JP5843228B2 - Split projection method of starry sky in planetarium - Google Patents

Split projection method of starry sky in planetarium Download PDF

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JP5843228B2
JP5843228B2 JP2010003023A JP2010003023A JP5843228B2 JP 5843228 B2 JP5843228 B2 JP 5843228B2 JP 2010003023 A JP2010003023 A JP 2010003023A JP 2010003023 A JP2010003023 A JP 2010003023A JP 5843228 B2 JP5843228 B2 JP 5843228B2
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誠 笠原
誠 笠原
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Goto Optical Manufacturing Co Ltd
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この発明はプラネタリウムに関し、より詳細には複数の投映機からの分割投映像をつなぎ合わせてドームスクリーン上に全天の星空を投映する場合の分割投映方法に関する。   The present invention relates to a planetarium, and more particularly to a division projection method in which division projection images from a plurality of projectors are connected to project the entire starry sky on a dome screen.

プラネタリウムにおいては、複数の投映機からの分割投映像の合成によりドームスクリーン上に全天の星空を投映している。すなわち、図9に示すように全天Aを複数の面に分割し、投映光学系上に投映原板を配した複数の投映機によりそれぞれの分割面に分割投映像Bを投映し、これらをつなぎ合わせて星空を再現している。この場合、分割投映像の形状は正多面体あるいは、切頂多面体であることが通常であり、分割投映像の形状は図形の各内角が180度を越えることのない凸多角形となる(特許文献1)。   In the planetarium, the entire starry sky is projected on a dome screen by combining divided projection images from multiple projectors. That is, as shown in FIG. 9, the whole sky A is divided into a plurality of surfaces, and the divided projection images B are projected onto the respective divided surfaces by a plurality of projectors in which a projection original plate is arranged on the projection optical system, and these are connected. The starry sky is reproduced together. In this case, the shape of the divided projected image is usually a regular polyhedron or a truncated polyhedron, and the shape of the divided projected image is a convex polygon in which each internal angle of the figure does not exceed 180 degrees (Patent Document). 1).

複数の投映機からの分割投映像のつなぎ合わせにより一つの映像を完成させることはプラネタリウムに特有の技術ではなく、映像一般の投映技術においては慣用されている技術であることは勿論である(例えば、特許文献2)。   It is a matter of course that the completion of one image by joining divided projection images from a plurality of projectors is not a technique specific to a planetarium, but is a technique commonly used in general image projection technology (for example, Patent Document 2).

前記のプラネタリウムにおけるつなぎ合わせにおいては、各投映像のつなぎ合わせ箇所は重複や隙間が生じることなく連続するように設計されることが原則であり、実際のプラネタリウムの製造においては設計通りつなぎ合わせが行われるように、各投映機同士の配置の誤差や投映倍率の誤差が生じないように最新の注意が払われた。   In the above-mentioned jointing in the planetarium, in principle, the jointed portions of the projected images are designed to be continuous without any overlap or gaps. In actual planetarium production, joining is performed as designed. As mentioned above, the latest attention was paid so as not to cause an error in the arrangement of the projectors and an error in the projection magnification.

これに対し、映像一般の投映技術においては、各投映像の隣接する一部を互いに重なるように配することにより、各投映機同士の配置や投映倍率の精度が多少甘くても、投映像同士に隙間が生ぜずつながりをスムーズとする方法が早くから提案されていた(例えば、特許文献3)。   On the other hand, in general projection technology, by arranging adjacent parts of each projected image so as to overlap each other, even if the placement of each projector and the accuracy of the projection magnification are somewhat less accurate, In the past, a method for smooth connection without causing gaps has been proposed (for example, Patent Document 3).

そして、この場合、重なり合う箇所において同一の画像をそれぞれの投映機から重畳して投映する方法(例えば、特許文献4)の他、画像をそれぞれの投映機に振り分けて投映する方法(例えば、特許文献5)が提案されており、後者の場合は画像のずれが防止される効果が得られていた。   In this case, in addition to a method (for example, Patent Document 4) in which the same image is superimposed and projected from each projector at an overlapping portion, a method for distributing and projecting an image to each projector (for example, Patent Document) 5) has been proposed, and in the latter case, an effect of preventing image shift was obtained.

一方、プラネタリウムの分野においては、天の川などの自然な再現を目的に通常肉眼では点像として認識できない6等星よりも暗い星を投映する場合のように投映する星の数が増加した際に、設計通りつなぎ合わせが行われていないと分割境界線が目立ってしまうという問題が顕在化した。そこで、重なり合う箇所において画像をそれぞれの投映機に振り分けて投映する前記の映像一般の投映技術を適用し、各投映像の隣接する一部を互いに重なるように配すると共に、重なり合う箇所に投映すべき恒星群をそれぞれの投映機に振り分けて投映する方法が提案されていた(特許文献6)。具体的には、この方法においては両投映原板に互いに重なり合う部分を設け、その部分の星の投映割合を重なり合う位置によってそれぞれの投映原板に振り分けることで、それぞれの投映原板の星の数を調整している。前記特許文献6では、分割投映像に関して開示されている実施例において角を持たない凸図形が示されている。   On the other hand, in the planetarium field, when the number of stars projected increases, as in the case of projecting stars that are darker than the 6th magnitude stars that cannot be recognized as point images by the naked eye for the purpose of natural reproduction of the Milky Way, The problem that the dividing boundary line becomes conspicuous if it is not connected as designed. Therefore, the above-mentioned general projection technology that distributes and projects images to each projector at the overlapping locations is applied, and adjacent portions of each projection image are arranged so as to overlap each other and should be projected to the overlapping locations. A method has been proposed in which a star cluster is distributed to each projector and projected (Patent Document 6). Specifically, in this method, both projection masters are provided with overlapping parts, and the projection ratio of the stars in that part is distributed to each projection master according to the overlapping position, thereby adjusting the number of stars on each projection master. ing. In Patent Document 6, a convex figure having no corners is shown in the embodiment disclosed with respect to the divided projected image.

また、特許文献7においても、隣り合う恒星原板に重複部分を設け、該当する星の等級によって描画する原板を決定する方法が開示されているが、分割投映像に関しては図5、6、8において凸図形であることが示されている。なお、同文献において図11、12は原板同士を重複させる部分の恒星の振り分け方を示すために重複箇所毎の恒星の限界等級を示したグラフである。
特開平9−218641号公報 特開平5−19347号公報 特開昭58−125986号公報 特開平3−58082号公報 特開昭64−27374号公報 特開2001−134172公報 特開2003−122247公報
Also, Patent Document 7 discloses a method for determining an original plate to be drawn according to the star grade provided with overlapping portions on adjacent star original plates. For divided projection images, FIGS. It is shown to be a convex figure. In this document, FIGS. 11 and 12 are graphs showing the limit grade of the stellar for each overlapping portion in order to show how to assign the stellar at the portion where the original plates overlap each other.
Japanese Patent Laid-Open No. 9-218641 Japanese Patent Laid-Open No. 5-19347 JP 58-125986 A Japanese Patent Laid-Open No. 3-58082 Japanese Unexamined Patent Publication No. 64-27374 JP 2001-134172 A JP 2003-122247 A

自然な星空を再現するために、多数の星を投映するようなプラネタリウムにおいて、特許文献6に記載の方法は非常に有効である。この方法によれば、投映機の組付け誤差が大きな場合や、プラネタリウム投映機の設置位置が正しくない場合においても、分割境界線がある程度目立たない星空を再現することができる。   In order to reproduce a natural starry sky, the method described in Patent Document 6 is very effective in a planetarium that projects a large number of stars. According to this method, even when the installation error of the projector is large or when the installation position of the planetarium projector is not correct, it is possible to reproduce a starry sky in which the division boundary line is not noticeable to some extent.

しかしながら、前記のような方法を用いて投映を行なう場合、正確な組立や調整、設置作業を行なわずとも分割境界線の目立たない星空が再現されてしまうため、適切な組立調整作業、あるいは据付作業が行なわれているのか目視により判断することが困難となった。そのため、正確な組立や調整、設置作業が困難になりドーム上に再現される星空の星々の位置が正確であるとは限らない問題が生じてしまう。   However, when projecting using the method as described above, the starry sky with inconspicuous divisional boundaries is reproduced without accurate assembly, adjustment, and installation work, so appropriate assembly adjustment work or installation work It has become difficult to visually determine whether or not Therefore, accurate assembly, adjustment, and installation work become difficult, and a problem arises that the positions of the stars in the starry sky reproduced on the dome are not always accurate.

組立や調整け精度が悪くなれば、投映機ごとに星々の相対的な位置が正しくないことになるし、設置位置が悪くなれば、ドームスクリーンとの相対的位置関係で、実際の星空よりも拡大されて投映される星空と縮小されて投映される星空が混在してしまうことになる。また、特許文献7に記載の方法においても、どのような等級ごとの分布を用いるかによって差はあるものの同様の問題が生じる。   If the accuracy of assembly and adjustment is poor, the relative position of the stars for each projector will be incorrect, and if the installation position is poor, the relative position with the dome screen will be higher than the actual starry sky. A starry sky that is projected after being enlarged and a starry sky that is projected when it is reduced will be mixed. In the method described in Patent Document 7, the same problem arises although there is a difference depending on what grade distribution is used.

星の数が数万個程度と少なかった従来のプラネタリウムにおいても、特別な場合を除いて、星の位置は可能な限り正確であることが要求されてきた。星の数が非常に多くなった場合の対策として提案されている前記の方法では、別途測定をしなければ、これまでと同等の精度で組立調整、あるいは据付けが行なわれたか否かが判断できない問題を生じた。   Even in the conventional planetarium where the number of stars is as small as tens of thousands, the position of the stars has been required to be as accurate as possible except in special cases. In the above-mentioned method proposed as a countermeasure when the number of stars becomes very large, it is impossible to determine whether assembly adjustment or installation has been performed with the same accuracy as before unless measurement is separately performed. Caused a problem.

この発明のプラネタリウムにおける星空の分割投映方法は以上の従来技術の問題点に鑑みて創作されたものであり、投映する星の数が非常に多い場合であっても、組立調整作業や据付調整作業を投映像の目視により容易に実現できることを可能とし、これまでと同等の星の位置の再現精度を確保すると共に、分割境界線の目立たない星空を再現することを目的とする。   The starry sky division projection method in the planetarium of the present invention was created in view of the above problems of the prior art, and even if the number of projected stars is very large, assembly adjustment work and installation adjustment work It is possible to easily realize this by visually observing the projected image, to ensure the same reproducibility of the star position as before, and to reproduce the starry sky in which the dividing boundary line is not conspicuous.

前記目的を達成するために、この発明の星空の分割投映方法は、全天を複数の面に分割し、投映光学系上に投映原板を配した複数の投映機によりそれぞれの分割面に分割投映像を投映し、これらをつなぎ合わせて星空を再現するように、複数の投映機からの分割投映像の合成によりドームスクリーン上に全天の星空を投映するに際し、分割投映像を相互に重複しないようにつなぎ合わせて投映すると共に、ドームスクリーン上に投映される分割投映像の形状を波形もしくは鋸歯状の線に囲まれた図形としたことを特徴とする In order to achieve the above object, according to the starlit sky division projection method of the present invention, the whole sky is divided into a plurality of planes, and a plurality of projectors each having a projection master plate arranged on a projection optical system are divided into projection planes. When projecting images and connecting them together to reproduce the starry sky, when projecting the entire sky on the dome screen by combining the divided projection images from multiple projectors, the divided projection images do not overlap each other The divided projection images projected onto the dome screen are formed into a figure surrounded by a waveform or a sawtooth line .

すなわち、この発明においては分割投映領域に設計上の重複する部分を設けずに、ドームスクリーン上に投映される分割投映像を旧来用いられてきた凸図形(図形内の任意の2点間を結ぶ線分が、すべて図形内に含まれるような図形)ではなく波形もしくは鋸歯状の線に囲まれた図形としている。In other words, in the present invention, the divided projected image projected on the dome screen is not provided with a design overlapping portion in the divided projected area, and the projected figure (which connects two arbitrary points in the figure) that has been used in the past. It is assumed that the line segment is a figure surrounded by a wavy or saw-toothed line rather than a figure that is all included in the figure.

プラネタリウムの製造上で生ずるわずかな誤差は、基本的に投映倍率の誤差と投映機相互の配置の誤差である。これらのうち、相互配置の誤差は調整によって最小限に抑えることが可能であるが、投映倍率の誤差を調整することは非常に困難である。   The slight error that occurs in the production of the planetarium is basically an error in the projection magnification and an error in the arrangement of the projectors. Among these, the mutual arrangement error can be minimized by adjustment, but it is very difficult to adjust the projection magnification error.

この誤差は投映像の大きさの違いとなって現れるから、各原板の投映範囲が凸図形である場合、倍率の誤差を直接反映した量の隙間あるいは重複箇所として投映ユニットの中心から放射状の方向に現れる。   Since this error appears as a difference in the size of the projected image, if the projection range of each original plate is a convex figure, the radial direction from the center of the projection unit as a gap or overlap of the amount that directly reflects the magnification error Appear in

しかしながら、波形もしくは鋸歯状の線に囲まれた図形である場合、その投映ユニットの中心からの放射状の方向では同量の値であっても、実際の隙間あるいは重複は、描画領域の接線方向と直交した成分となるため、放射状方向とのなす角が小さいほど少なくなり、原板の分割境界線は目立たなくなる。 However, in the case of a figure surrounded by a wavy or sawtooth line , the actual gap or overlap is the same as the tangential direction of the drawing area, even if the value is the same amount in the radial direction from the center of the projection unit. Since the components are orthogonal to each other, the smaller the angle formed with the radial direction, the smaller the angle, and the dividing boundary line of the original plate becomes less noticeable.

一方で、中心位置の誤差は、許容される量を超えると、ズレている方向では、周囲に比べて明るくなり、逆の方向では暗くなり、また直交する方向では繰返される明暗の模様になり目立つようになることから、調整すべき方向を容易に判断することができる。   On the other hand, if the error of the center position exceeds the allowable amount, the deviation direction becomes brighter than the surrounding area, becomes darker in the opposite direction, and becomes a bright and dark pattern that repeats in the orthogonal direction. As a result, the direction to be adjusted can be easily determined.

以上のように、この発明による分割投映方法によれば、投映恒星数を著しく増加させた場合においても、これまでと同様の星の位置精度を保った上で分割境界線が目立たず、自然で正確な星空を再現することが可能となり、一方、位置精度が許容範囲を超えた場合は目視により調整すべき方向を容易に判断して調整することが可能となる効果を奏する。   As described above, according to the divided projection method according to the present invention, even when the number of projection stars is remarkably increased, the division boundary line is not conspicuous while maintaining the same star position accuracy as before, and it is natural. It is possible to reproduce an accurate starry sky. On the other hand, when the position accuracy exceeds the allowable range, it is possible to easily determine and adjust the direction to be adjusted visually.

この発明の投映方法による分割投映像を示す概念図。The conceptual diagram which shows the division | segmentation projection image by the projection method of this invention. この発明の投映方法による分割投映像を示す概念図。The conceptual diagram which shows the division | segmentation projection image by the projection method of this invention. この発明の投映方法と従来技術の投映方法の作用を示す概念図。The conceptual diagram which shows the effect | action of the projection method of this invention, and the projection method of a prior art. この発明の投映方法の作用を示す概念図。The conceptual diagram which shows the effect | action of the projection method of this invention. この発明の投映方法と従来技術の投映方法の作用を示す概念図。The conceptual diagram which shows the effect | action of the projection method of this invention, and the projection method of a prior art. この発明の投映方法と従来技術の投映方法の作用を示す概念図。The conceptual diagram which shows the effect | action of the projection method of this invention, and the projection method of a prior art. この発明の投映方法と従来技術の投映方法の作用を示す概念図。The conceptual diagram which shows the effect | action of the projection method of this invention, and the projection method of a prior art. この発明の投映方法と従来技術の投映方法の作用を示す概念図。The conceptual diagram which shows the effect | action of the projection method of this invention, and the projection method of a prior art. 従来技術の投映方法の構成図。The block diagram of the projection method of a prior art.

図1に、例として波形もしくは鋸歯状の線に囲まれた図形で構成される分割投映像1の一部を示す。この分割投映像1の形状は曲線(sinカーブ)で構成されており、図に示すように波型となっている。図2は分割投映像1同士が相互に重複しないように正確につなぎ合わせられた状態を示す図である。 FIG. 1 shows a part of a divided projected image 1 composed of a figure surrounded by a waveform or a sawtooth line as an example. The shape of the divided projected image 1 is composed of a curve (sin curve), and has a wave shape as shown in the figure. FIG. 2 is a diagram showing a state in which the divided projected images 1 are accurately connected so as not to overlap each other.

図3はこの発明の投映方法と従来技術の投映方法の作用の違いを示す図である。図中符号(3−1)はこの発明の投映方法による分割投映像1、(3−2)は従来技術の投映方法による分割投映像10であり、ここでは分割投映像同士に隙間が生じた状態を図示している。   FIG. 3 is a diagram showing the difference in operation between the projection method of the present invention and the projection method of the prior art. In the figure, reference numeral (3-1) is a divided projected image 1 by the projection method of the present invention, and (3-2) is a divided projected image 10 by the projection method of the prior art. Here, there is a gap between the divided projected images. The state is illustrated.

前記の(3−1)、(3−2)はそれぞれ同量の隙間が生じた例を示すものであるが、従来技術の凸図形で構成される分割投映像10同士の隙間S10は直線であるが、この発明の分割投映像1同士の隙間はsinカーブとすることでカーブの頂上/底部では直線の場合と同じだけの隙間S1が生じるが、それ以外のカーブの斜面の隙間S2は小さくなる。すなわち、図4に示すようにカーブの頂上/底部の隙間S1の間隔A−Aに比べて、斜面の隙間S2の間隔B−Bは狭くなる。   The above (3-1) and (3-2) show examples in which the same amount of gaps are generated, but the gap S10 between the divided projected images 10 composed of the convex figures of the prior art is a straight line. Although the gap between the divided projected images 1 of the present invention is a sin curve, the same gap S1 is generated at the top / bottom of the curve as in the case of a straight line, but the gap S2 on the slope of the other curves is small. Become. That is, as shown in FIG. 4, the interval BB of the gap S2 on the slope is narrower than the interval AA of the gap S1 at the top / bottom of the curve.

プラネタリウムは、ドームの中心に設置される機械であるため、映し出される星の位置が正しくない最大の要因は、投映する倍率に誤差がある場合がもっとも大きくなる。この発明の投映方法においては、sinカーブとすることでこの方向(投映中心から放射方向) の誤差が目立たなくなる。   Since the planetarium is a machine installed at the center of the dome, the biggest cause of the incorrect position of the projected star is the largest when there is an error in the projection magnification. In the projection method of the present invention, an error in this direction (from the projection center to the radiation direction) becomes inconspicuous by using a sin curve.

図5はこの発明の投映方法と従来技術の投映方法を示す図である。図中符号(5−1)はこの発明の投映方法による分割投映像1、(5−2)は従来技術の投映方法による分割投映像10であり、ここではこの発明の投映方法による波形もしくは鋸歯状の線に囲まれた図形で構成される分割投映像1として、直線で構成された鋸刃状のものを図示している。 FIG. 5 is a diagram showing the projection method of the present invention and the projection method of the prior art. In the figure, reference numeral (5-1) denotes a divided projected image 1 by the projection method of the present invention, and (5-2) represents a divided projected image 10 by the projection method of the prior art. Here, the waveform or sawtooth by the projection method of the present invention is shown. As a divided projection image 1 composed of figures surrounded by a line, a saw blade-shaped one composed of straight lines is shown.

図6は前記の図5に示したこの発明の投映方法の分割投映像同士、従来技術の投映方法の分割投映像同士に投映機相互の配置の誤差により位置ずれが生じた状態を示す図である。この場合、(6−2)の従来技術の投映方法の分割投映像10同士には重複箇所S11と隙間S10が生じ、重複箇所は明るい線として、隙間は暗い線としてはっきりと見えてしまう。   FIG. 6 is a diagram showing a state in which a positional deviation has occurred between the divided projection images of the projection method of the present invention shown in FIG. 5 and between the divided projection images of the projection method of the prior art due to an error in the arrangement of the projectors. is there. In this case, an overlapping portion S11 and a gap S10 are generated between the divided projected images 10 of the conventional projection method (6-2), and the overlapping portion is clearly seen as a bright line and the gap is clearly seen as a dark line.

これに対し、この発明の投映方法による分割投映像1同士の重複箇所と隙間は狭く目立たちにくい。この場合、波形もしくは鋸歯状の線に囲まれた図形を曲線により構成すると重複箇所や隙間の濃淡が変化してより目立ちにくくなる。 On the other hand, the overlapping portions and gaps between the divided projected images 1 according to the projection method of the present invention are narrow and not easily noticeable. In this case, if a figure surrounded by a waveform or a sawtooth line is formed by a curve, the overlapping portion and the density of the gap change and become less noticeable.

一方、この発明の投映方法において、ずれが許容範囲を超えて重複箇所や隙間がはっきりと目立つ場合は分割投映像の頂点あるいは底部の組み合わさる部分が投映中心から放射状の方向の精度を示し、斜面がそれと直交する方向の精度を示し、この図における分割投映像のように波形もしくは鋸歯状の線に囲まれた図形を回転対称なものとすれば、どの方向に大きくずれているのかが一目瞭然となり、目視による調整を容易に行うことができる。 On the other hand, in the projection method of the present invention, when the deviation exceeds the allowable range and the overlapping portion or the gap is clearly noticeable, the combined portion of the top or bottom of the divided projection image shows the accuracy in the radial direction from the projection center, and the slope Indicates the accuracy in the direction orthogonal to it, and if the figure surrounded by the waveform or sawtooth line is rotationally symmetric as in the divided projection image in this figure, it is obvious at which direction it is greatly shifted. The visual adjustment can be easily performed.

図7は前記の図5に示したこの発明の投映方法の分割投映像同士、従来技術の投映方法の分割投映像同士に投映機相互の投映倍率の誤差により隙間が生じた状態を示す図である。この場合、(7−1)のこの発明の投映方法による分割投映像1同士の隙間は前記したように目立たちにくいが、(7−2)の従来技術の投映方法の分割投映像10同士には、投映倍率が小さい分割投映像10Aの周囲に環状の隙間S10が生じ、暗い線としてはっきりと見えてしまうことなる。   FIG. 7 is a diagram showing a state in which a gap is generated between the divided projection images of the projection method of the present invention shown in FIG. 5 and the divided projection images of the projection method of the prior art due to the projection magnification error between the projectors. is there. In this case, the gap between the divided projected images 1 according to the projection method of the present invention of (7-1) is not conspicuous as described above, but between the divided projected images 10 of the conventional projection method of (7-2). In this case, an annular gap S10 is generated around the divided projected image 10A having a small projection magnification, so that it can be clearly seen as a dark line.

図8は前記の図5に示したこの発明の投映方法の分割投映像同士、従来技術の投映方法の分割投映像同士に投映機相互の投映倍率の誤差により重複箇所が生じた状態を示す図である。この場合、(8−1)のこの発明の投映方法による分割投映像1同士の重複箇所は前記したように目立たちにくいが、(8−2)の従来技術の投映方法の分割投映像10同士には、投映倍率が大きい分割投映像10Aの周囲に環状の重複箇所S11が生じ、明るい線としてはっきりと見えてしまうことなる。   FIG. 8 is a diagram showing a state where overlapping portions are generated between the divided projection images of the projection method of the present invention shown in FIG. 5 and the divided projection images of the projection method of the prior art due to the projection magnification error between the projectors. It is. In this case, the overlapping portion of the divided projection images 1 by the projection method of the present invention of (8-1) is not conspicuous as described above, but the divided projection images 10 of the conventional projection method of (8-2) In this case, an annular overlapping portion S11 is generated around the divided projected image 10A having a large projection magnification, and is clearly visible as a bright line.

Claims (1)

全天を複数の面に分割し、投映光学系上に投映原板を配した複数の投映機によりそれぞれの分割面に分割投映像を投映し、これらをつなぎ合わせて星空を再現するように、複数の投映機からの分割投映像の合成によりドームスクリーン上に全天の星空を投映するに際し、分割投映像を相互に重複しないようにつなぎ合わせて投映すると共に、ドームスクリーン上に投映される分割投映像の形状を波形もしくは鋸歯状の線に囲まれた図形としたことを特徴とするプラネタリウムにおける星空の分割投映方法。 Divide the whole sky into multiple planes, project multiple projected images on each divided plane using multiple projectors with a projection master plate on the projection optical system, and connect them to reproduce the starry sky. When projecting the entire starry sky on the dome screen by synthesizing the divided projection images from other projectors, the divided projection images are projected together so as not to overlap each other, and the divided projections projected on the dome screen are projected. A method for dividing the starry sky in a planetarium, characterized in that the shape of the image is a waveform or a figure surrounded by a sawtooth line.
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