sphere radius (R) is 1.37 ± 20%, and - The length Q of the great circle (20) between the third nearest neighbor P9 and its nearest neighbor P6 18,3 The ratio to the sphere radius (R) is 1.62 ± 20%.

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