JP7149981B2 - lighting equipment - Google Patents

Description

The present invention mainly relates to an illumination device using light sources of a plurality of light source colors and a lens suitable for mixing the light source colors.

With the spread of LED lighting using small-sized LEDs, it has become possible to easily realize a lighting device that can freely change colors by mixing two or three colors of LEDs. Color-tunable lighting equipment mimics the changes in natural colors, such as the bluish-white light of the blue sky during the day and the red light of the setting sun in the evening, and can create light that matches the natural biorhythms of humans. It is thought that it can contribute to improving the efficiency of the body, relaxing, and improving the quality of sleep, which in turn contributes to improving human health.

When spotlights and downlights are created using such light sources of multiple colors, a lens is used to control the light distribution characteristics, and the light emitted by the LEDs of multiple colors is mixed to achieve uniform illumination. There is a need to. However, methods that give priority to color mixture often impair the efficiency of light utilization.

As a prior art related to lenses, Patent Literature 1 discloses a lens-equipped light-emitting unit including a substrate, LED elements of a plurality of light colors arranged on the substrate, and a rotating body-shaped lens. This lens is in contact with an exit surface, a reflecting surface that extends convexly from the exit surface toward the substrate, a side entrance surface that folds back from the substrate side of the reflection surface to the exit surface, and the reflection surface of the side entrance surface. and an upper incident surface connecting the non-polarized ends, wherein the divergence angle of the exit surface inside the predetermined circle is larger than the divergence angle of the exit surface outside the predetermined circle.

Further, Patent Document 2 discloses a lens in which a concave portion is formed in the central portion of the bottom of the lens and a concavo-convex structure is provided on the side surface of the concave portion.

As a prior art related to LED, there is a CSP (Chip Scale Package)-LED module described in Patent Document 3. A conventional surface-mounted white LED package is obtained by sealing a GaN-based blue LED chip with a phosphor containing yellow or other phosphor particles. The length of one side of the package to be mounted was 3 mm. CSP is literally a chip-scale package, and the CSP-type LED described in Patent Document 2 has a side length of 1.0 mm for the LED chip, and a side length of 1 mm for the package in which it is mounted. .3 mm, which is close to the chip size, and an LED module is described in which 24 such CSPs (daylight and warm colors) are arranged.

As a prior art related to toning, FIG. 7 discloses an illumination device for toning within chromaticity coordinates surrounded by three color LEDs (BSY1, BSY2, R, where BSY stands for blue-shifted-yellow).

Japanese Patent No. 5279329 JP 2019-211736 A Japanese Patent No. 6095479 U.S. Pat. No. 8,598,809

An object of the present invention is to suppress color unevenness in a lighting device that uses a plurality of LEDs with different emission colors to distribute light with a lens.

In particular, when a plurality of LEDs with different chromaticities are used so as to irradiate a wide range of chromaticities, color unevenness is likely to be conspicuous. Color unevenness can be reduced by narrowing the distance between the light sources or enlarging the lens. If diffusion in the lens is increased to reduce color unevenness, it becomes difficult to obtain desired light distribution characteristics such as a narrow angle, and luminous efficiency (luminous flux (lm) per input power (W)) decreases. An object of the present invention is to suppress color unevenness while minimizing these side effects.

The present invention is a lighting device for distributing light emitted from a light source with a lens, wherein the light source comprises LEDs of a plurality of emission colors, the lens comprises an entrance hole with an entrance side surface and an entrance bottom surface, a reflector An illumination device comprising a side surface, an exit top surface, and a spiral convex or concave stripe on the incident side surface.

In the present invention, it is preferable that the light source includes LEDs emitting light of three colors .

In the present invention, it is preferable that the light source has a portion in which three or more LEDs of the same emission color are arranged in a straight line.

In the present invention, it is preferable that the light source is a light source in which CSP-type LEDs are arranged on a substrate as the LEDs of the plurality of emission colors.

In the present invention, it is preferable that the light emitting upper surface has a textured structure covered with scalene polygons.

In the present invention, the incident bottom surface preferably has a textured structure covered with scalene polygons.

Advantageous Effects of Invention According to the present invention, it is possible to suppress color unevenness while maintaining desired light distribution characteristics and good luminous efficiency in a lighting device using LEDs of a plurality of luminescent colors.

1A and 1B are a front view and a cross-sectional view of a lighting device according to Embodiment 1. FIG. 2 is a plan view of a light source used in Embodiment 1. FIG. 4 is a chromaticity diagram of each CSP-type LED used in Embodiment 1. FIG. FIG. 2 is a side cross-sectional view of a lens and a light source in Embodiment 1; 2 is a front cross-sectional view of a lens and a light source in Embodiment 1. FIG. 2 is a top view of the lens in Embodiment 1. FIG. FIG. 4 is a plan view of a light source used in Embodiment 2; 8 is a plan view of another light source used in Embodiment 2. FIG. FIG. 8 is a side cross-sectional view of a lens and a light source in Embodiment 2; FIG. 10 is a top view of a lens according to Embodiment 2; FIG. 10 is a plan view of the entrance hole of the lens in Embodiment 2;

<Embodiment 1>
<Basic configuration>
A front view and a cross-sectional view of a lighting device 300 according to this embodiment are shown in FIGS. 1(a) and 1(b), respectively. The illumination device 300 is a downlight, and includes a heat sink 310 , a lamp body 315 (including a light source 320 in which a plurality of CSP type LEDs are arranged and a lens 330 ), a cone 340 and three mounting springs 350 .

The illumination device 300 is supplied with driving power from a power source 360 (not shown), and the power source 360 is controlled by a control device 370 (not shown) for dimming and toning. The power supply 360 has 3ch driving outputs for controlling the 3-color LEDs in this embodiment. A smartphone, tablet, or PC installed with lighting control software is used as the control device 370 , a control signal is wirelessly transmitted to a receiving device connected to the power source 360 , and the receiving device transmits the control signal to the power source 360 . , the driving outputs of the 3ch of the power supply 360 are controlled respectively.

<CSP type LED light source>
FIG. 2 shows a plan view of the light source 320 used in this embodiment (although the light source 320 is viewed from below in the normal arrangement shown in FIG. 1, this surface is the top surface in the description of the light source and the CSP). . The light source 320 has 22 each of three-color CSP type LEDs 323A, 323B, and 323C arranged on a substrate 322 as shown in FIG. 2 and connected in series by wiring inside the substrate. The light source 320 is provided with a mounting hole 325 and fastened to the heat sink 310 by passing a screw through the mounting hole 325 . The light source 320 has wiring terminals 326A (for CSP-type LED 323A), 326B (for CSP-type LED 323B), 326C (for CSP-type LED 323C), and 326P (common terminal), and is connected to a power supply 360 having three outputs. The light emission is dimming controlled.

In this case, the arrangement of the CSPs is such that the numbers of the CSP LEDs 323A, 323A, 323B, and 323C are the same but are not biased. In other words, "AAAA" is not arranged in the same row or column. However, three or more CSP-type LEDs labeled "AAA" shown in 323AX may be arranged in an oblique direction. In this light source 320, instead of a simple lattice arrangement, some rows described on the left side of the CSP type LED, that is, +1, 0, -1 rows in the figure (0th row is "BACBACBACB" arrangement) are They are arranged to be shifted by half from the other rows, that is, the +4, +3, +2 rows and the -2, -3, and -4 rows. The diameter of the light source area (from the left end to the right end of the 0th row) is 13 mm. As a result, LEDs of the same color are not arranged in the horizontal direction, the vertical direction, and from the upper left to the lower right in the drawing, but the LEDs of the same color may be arranged from the upper right to the lower left, for example, as indicated by 323AX.

<Chromaticity of CSP type LED>
As the CSP type LED packages 323A, B, and C used in this embodiment, any one of the bluish white LED "Bw", the red LED "R", and the yellowish white LED "Yw" is used. Although Yw is a color close to yellow on the chromaticity diagram, it looks like a color close to green when the other Bw and R are turned on at the same time. FIG. 3 is a chromaticity diagram (CIE1931 chromaticity coordinate diagram) for explaining the chromaticity of these LEDs, and for reference, a dotted line shows a line connecting the chromaticity of black body radiation at each color temperature. .

Bw, which is a blue-white LED, is (0.336, 0.24), (0.352, 0.44), (0.15, 0.2), (0.2) in the chromaticity coordinates of FIG. , 0.1), for example (0.23, 0.26).

R, which is a red LED, is (0.66, 0.23), (0.423, 0.355), (0.5, 0.5) and the chromaticity boundary line E The light is emitted with a chromaticity in the range enclosed by (0.60.0.38) as an example. Note that it is not the same as the general definition of red.

Yw, which is a yellow-white LED, is (0.5, 0.5), (0.423, 0.355), (0.342, 0.312), (0.352) in the chromaticity coordinates of FIG. , 0.44), (0.37, 0.63) and chromaticity in the range surrounded by the chromaticity boundary line E, for example (0.44, 0.47).

Within the chromaticity range of Yw, a range in which d _uv is positive from a line connecting the chromaticity of black body radiation at each color temperature is preferable, and a range in which d _uv is plus 0.03 to 0 is particularly preferable.

Among the chromaticity ranges of Bw and R, a range of _+0.03 to -0.03 in duv from the line connecting the chromaticities of blackbody radiation at each color temperature is particularly preferable.

It should be noted that the chromaticity of each LED can be represented by chromaticity coordinates (u', v') in CIE1976 instead of chromaticity coordinates (x, y) in CIE1931, and both are u'=4x/(-2x+12y+3 ) and v′=9y/(−2x+12y+3). It may be displayed in another chromaticity coordinate system.

<Structure of CSP type LED>
The CSP-type LEDs 323A, 323B, and 323C used in the embodiment have an anode electrode and a cathode electrode on the bottom surface of a blue LED chip whose light-emitting layer is made of InGaN. These electrodes of the LED chip are connected to terminals on the substrate 322 . The width of the CSP type LED is about 1 mm.

Bw (for example, 323C, but may be 323A or 323B) is an InGaN blue LED chip whose upper and side surfaces are covered with resin containing greenish phosphor particles or yellowish phosphor. Moreover, red phosphor particles may be further included.

Yw (for example, 323B, but may be 323A or 323C) is similar to Bw in that an InGaN-based blue LED chip is placed at the bottom of the package, and the top and side surfaces of the InGaN-based blue LED chip are coated with green phosphor particles or yellow. It is covered with a resin containing a system phosphor, but the concentration of the phosphor is higher than that of Bw. Moreover, red phosphor particles may be further included.

R (for example, 323A, but may be 323B or 323C) is an InGaN blue LED chip whose top and side surfaces are covered with resin containing red phosphor particles.

As for R, an AlGaInP-based LED chip may be used in place of the InGaN-based blue LED chip and covered with a resin containing no phosphor. An LED package using an AlGaInP-based LED chip as R is preferable in that the emission spectrum does not include blue, but the drive circuit is complicated because the drive voltage is different from Bw and Yw. In the case of an LED package in which a blue LED chip and a red phosphor are combined as R, a process for reducing the blue color included in the emission spectrum is required. For example, it is preferable to increase the concentration of the red phosphor to reduce the ratio of light emitted from the blue LED chip to the outside, but a filter that absorbs blue may be used.

In each of the CSP-type LEDs described above, the yellow phosphor particles are, for example, (Y _1−x Gd _x ) ₃ Al ₅ O ₁₂ :Ce ²⁺ (0≦x≦1), and the green phosphor particles are, For example, Lu ₃ Al ₅ O ₁₂ :Ce ²⁺ , Sr _x Ca _1−x AlSiN ₃ :Eu ³⁺ (0≦x≦1) phosphor, Sr[LiAl ₃ N ₄ ]:Eu ²⁺ as a red phosphor. or K ₂ SiF ₆ :Mn ⁴⁺ phosphor can be preferably used. Quantum dots can also be suitably used.

<Lens>
In the present invention, a lens has been developed in which even if a plurality of CSP-type LEDs having different colors are used, color unevenness does not occur practically and light loss is suppressed as much as possible.

As shown in FIG. 4, which is a cross-sectional view in the side direction, the lens 330 has an incident hole 331 in the center of a lower surface 334 (the side facing the light source 320). Prepare. The diameter of the incident hole 331 on the lower surface side is 14.7 mm. Lens 330 has a reflective side surface 335 and an exit top surface 339 . At the center of the top exit surface 339 is an exit hole 336 with a bottom exit surface 337 and a side exit surface 338 . The diameter of the exit top surface 339 (the portion that functions as a lens) is 70 mm, and the height of the lens is 49.3 mm from the center of the light source 320 .

The entrance bottom surface 332, the entrance side surface 333, and the exit top surface 339 of the entrance hole 331 are subjected to texturing (processing to divide the surface into convex or concave regions) for intricately changing the direction of the emitted light with respect to the direction of the incident light. is applied. The reflective side surface 335 is also faceted to such an extent that the function of total reflection is not impaired.

FIG. 4 shows an example of rays emitted from the center P of the light source 320 (when not affected by texturing). The light is incident on the incident side surface 333 at the point Q from the point P, is totally reflected at the reflecting side surface 335 at the point R, is emitted to the outside from the emitting upper surface 339, and becomes a ray in the optical axis direction. The length of line segment PQ is 11.06 mm, and the length of line segment QR is 10.9 mm. If the length of the line segment QR is increased to, for example, 90% or more with respect to the line segment PQ, the diffusion effect of the reflecting side surface 335, which will be described later, can be increased, but the diffusion effect of the reflecting side surface 335 is unlikely to lead to the spread of the radiation angle. Therefore, it is convenient to suppress color unevenness while maintaining a narrow radiation angle. The inner angle θ of the incident side surface 333 with respect to the lower surface 334 is 78°, and since there is a tendency that color unevenness can be suppressed when this angle is large, it is preferably 70° or more and 100° or less, and 75° or more and 88° or less. is even more desirable.

The diffusion structure formed on the entrance side surface 333 of the entrance hole 331 will be described with reference to FIG. 5, which is a cross-sectional view taken along line AA of FIG. 333G is a convex streak portion 333G formed in a spiral shape from the bottom toward the upper surface direction, and this convex streak portion 333G is formed by spirally connecting facets 333F from the bottom portion toward the upper surface direction. The convex streak 333G has a structure for diffusing light rays in the circumferential direction c in the figure. Therefore, the light can be diffused without widening the radiation angle of the light emitted from the lens 330 too much.

For the purpose of explaining the reason why the convex streaks 333G have a spiral structure instead of a diffusion structure along the radial direction, 323AX′ (the above-mentioned 323AX is used for explanation), which is a schematic arrangement of CSP-type LEDs having the same emission color. 323BX' (schematic representation of the arrangement of CSP type LEDs 323B). The light beams emitted in parallel at the same angle from each CSP-type LED are refracted by hitting the convex stripes 333G of different heights. The emitted light is refracted as ray ra1 in the drawing, and the light emitted from other CSPs is refracted as ray ra2 in the drawing (because it is a schematic illustration, the refraction of the ray is slightly emphasized). On the other hand, light emitted from 323BX', which is a CSP of another color, is refracted as light rays rb1 and rb2. For example, light rays ra1 and rb2 of different colors travel in the same direction, thereby mixing colors and improving color unevenness. With such a structure, the light emitted from the linearly arranged 323A of the same color is guided in a different direction, so that the color mixing property is improved.

A top view of lens 330 is shown in FIG. The exit upper surface 339 of the lens is entirely covered with a crimped structure (scale-like projections) as shown in FIG. This scale-like projection does not have a structure along the radial direction, but has various outlines of trapezoidal polygons (such as trapezoid 339(4) and trapezoid 339(6)) arranged in a spiral. ing. By not arranging them in a regular arrangement such as a lattice or radial pattern, the effect of the regular arrangement of the CSP type LEDs is suppressed, and color unevenness is suppressed.

<Radiation pattern>
The illuminating device 300 had a 1/2 beam angle of 21°, no color unevenness, and obtained a pattern in which the illuminance gradually decreased from the central portion to the peripheral portion of the illuminated surface. Slight color unevenness was observed when the wall surface away from the downlight installation position was enlarged, but no color unevenness was observed on the floor surface in front of the fixture installation position.

<Embodiment 2>
<Basic configuration>
A lighting device 400 (downlight) according to the present embodiment is obtained by replacing the light source 320 in the lighting device 300 with the light source 420 and replacing the lens 330 with the lens 430 .

<CSP type LED>
FIG. 7 shows a top view of the light source 420 used in this embodiment. The light source 420 has seven three-color CSP type LEDs 423A, 423B, and 423C arranged on a substrate 422 as shown in FIG. 7 and connected in series by wiring inside the substrate. Mounting holes 425 are provided, and screws are passed through the mounting holes 425 and fastened to the heat sink 410 (which is smaller than the heat sink 310). A power supply 460 having wiring terminals 426A (for CSP type LED 423A), 426B (for CSP type LED 423B), 426C (for CSP type LED 423C), and 426P (common terminal) and having three outputs (different rated output from power supply 360) ) to control light emission of each CSP.

In this case, the arrangement of the CSPs is such that the numbers of the CSP type LEDs 423A, 423A, 423B, and 423C are the same and are evenly arranged. In other words, "AAAA" is not arranged in the same row or column. However, in some cases, three or more CPS-type LEDs are arranged in a straight line in an oblique direction, indicated by "AAA" indicated by 423AX. In this light source 420, all of the +2 to -2 rows are arranged in a grid pattern. The diameter of the light source area is 8 mm.

As the CSP type LED packages 423A, 423B, and 423C used in this embodiment, any one of the bluish white LED "Bw", the red LED "R", and the yellowish white LED "Yw" is used.

A light source 520 can also be used in place of the light source 420 in this embodiment. This top view is shown in FIG. The light source 520 has six CSP-type LEDs 523A and 523C and seven CSP-type LEDs 523B arranged on a substrate 522 as shown in FIG. 8 and connected in series by wiring inside the substrate. A mounting hole 525 is provided, and a screw is passed through the mounting hole 525 to be fastened to the heat sink 410 . Wiring terminals 526A (for CSP type LED 523A), 526B (for CSP type LED 523B), 526C (for CSP type LED 523C), and 526P (common terminal) are connected to a power supply 460 having three outputs, and each CSP type LED Light emission is dimming controlled.

In this case, the arrangement of the CSPs is such that the numbers of the CSP LEDs 523A, 523A, 523B, and 523C are almost the same and are evenly arranged. In other words, "AAAA" is not arranged in the same row or column. However, a linear arrangement of three or more CPS-type LEDs called "BBB" shown in 523BX may occur in an oblique direction. In this light source 520, the +2 rows, 0 rows, and -2 rows are arranged in a grid pattern, and the +1 rows and -1 rows are arranged in a grid pattern shifted by half. In addition, each CSP type LED is arranged at the vertex of the triangle as a whole. The diameter of the light source area is 8 mm.

As the CSP-type LED packages 523A, B, and C used in this embodiment, any one of blue-white LED "Bw", yellow-white LED "Yw", and red LED "R" is used.

<Lens>
As shown in FIG. 9 , which is a cross-sectional view, the lens 430 has an entrance hole 431 in the center of a lower surface 434 (the side facing the light source 420 ), and the entrance hole 431 has an entrance bottom surface 432 and an entrance side surface 433 . The diameter of the incident hole 431 on the lower surface side is 9 mm. Lens 430 also includes a reflective side surface 435 and an exit top surface 439 . A central portion surrounded by the upper surface 439 of emission has an emission convex portion 436 . Lens 430 also includes support 438 . The effective diameter d _e (the part that functions as a lens) of the upper surface of the exit 439 is 20 mm, the diameter including the support portion 438 is 40 mm, and the effective height of the lens is 15 mm from the center of the light source 420 .

The entrance bottom surface 432, the entrance side surface 433, and the exit top surface 439 of the entrance hole 431 are textured (non-smooth surface processing) for intricately changing the direction of emitted light with respect to the direction of incident light. Specifically, the incident side surface 433 is provided with a spiral convex stripe portion 433G, and the convex stripe portion 433G is further provided with a facet 433F. The reflective side surface 435 is also faceted.

The inner angle θ of the incident side surface 433 with respect to the lower surface 434 is 85°, and since there is a tendency that color unevenness can be suppressed when this angle is large, 70° to 100° is preferable, and 75° to 88° is even more desirable.

An entrance side surface 433 of the entrance hole 431 is formed with a spiral convex stripe 433G, which is further made up of a series of facets 433F. With such a diffusion structure that does not extend along the radial direction, it is possible to effectively reduce color unevenness even if there is a row of CSPs 423AX or the like.

A top view of lens 430 is shown in FIG. The exit upper surface 439 of the lens is entirely covered with a crimped structure (scale-like projections) as shown in FIG. This scale-like protrusion is a non-radial diffusion structure in which various contoured trapezoidal polygons (such as the trapezoidal pentagon 439(5) and the trapezoidal hexagon 439(6)) are arranged roughly in a spiral. is. By not arranging them in a regular arrangement such as a lattice or radial pattern, the effect of the regular arrangement of the CSP type LEDs is suppressed, and color unevenness is suppressed.

FIG. 11 shows an incident hole 431 of the lens 430 viewed from the direction of the light source 420. FIG. The entrance bottom surface 432 and the entrance side surface 433 of the lens are entirely covered with a textured structure (scale-like convex portions or concave portions) as shown in FIG. On the incident bottom surface 432, various outlines of trapezoidal polygons (such as a trapezoidal pentagon 432(5) and a trapezoidal quadrangle 432(4)) are arranged in a spiral. By not arranging them in a regular arrangement such as a lattice or radial pattern, the effect of the regular arrangement of the CSP type LEDs is suppressed, and color unevenness is suppressed.

<Variation>
(1) Although three-color LEDs are shown as examples of LEDs with a plurality of emission colors, two colors, four colors or more may be used. For example, it may be an LED that performs toning by combining a daylight color LED and a light bulb color LED.

(2) Although R, Yw, and Bw have been described as three-color LEDs, other three-color LEDs may be used. For example, R, Yw, and B (LEDs with chromaticity coordinates of x≦0.2 and y≦0.2) may be used. Three-color LEDs of R, G (LEDs with chromaticity coordinates x≦0.35 and y≧0.4) and B may be used. You may combine R, G, B and white (for example, daylight white LED or daylight LED).

(3) As the regular arrangement of the LEDs, the lattice arrangement, the lattice arrangement in which some rows are shifted, and the triangular arrangement are shown. Not limited. In particular, the present invention can effectively reduce color unevenness when three or more LEDs are arranged in a straight line.

(4) As the lighting equipment, a ceiling-embedded downlight has been exemplified, but a spotlight or a universal (variable direction) downlight may be used.

(5) The convex streaks 333G and 433G may be replaced with "concave streaks", and in either structure, the direction of light rays can be changed to reduce color unevenness.

It should be noted that the above-described embodiment disclosed this time is an example in all respects and does not serve as a basis for restrictive interpretation. Therefore, the technical scope of the present invention is not to be interpreted only by the above-described embodiments, but is defined based on the claims. In addition, all changes within the meaning and range of equivalents to the scope of claims are included.

300, 400 lighting devices 310, 410 heat sink 315 lamp bodies 320, 420 light sources 322, 422 substrates 323A, 323B, 323C, 423A, 423B, 423C, 523A, 523B, 523C CSP type LED
323AX, 323AX', 423AX, 523BX Rows of CSP type LEDs 325, 425 Mounting holes 326A, 326B, 326C, 326P, 426A, 426B, 426C, 426P, 526A, 526B, 526C, 526P Wiring terminals 330, 430 Lenses 331, 431 Entrance holes 332, 432 Entrance bottom surfaces 333, 433 Entrance side surfaces 333G, 433G Convex grooves 333F, 433F Facets 334, 434 Lower surfaces 335, 435 Reflection side surface 336 Output hole 337 Output bottom surface 338 Output side surfaces 339, 439 Output top surface 340 Cone 350 Attachment Spring 360 Power supply 370 Control device 438 Support part 429 Terminal Bw Blue-white LED
Yw yellowish white LED
R Red LED

Claims (6)

A lighting device that distributes light emitted from a light source with a lens,
The light source comprises LEDs of a plurality of emission colors,
the lens has an entrance hole with an entrance side and an entrance bottom, a reflective side, and an exit top;
The illumination device , wherein the incident side surface is provided with a spiral convex or concave muscle . 2. The illumination device of claim 1 , wherein the light source comprises LEDs with three emission colors . The lighting device according to claim 1 or 2 , wherein the light source has a portion in which three or more LEDs of the same emission color are arranged in a straight line. The lighting device according to any one of claims 1 to 3, wherein the light source is a light source in which CSP type LEDs are arranged on a substrate as the LEDs of the plurality of emission colors. 5. A lighting device according to any one of the preceding claims, wherein the top exit surface comprises a textured structure covered with trapezoidal polygons. 6. A lighting device according to any one of the preceding claims, wherein the entrance bottom surface comprises a textured structure covered with scalene polygons.

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