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制作宇宙飞船的方法英语作文The Fabrication of a Spacecraft.The journey towards the unknown depths of space has always fascinated mankind. Spacecrafts, the vehicles that carry our dreams and aspirations, are the epitome of human innovation and technological prowess. Thefabrication of a spacecraft is a complex and meticulous process, requiring a blend of precision engineering, advanced technology, and rigorous testing.1. Design and Planning.The journey begins with a thorough design and planning phase. Engineers and scientists collaborate to create a blueprint that not only fulfills the mission objectives but also ensures the safety and comfort of the crew. The design takes into account the specific requirements of the mission, such as the duration of the flight, the destination, andthe scientific equipment that needs to be carried.2. Materials Selection.The next step is the selection of appropriate materials. Spacecrafts must be built to withstand the extremeconditions of space, including the vacuum, radiation, and extreme temperature fluctuations. Materials like titanium, aluminum, and composites are chosen for their strength, durability, and lightweight properties.3. Manufacturing Process.The manufacturing process is highly specialized andoften involves precision machining, welding, and assembly. Components like the hull, engines, and life support systems are meticulously crafted to ensure their reliability and durability. The hull, for instance, must be able to withstand the immense pressure differences between theEarth's atmosphere and the vacuum of space.4. Integration and Testing.Once the individual components are ready, they are integrated into a complete spacecraft. This phase involves meticulous planning and careful execution to ensure thatall systems work together seamlessly. Following integration, rigorous testing begins. This includes ground tests to simulate the conditions of space, as well as flight simulations to test the spacecraft's performance underreal-time conditions.5. Launch Preparation.As the launch date nears, the spacecraft undergoesfinal checks and preparations. This includes ensuring that all systems are operational, loading the necessary fuel and supplies, and conducting final safety inspections. The launch vehicle, which will carry the spacecraft into orbit, is also prepared during this phase.6. Launch and Deployment.The launch is a momentous occasion, marked byexcitement and anticipation. As the launch vehicle rocketsinto the sky, the spacecraft separates from the launch vehicle and begins its journey into orbit. Once in orbit, the spacecraft undergoes further checks and deployments, such as activating its solar panels or deploying scientific instruments.7. On-Orbit Operations.Once in orbit, the spacecraft carries out its mission objectives. This could involve conducting experiments, observing celestial objects, or even supporting human habitation. The spacecraft's systems are continuously monitored and maintained to ensure its continued operation.8. Decommissioning and Disposal.After completing its mission, the spacecraft is decommissioned and disposed of responsibly. This could involve de-orbiting the spacecraft and allowing it to burn up in the Earth's atmosphere or sending it to a graveyard orbit. The decommissioning process ensures that the spacecraft does not pose a threat to other spacecraft orthe Earth's environment.In conclusion, the fabrication of a spacecraft is a complex and multifaceted process that requires the expertise of numerous professionals and the latest in technological advancements. It is a testament to human ingenuity and determination, paving the way for future explorations and scientific discoveries in the vast and mysterious universe.。
打造艺术空间英文作文英文：Creating an art space is a challenging yet rewarding task. It requires careful planning, attention to detail, and a deep understanding of art and its various forms. As an art enthusiast, I have always dreamed of creating a space where artists can showcase their work and art lovers can come together to appreciate it.The first step in creating an art space is to identify the purpose of the space. Will it be a gallery, a studio, or a combination of both? Once the purpose is established, the next step is to choose a location that is easily accessible and has ample space to accommodate the art and the visitors. The space should be well-lit, ventilated, and have a welcoming ambiance.Next, it is important to create a layout that is both functional and aesthetically pleasing. The art should bedisplayed in a way that enhances its beauty and allows visitors to appreciate it from different angles. The space should also have areas for artists to work and interactwith each other and visitors.In addition to the physical space, it is important to create a digital presence for the art space. This can be done through a website, social media, and online exhibitions. This will allow the art to reach a wider audience and attract more visitors to the physical space.Creating an art space also requires a strong communityof artists, art lovers, and supporters. It is important to build relationships with local artists and organizations, and to host events and exhibitions that engage the community and promote the arts.In conclusion, creating an art space is a complex yet fulfilling endeavor. It requires a deep passion for art, a strong understanding of its various forms, and a commitment to building a community that supports and promotes the arts. With careful planning and attention to detail, anyone cancreate an art space that inspires and enriches the lives of those who visit.中文：打造艺术空间是一项具有挑战性但又有回报的任务。
泉州2024年统编版小学3年级英语第三单元期中试卷考试时间：100分钟（总分:100）A卷考试人：_________题号一二三四五总分得分一、综合题(共计100题共100分)1. 选择题：What is the opposite of happy?A. SadB. AngryC. ExcitedD. Tired2. 填空题：I like to _______ my own lunch.3. 听力填空题：I prefer __________ over __________ because it is __________.4. 听力题：A ____ is known for its quick movements and long legs.5. 听力题：The Earth's surface is shaped by both human and ______ factors.6. 听力题：A telescope helps us see _____ objects in space.7. 选择题：What do you call the soft part of the head of a baby?A. SkullB. BrainC. FontanelD. Cranium答案:C8. ssance was a time of great ________ (文化繁荣). 填空题：The RenaShe is a great ________.10. 填空题：The park is ________ (适合家庭).11. 填空题：On sunny days, I love to go __________ with my family. (徒步旅行)12. 选择题：What do we call the time it takes for the Earth to go around the sun?A. DayB. MonthC. YearD. Week答案:C13. 听力题：I like ________ (exploring) new ideas.14. 填空题：We visited a __________ (图书馆) full of maps.15. 选择题：Which month has 28 days?A. FebruaryB. JanuaryC. MarchD. April16. 听力题：An island is a piece of land that is completely __________ by water.17. 选择题：What do we call a person who studies the history of revolutions?A. Revolutionary HistorianB. SociologistC. AnthropologistD. Political Scientist答案: A18. 听力题：The __________ can be affected by natural disasters.19. 听力题：The ____ is often seen in parks searching for food.Dolphins are very ______ (聪明) animals that live in the ocean.21. 填空题：A turtle hides safely in its _______ when it feels scared.22. 填空题：A rabbit can produce many ______ (小兔子) in a year.23. 听力题：The chemical formula for manganese dioxide is _____.24. 听力题：An electrolyte is a substance that conducts electricity when _____.25. 听力题：I like _____ (to cook/to eat).26. 填空题：The zebra has black and white _______ (条纹).27. 听力题：A _______ can be used to measure the temperature of the air.28. 听力题：He is a good ___. (friend)29. 听力题：The main gas produced during respiration is ______ dioxide.30. 选择题：What is the name of the chemical element with the symbol H?A. HeliumB. HydrogenC. OxygenD. Nitrogen答案: B31. 听力题：The chemical formula for cadmium sulfide is _______.32. 选择题：What is the name of the famous Egyptian structure?A. ColosseumB. Great WallC. PyramidsD. Eiffel Tower答案: C33. 听力题：My grandma enjoys baking ____ (cakes) for birthdays.34. 填空题：The dolphin is a friendly _________ (生物).35. 听力题：The process of oxidation involves __________ losing electrons.36. 选择题：What is the capital of Oman?A. MuscatB. SalalahC. SoharD. Nizwa答案:A. Muscat37. 听力题：The ice cream is _____ (cold/warm) and delicious.38. 听力题：Hubble's law explains the expansion of the ______.39. 选择题：What is the main source of energy for the Earth?A. WaterB. WindC. SunD. Coal答案：C40. 填空题：My neighbor, ______ (我的邻居), has a big garden.41. 听力题：A biome is a large geographical area with similar ______ conditions.42. 听力题：A _______ is a small particle made up of atoms.43. 填空题：The _____ (植物学家) studies various plant species.A _______ can provide food for butterflies.45. tine Empire was the continuation of the ________ (罗马帝国). 填空题：The Chin46. ts can grow in ______ (阴凉) places. 填空题：Some pla47. 听力题：The Sahara is a large __________.48. 填空题：Pollinators like butterflies and bees are crucial for ______ (授粉).49. 听力题：The flowers are ________ and colorful.50. 听力题：A polymer is a large molecule made up of many ______ units.51. 填空题：My hamster runs on its ______ (轮子) all night.52. 填空题：Every Sunday, my family goes for a ________ (散步) in the park to enjoy nature.53. 填空题：The walrus has long _______ (獠牙).54. 听力题：In the periodic table, elements are organized by their ________ properties.55. 听力题：The process of making vinegar involves the fermentation of _______.56. 填空题：She is _______ (在读书).57. 填空题：The fox is very _______ (狐狸非常_______).58. 填空题：The ________ (公共设施) support community needs.We will go ______ next week. (shopping)60. 听力题：A mixture that is not uniform throughout is called a ______ mixture.61. 听力题：I like to go ______ (swimming) in the ocean.62. 听力题：Quasars are extremely bright objects powered by supermassive ______.63. 听力题：The _______ of sound can travel through solids, liquids, and gases.64. 听力题：I have a ___ in my backpack. (notebook)65. 选择题：What is the color of snow?A. WhiteB. BlackC. GrayD. Blue66. 填空题：The ancient Greeks wrote _______ that are still read today. (戏剧)67. 填空题：My dad is the best __________ (厨师) in our family.68. 填空题：The zebra has black and _________ stripes. (白色)69. 听力题：The monkey is ______ (swinging) from tree to tree.70. 听力题：A ____ is a loyal companion and loves to be around people.71. 选择题：What do you call a large body of ice that moves slowly?A. GlacierB. IcebergC. Snow答案:A72. 听力题：Fermentation is an anaerobic process that produces _____ and alcohol.73. 填空题：My sister is _______ than I am.74. 填空题:I love going on ________ during the holidays.75. 听力题：The __________ can help improve understanding of geological processes.76. 填空题：The ________ (农业) is crucial for feeding people.77. 选择题：What do we call a place where we keep books?A. LibraryB. MuseumC. SchoolD. Park答案: A78. 听力题：In a chemical reaction, the energy can be stored or _____.79. 填空题：In my dream, I can fly like a ______ (鸟). It would be amazing to see the world from above.80. 选择题：What do bees produce?A. MilkB. HoneyC. ButterD. Eggs答案:B81. 听力题：The classroom is ___ (clean/messy).82. 选择题：What is the largest animal in the ocean?B. DolphinC. WhaleD. Octopus答案:C83. 听力题：The chemical formula for uric acid is ______.84. 填空题：The __________ (历史的交互作用) enhance learning experiences.85. 听力题：The bear searches for food in the ____.86. 选择题：What do we call the area where a river meets the ocean?A. DeltaB. EstuaryC. BayD. Coast87. ts are _____ (medicinal) and help us stay healthy. 填空题：Some pla88. 填空题：The golden retriever is a popular _________ (宠物).89. 选择题：What is the largest organ inside the human body?A. LiverB. HeartC. BrainD. Lung答案: A90. 选择题：What do you call a large, round fruit that is typically orange?A. AppleB. PeachC. OrangeD. Banana91. 选择题：What do we call a large body of saltwater?A. LakeC. OceanD. Pond92. 填空题：The coyote hunts in ______ (群体).93. 听力题：A ______ is a system used to keep track of elements and their properties.94. 填空题：The first modern Olympics were held in ________ (雅典).95. 选择题：What is the capital of France?A. RomeB. BerlinC. ParisD. Madrid96. 填空题：The rabbit's favorite treat is ______ (水果).97. 填空题：The _____ (植物生长阶段) varies from species to species.98. 填空题：World War II lasted from ________ (1939) to 1945.99. 听力题：I have two ___. (sisters)100. s are deciduous and lose their leaves in ______.(有些树是落叶树，秋天会掉叶子。
2024届高三“天使计划”第二轮英语注意事项：1.答卷前，考生务必将自己的姓名、准考证号填写在答题卡上。
2.回答选择题时，选出每小题答案后，用铅笔把答题卡上对应题目的答案标号涂黑。
如准考证号：如需改动，用橡皮擦干净后，再选涂其他答案标号。
回答非选择题时，将答案写在答题卡上。
写在本试卷上无效。
3.考试结束后，将本试卷和答题卡一并交回。
第一部分听力(共两节，满分30分)第一节(共5小题;每小题1.5分,满分7.5分)听下面5段对话。
每段对话后有一个小题，从题中所给的A、B、C三个选项中选出最佳选项。
听完每段对话后，你都有10秒钟的时间来回答有关小题和阅读下一小题。
每段对话仅读一遍。
例:How much is the shirt?A.£19.15.B.£9.18.C.£9.15.答案是C。
1.What will the man do?A.Change the plan.B.Wait for a phone call.C.Sort things out.2.What does the woman want to do?A.See a film with the man.B.Offer the man some help.C.Listen to some great music.3.Which place are the speakers trying to find?A.A hotel.B.A bank.C.A restaurant.4.What does the man like about the play?A.The story.B.The ending.C.The actor.5.At what time will the two speakers meet?A.5:20.B.5:10.C.4:40.第二节(共15小题;每小题1.5分,满分22.5分)听下面5段对话或独白。
Unit 4 Space exploration单元分析本单元主题：人与自然——宇宙探索单元内容分析本单元围绕“太空探索”这一主题展开，内容涉及人类探索太空的历史与成就（包括我国航天事业的发展和成就）宇航员的选拔、太空中的生活、火星探索计划、天文百科知识，以及关于“人类耗费时间和金钱去探索太空是否值得”这一话题的会科学家以及宇航员们为航空航天事业的发展不断努力、勇于开拓的精神，从而激励青少年勤奋学习、刻苦钻研、不畏挫折，努力探索自己的发展道路，立下远大志向，为国家科技事业的发展奉献自己的智慧和力量。
以下为教材各部分教学内容简要分析及教学活动实施建议：1．Opening Page开篇页的主题图展示了2019年1月3日“玉兔二号”（即嫦娥四号月球车）与嫦娥四号着陆器分离后驶抵月球表面的画面。
嫦娥四号首次实现在月球背面着陆，是中国航天事业发展的一座里程碑，是值得每个中国人铭记的时刻。
这一图片不仅能够增强学生的民族自豪感，还能激发学生讨论的热情，让他们能够很快投入到本单元的学习中去。
本单元的名言警句是“Mystery creates wonder and wonder is the basis of man＇s desire to understand.-Neil Armstrong”这是第一位登上月球的美国宇航员尼尔·阿姆斯特朗的名言：“神秘感激发好奇心，而好奇心则是人们探索未知事物的前提。
”人类太空探索的原始动机主要源于与生俱来的好奇心，而这种好奇心和求知欲也是推动人类发展的巨大动力。
太空探索的成就背后是人类不断进取的精神，而只有将这种精神延续下去，人类才能突破极限，走向广阔的宇宙。
2．Listening and Speaking：Talk about how to become an astronaut 本部分的主题是“谈论如何成为一名宇航员”。
关于探索太空的奥秘，能够获得最直接感受的非宇航员莫属，因此，宇航员这一职业对于青少年而言充满了魅力，既带着神秘感，又带着使命感。
2022年考研考博-考博英语-燕山大学考试全真模拟全知识点汇编押题第五期（含答案）一.综合题(共15题)1.单选题In the ______ of the project not being a success, the investors stand to lose up to $30 million.问题1选项A.faceB.timeC.eventD.course【答案】C【解析】【试题解析】考查固定搭配。
A选项in the face of“面对”；B选项in the time of“在……时间段”；C选项in the event of“万一，倘若”；D选项in the course of“在……过程中”。
句意：______项目不成功，投资者将损失高达3000万美元。
根据语境，这里指项目失败的结果是损失3000万美金，C选项in the event of“万一，倘若”符合题意。
因此C选项正确。
2.单选题This is the Chinese ______, translated from English.问题1选项A.publicationB.editorC.printingD.version【答案】D【解析】【试题解析】考查名词辨析。
A选项publication“发行，出版物”；B选项editor“（报刊、杂志的）主编”；C选项printing“（书籍的）一次印刷”；D选项version“（电影、剧本、乐曲等的）版本”。
句意：这是英文翻译过来的中文______。
根据语境，这里指的是这原来是英文的，被翻译成了中文，D选项versio n“（电影、剧本、乐曲等的）版本”符合题意。
因此D选项正确。
3.单选题The band ______ and were beginning to throw their instrument down.问题1选项A.broke downB.broke offC.broke rankD.broke out【答案】A【解析】【试题解析】考查词组辨析。
如何制作宇宙飞船模型英文作文Title: Building a Model SpaceshipBuilding a model spaceship is a fascinating endeavor that allows one to explore the depths of imagination while honing craftsmanship skills. From selecting materials to adding intricate details, every step in the process contributes to the creation of a miniature marvel that embodies the essence of space exploration.The first step in constructing a model spaceship is gathering the necessary materials. Depending on the desired level of detail and authenticity, one may choose from a variety of materials such as plastic, wood, metal, or even recycled materials. Each material offers unique advantages and challenges, influencing the final outcome of the model.Once the materials are assembled, the next step is planning the design of the spaceship. Whether inspired byexisting spacecraft or completely original, careful consideration must be given to proportions, structural integrity, and aesthetic appeal. Detailed sketches or computer-aided design (CAD) software can aid in visualizing the final product and identifying any potential issues before construction begins.With a clear design in mind, construction can commence. This often involves cutting, shaping, and assembling the chosen materials according to the design specifications. Precision is key, as even minor discrepancies can affect the overall appearance and functionality of the model. Advanced techniques such as 3D printing or laser cutting may be employed to achieve intricate details with accuracy.As the spaceship takes shape, attention turns to adding fine details and embellishments. This stage allows for personalization and creativity, as each builder can incorporate unique features and decorations to enhance therealism and appeal of the model. From miniature control panels to tiny thrusters, every detail contributes to the authenticity of the final product.Throughout the construction process, patience and perseverance are essential virtues. Building a model spaceship is a labor-intensive endeavor that requires careful attention to detail and a willingness to overcome challenges. From intricate assembly techniques to troubleshooting unforeseen issues, each obstacle presents an opportunity for growth and learning.Finally, as the last piece is put in place and the finishing touches applied, a sense of satisfaction washes over the builder. Beholding the completed model spaceship, one cannot help but marvel at the journey from imagination to reality. Whether displayed proudly on a shelf or used as a prop in imaginative play, the model spaceship serves as atestament to the boundless creativity and ingenuity of the human spirit.In conclusion, building a model spaceship is a rewarding pursuit that combines technical skill with creative vision. From the initial concept to the final product, each step in the process offers opportunities for exploration and discovery. Whether embarked upon as a solitary endeavor or shared with fellow enthusiasts, the creation of a model spaceship is a testament to the enduring fascination with the mysteries of the cosmos.。
UNIT 4 单元主题训练Ⅰ.阅读理解A(2022·佛山二模)The future of space exploration may depend on an art form from the past: origami (折纸艺术), the ancient art of paper folding.Researchers from Washington State University (WSU), US, have used origami to possibly solve the problem of storing and moving fuel to rocket engines, a key challenge in space travel, according to Newswise.They've developed a foldable plastic fuel “bladder (囊状物)” resistant to super cold temperatures, which could be used to store and pump fuel in spacecrafts of the future.Their findings have recently been published in the journal Cryogenics.“Folks have been trying to make bags for rocket fuel for a long time，” said Jake Leachman, one of the lead researchers.“We currently don't do large, long-duration trips because we can't store fuel long enough in space.”Meanwhile, NASA is also looking to paper folding to help observe distant planets.The agency is currently developing Starshade, a foldable, sunflower-shaped piece of hardware that would help block starlight and enable telescopes to view distant objects more clearly in space.“A huge part of my job is looking at something on paper and asking, ‘Can we fly this？’” Manan Arya, a technologist in California, said.“Once I realized this is how you fold spacecraft structures, I became interested in origami.I realized I was good at it and enjoyed it.Now, I fold constantly.”Using origami for space purposes isn't new, however.Solar arrays (太阳能阵列), experimental wings for space shuttle programs and an inflatable (可充气的) satellite were also inspired by origami in both past and present space projects.“With most origami, the magic comes from the folding，” Robert Salazar, who helped design the Starshade and now works on the Transformers project, said in a statement.“There are so many patterns to still be explored.”语篇解读：本文是一篇说明文。
CHORDAL SPACE STRUCTURES, SHAPED FROMVORONOI DIAGRAMSCésar Otero1 , José Andrés Díaz1, Reinaldo Togores1, Cristina Manchado1ABSTRACTThe definition of the shape of the enclosures of large spaces by means of large span structures, where the absence of intermediate supports is a strong conditioning factor, is an interdisciplinary activity where geometry, biology, topology, architecture and engineering have complemented themselves. A relatively recent discipline, COMPUTATIONAL GEOMETRY, has permitted a new formulation of the geometric basis and the numerical procedures that permit to generate a spatial dome, regardless of its type.The geometric and topologic configuration of any Spatial Mesh or Structure (including typologies like Lattice, Geotangent or any other patented or published structural form) does not suppose anything else than the creation of a polyhedron that approximates the shape of the ideal surface. Thanks to the methods of Computational Geometry, we are able to demonstrate that this problem has a purely and exclusively two-dimensional nature and treatment.KEYWORDSComputational Geometry, Spatial Structures, Voronoi Diagrams, Chordal Space Structures, Structural MorphologyINTRODUCTION. DEFINITION AND TYPOLOGY OF SPACE STRUCTURESA space frame is a structural system assembled from linear elements so arranged that forces are transferred in a three-dimensional manner. In some cases, the constituent elements may be two-dimensional. Macroscopically a space frame often takes the form of a flat or curved surface (Tsuboi,1984). That classical definition can be extended according to the following classification of space structures (Wester,1990):•Lattice archetype: frames composed by bars (one-dimensional elements) interconnected at nodes (zero-dimensional point objects). The structure is stabilizedby the interaction of axial forces that concur at the nodes (fig.1, left).1 Research Group EGICAD, www.egicad.unican.es. University of Cantabria, Dept. of Geographical andGraphical Engineering, Civil Engineering Faculty, Avda de Los Castros s/n 39005 Santander, Spain, Phone 942201794, FAX 942201703, oteroc@unican.es (professor), diazja@unican.es (professor), togoresr@uican.es (professor), manchadoc@unican.es (research engineer).•Plate archetype: plates (bi-dimensional elements) that conform a polyhedron’s faces stabilized by the shear forces acting along its edges (one-dimensional hinges) (fig. 2, left).•Solid archetype: structures composed by three-dimensional elements which are stabilized by the action of forces transferred between the planar faces of the solids. GEOMETRIC GENERATION OF SPACE STRUCTURESThe design of space structures can be approached from different points of view. We now review three methods developed during the second half of the 20th century that suggest different ways for approximating a quadric surface taken as reference.Geodesic Dome: lattice type structure with a configuration derived from regular or semi-regular polyhedra in which the edges are subdivided into equal number of parts (“frequency”, Fuller, 1954); making use of these subdivisions, a three-way grid can be induced upon the faces of the original polyhedron. The central projection of these grid’s vertices on the polyhedron’s circumsphere (see fig. 1), leads to a polyhedron approximating the sphere in which only the lattice’s nodes lie on the sphere’s surface (Kitrick,1990).Fig. 1.Left: Generation of the Geodesic Dome through theprojection of the three-way grid on the circumscribed sphere. Right:U.S. Pavilion, Montreal Universal Exposition (1967).Geotangent Dome: a plate type polyhedral structure in which the edges are tangent to a sphere. Such a sphere is sectioned by the polyhedron’s faces in such a way (fig. 2, above) that the faces’ inscribed circles are tangent to the inscribed circles of neighboring faces. Following this rule it is possible to determine the planes containing the circles generating the polyhedron’s edges from their intersection (Yacoe, 1987). The procedure is involved and its calculations imply the solution of a non-linear equation system through an iterative process based on successive approximations.Panel Structure: these plate type structures (Wester, 1990) derive from lattice type geometries by applying the principle of structural and geometric duality (based on the concept of a point’s polarity regarding a sphere). Taking as a starting point the geodesic dome’s circumsphere, it is possible to transform the lattice’s nodes in the faces of its dual structure (fig. 2, below); the primitive sphere remains as the new structure’s insphere.Fig. 2. Above:(left) Geotangent Polyhedron elevation. (R ight). Nine meter diameter geotangent dome crowning Canopy Tower, Cerro Semáforo, Panamá (1963).Below: (left) Panel structure, derived as the dual polyhedron of a Schwedler type dome. (R ight) Structures suggesting the plate typology. Eden Project, Cornwall, UK. COMPUTATIONAL GEOMETRY: VORONOI DIAGRAM AND DELAUNAY TRIANGULATION ON THE PLANEVoronoi Diagrams and Delaunay Triangulations belong to Computational Geometry, a modern field of study that can supply some new ideas to structural design. We will make a brief introduction of Voronoi Diagram and Delaunay Triangulation in plane 2D (Preparata, 1985). Let us consider a set S of points in the plane, S = {S1, S2, . . . S n}, where S i = (x i, y i). Given a member S i of this set, the polygon of Voronoi of S i, V(S i), is defined as the set of points P i on the plane that are closer to S i than to any other point S j of S. A Voronoi polygon is shown in figure 3. Each point S i of S defines a Voronoi polygon; the n regions arising from the set S partition the plane into a convex net which is known as the Voronoi Diagram of S (see figure 3). Each line segment of the diagram is a Voronoi edge, and its end points are called Voronoi Vertices. Notice that the S i points are not vertices of the diagram: we will refer to them as Generator Points. Next, we list a number of important properties of the Voronoi Diagram: (i) every Voronoi Polygon V(S i) contains only one S i generator point , (ii) every V(S i) is a convex polygon, (iii) every edge of the Voronoi Diagram is the perpendicular bisector of two generator points S i and S j, (iv) Every vertex of Voronoi is the common intersection of exactly three edges of the diagram. Only when four generator points are cocircular, does the related Voronoi vertex join 4 edges, (v) the straight-line dual of the Voronoi Diagram is a triangulation of S. (This means a planar subdivision of the plane, where every polygon is a triangle and where each vertex is a generator point; see fig. 3).Fig. 3. Voronoi Diagram, Delaunay Triangulation and their duality.SOME USEFUL PROPERTIES OF THE VORONOI DIAGRAM. AN APPLICATION TO THE GEOMETRY OF SPATIAL STRUCTURES.Let then S’={P’1, P’2, . . . , P’n} be a set of points in the plane z=1 and let VD be its related Voronoi Diagram. Let us consider an inversive transformation on R3, with center on O(0,0,0) and power k=1; this transformation maps the plane z=1 into the sphere E [x2+y2+(z-1/2)2=1/4] and the set S’ into another one S ={P1, P2, . . . , P n}, were any P i belongs to the sphere E. We have demonstrated (Otero, 2000) that:PROP. 1. The mapped image of the Voronoi Diagram of S’ is a polyhedron that approximates the sphere E [x2+y2+(z-1/2)2=1/4] in such a way that each one of its faces is tangential to the sphere. There is a symmetric correspondence between each Voronoi polygon and each face of the polyhedron. Note that the vertices of these polyhedron’s faces do not belong to the sphere.PROP 2. The projection from the point of coordinates (0,0, 1/(2Z c-1)) of the Voronoi Polygons of S’ onto the second order surface SF [X2 + Y2 - Z2(1 – 2Z c) – 2ZcZ + 1 = 0] makes up an approximating polyhedron the faces of which are tangent to SF. Each point of the set S’ is transformed to the point of contact between the face of the polyhedron and the surface. The edges of the polyhedron are those transformed from the edges of the Voronoi Diagram of S’: (i) When zc =1/2, SF is a rotated paraboloid, (ii) when zc <1/2, SF is a rotated (two folds) hyperboloid, (iii) when zc >1/2, SF is a rotated ellipsoid.PROP2-BIS. It is enough to move the sphere E anywhere, but keeping it tangent to the plane z=1, to obtain, by the way stated in Prop. 2, polyhedra approximating non-revolution quadrics (Otero, 2002).A 2D PROCEDURE FOR CREATING MESHES MADE UP BY NONTRIANGULAR FACESWe have arrived at a very different family of approaches to the sphere, where the property of tangency is not produced on the edges of the polyhedron but in their faces. This supposes a more intuitive way for choosing the final shape of the body because its topology can be proposed by means of a 2D Voronoi Diagram at the plane z=1. We can handle different hypotheses easily, as we illustrate in the figure 4. On the other hand, all the properties are valid when the Voronoi Diagram is replaced by its dual transformation, the Delaunay Triangulation DT. So, not only panel structures, but lattice structures too, can be generated from a simple set of points on z=1, being possible to shape different types of quadric domes.Fig. 4. (Above): (left) a set of point on the plane z=1, (center) makes automatically arise a Voronoi Diagram, VD, on this plane; (right) an inversive transformation with center on (0,0,0) maps this VD into a polyhedron circumscribed to the sphere with center on (0,0, ½) and radius ½. (Below): (left) each face on VD is mapped into a face on the polyhedron, keeping that each edge of VD is an edge on the aforementioned polyhedron. (Right) A proyective transformation maps each face of the polyhedron into new one, approachingdifferent types of quadric surface.POWER DIAGRAMSGiven a collection of circumferences lying on a plane (Togores, 2003), it is well known that a Planar Division derived from this set exists: it is obtained by the intersection of the power lines for each pair of properly chosen neighboring circles (fig. 5, left). To each circumference, a convex region of the plane is associated (Aurenhammer, 1991), which is defined by the intersection of half planes containing those points with the least circle power. This region is known as the power cell, and the set of cells for the said collection of circumferences is known as its associated power diagram.The Power Diagram of a set of circles can be handled in a similar manner than we have done with the Voronoi Diagram in the previous points (Díaz, 2004). The result can be resumed in the next properties.PROP 3: The contact curve of the cone circumscribed to a quadric from an exterior point P is the conic section generated by the polar plane of point P. If among all the possible quadrics we select the paraboloid Ω: [z = x2 + y2], it is also true that the orthogonal projection of this section on any horizontal plane z=K is a circle.Fig. 5. (Left) Power Diagram of a set of 7 circumferences. (Right) Spatialinterpretation of a chordalePROP 4: Given two points P and Q outside of the paraboloid Ω, it results then that their respective polar planes: (i) generate on Ω two ellipses that are projected on z=1 as two circles; (ii) the intersection of these polar planes is projected as the power line (radical axis) of the aforementioned circles. See figure 5, right.An immediate consequence is that:PROP 5: E very power diagram is the equivalent of the orthogonal projection of the boundaries of a convex polyhedral surface (resulting from the intersection of the half-spaces defined by polar planes). This surface can be regarded as a polyhedron that approximates the quadric.DESIGN OF CHORDAL SPACE STRUCTURESThese last properties provide us with a mechanism to associate a cloud of points in space with the faces of a polyhedron approximating the paraboloid Ω; the obtained polyhedral body can be partially inscribed, partially circumscribed, partially tangent to its edges and partially secant to it (fig. 6). Again, it is demonstrated (Díaz, 2004) that not only for the paraboloid Ω, but for any other not ruled quadric this procedure is valid and useful.Fig. 6. A one-to-one correspondence (C2→ E3) as the mechanism of definition of the polyhedron’s faces that approximate the quadric. The relative positions of points with respect to Paraboloid Ω conditions the typology of the resulting structure, which can bepredicted from the associated power diagram.The field of knowledge related with the processes by which spatial structures are obtained is plagued with innumerable typologies, procedures, classes, subclasses and patents that Computational Geometry can synthesize in one single category: that we have named as Chordal Space Structures. This proposal simplifies and widens the scope of this technical activity. Nothing like this has been claimed before, because the intimate relation between Computational Geometry and the design of big lightweight structures remained unnoticed. Both fields are representative of progress in the XX th century and can go forward hand in hand in the XXI st.GLOBULAR STRUCTURES AND FLATTENED STRUCTURESLet us consider now the “parallel trihedra” procedure defined in Alvaro (2000). It is not difficult to notice that: (i) the operation of adding parallel trihedra around a center is feasible starting from both regular and semi-regular polyhedrons; and, (ii) the spherical patches will prove to be more convex or tauter (Fig. 7) than the sphere's surface circumscribing the initial polyhedron according to the position of point O i in relation to the only degree of freedom that has this composition of patches of sphere (see the reference for more detail).Fig. 7. Surfaces resulting from the aggregation of parallel trihedrons according to an icosahedral symmetry: globular (left) and flattened (right).Fig.8. Examples of flattened (left) and globular meshes (center and right).Resorting to the previous paragraph, it is possible to expand the catalog of Chordal Structures with forms that we shall call Flattened or Globular (see Fig. 8): the resulting quadric (ellipsoid, hyperboloid or paraboloid), as well as the intermediate polyhedron approximating the sphere is included.Nevertheless, as engineers, our interest will not lie so much in approximating whole quadrics, but in portions of them that will permit us to define, for example, space enclosures. As a conclusion, we now include some illustrative examples (see Fig. 9). But at the end, it is important to not forget the main feature of the process: the designer just selected a single set of points in a plane!!Fig. 9. Portions of quadrics used as space enclosures.CONCLUSIONIn the last years, a special effort is being made by the authors in order to spread out the potentialities of Computational Geometry in relation to the design of Spatial Structures. Under this premise, a central idea animates this article: the adequate interpretation in the physical three-dimensional space of the mathematical formulation that relates power diagrams with certain polyhedral structures is enough for promoting a novel way to approach the always complicated process of definition of space meshes. This is not the only way of relating the circumferences lying in a plane with the points and planes in space, but being so simple and intuitive, and due to the fact that it is developed from well established procedures, it results in a contribution of an especial interest for everyone that has in mind a research in the realm of the space enclosures.ACKNOWLEDGMENTSThis line of work is being object of our study since 1998. It was deeply encouraged by the IASS TSUBOY AWARD to the best article published in the IASS Journal, that the International Association for Shell and Spatial Structures, IASS, granted to this Group in 2001.REFERENCESAlvaro J.I. et al (2000). Designing optimal spatial meshes: cutting by parallel trihedra procedure. IASS Journal, vol 41, nº 2, 2001. Pp 101 a 110.Aurenhammer, F. (1991). Voronoi Diagrams – A survey of a fundamental geometric data structure. ACM Computing Surveys, Vol. 23, Nº. 3.Diaz J A, Togores R, Otero C (2004). Geometry applied to designing spatial structures: Joining two worlds. Lecture Notes in Computer Science. Vol. 3045 Pp: 158-167. Springer-Verlag Heidelberg.Fuller, R. B. (1954). Building Construction. U.S. Patent 2,682,235, p. 9.Kitrick, C. J. (1990). A Unified Approach to Class I, II & III Geodesic Domes. International Journal of Space Structures, vol. 5, Nº. 3&4, pp. 223-246.Otero, C., Gil, V., Álvaro J. I. (2000) CR-Tangent Meshes. 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