Balanced Aspect Ratio Trees and Their Use for Drawing Very Large Graphs
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floor PET value, and the surface of the water can effectivelyreduce the air temperature, relative humidity, and mean radiation temperature to decrease the PET markedly and to improve the thermal comfort of overhead floor. Increasing tree size (HW), the leaf area density (LAD) and aspect ratio of trees (ART) could increase relative humidity and decrease mean radiation temperature, and improve the thermal comfort of the overhead floor. Therefore, selecting reasonable underlying surface material and increasing tree size, leaf area density or aspect ratio of trees can significantly decrease the PET value and improve the thermal comfort of the overhead flo or.Key words overhead floor; thermal comfort; ENVI-met; landscape elements; physiological equivalent temperature21世纪以来,我国城市化进程加快,人口密度与建筑密度的持续增加导致城市热环境不断恶化,加上近年来气候变化极端天气的频发,以及“双碳”目标的提出,建筑热环境受到了越来越广泛的关注。
树木英语作文提纲模板Title: The Importance of Trees。
Introduction。
Introduce the topic of trees and their significance in our environment。
Mention the role of trees in providing oxygen, reducing air pollution, and supporting biodiversity。
Thesis statement: Trees play a crucial role in maintaining the balance of our ecosystem and are essential for the well-being of all living organisms.The Benefits of Trees。
Discuss the environmental benefits of trees, such as absorbing carbon dioxide and releasing oxygen through the process of photosynthesis。
Explain how trees help in reducing air pollution by trapping harmful gases and particles。
Highlight the role of trees in preventing soil erosion and maintaining water quality。
Discuss the importance of trees in providing habitat and food for various wildlife species。
The Social and Economic Importance of Trees。
Trees are an integral part of our ecosystem,providing a multitude of benefits to the environment and human life.They play a crucial role in maintaining the balance of nature and offer a serene beauty that enriches our landscapes.The Importance of Trees1.Oxygen Production:Trees are often referred to as the lungs of the Earth because they produce oxygen through the process of photosynthesis.This process absorbs carbon dioxide and releases oxygen,which is essential for the survival of all living beings.2.Carbon Sequestration:Trees absorb carbon dioxide from the atmosphere,which helps in mitigating the effects of climate change.They store carbon in their tissues,acting as a natural carbon sink.3.Biodiversity:Trees provide habitats for a wide range of species,from insects and birds to mammals.They support complex ecosystems by offering food,shelter,and breeding grounds for various forms of life.4.Soil Conservation:Tree roots help in binding the soil,preventing erosion and maintaining soil fertility.They also recycle nutrients back into the soil,promoting healthy plant growth.5.Water Cycle Regulation:Trees play a significant role in the water cycle by absorbing and releasing water through transpiration.They help in regulating the local climate by reducing the evaporation rate and increasing humidity.6.Economic Value:Trees are a source of timber,paper,and other forest products.They also contribute to the tourism industry by providing scenic beauty that attracts visitors.Caring for Trees1.Planting:Encouraging the planting of trees is a proactive step towards environmental conservation.It not only increases the green cover but also helps in creating a healthier environment.2.Conservation Efforts:It is essential to protect existing forests and trees from deforestation and illegal logging.Conservation efforts should be supported to preserve these natural resources.3.Urban Greening:Incorporating trees in urban planning can help in reducing the urbanheat island effect,improving air quality,and providing recreational spaces for city dwellers.cation and Awareness:Educating the public about the importance of trees and how to care for them is crucial.This includes understanding the correct methods of planting, watering,and pruning.5.Sustainable Practices:Promoting sustainable forestry practices ensures that trees are harvested in a way that does not deplete the forest or harm the environment.ConclusionTrees are not just silent witnesses to the worlds changes they are active participants in the health of our planet.By understanding and appreciating their value,we can take steps to protect and nurture them,ensuring a greener and more sustainable future for all.。
为什么树重要英语作文Title: The Significance of Trees。
Trees play a pivotal role in our ecosystem, and their importance cannot be overstated. From environmentalbenefits to economic advantages and their impact on human well-being, trees are indispensable. In this essay, we will delve into why trees are crucial.First and foremost, trees are vital for maintaining a healthy environment. Through the process of photosynthesis, trees absorb carbon dioxide and release oxygen, thereby purifying the air we breathe. This process helps mitigate the effects of climate change by reducing the concentration of greenhouse gases in the atmosphere. Additionally, trees act as natural air filters, trapping pollutants and particulate matter, which helps improve air quality and reduces the risk of respiratory diseases.Furthermore, trees play a crucial role in preservingbiodiversity. Forests provide habitat for countless species of plants, animals, and microorganisms. The diverse arrayof flora and fauna found in forests contributes to the stability of ecosystems and ensures the survival of many species. Deforestation, on the other hand, leads to habitat loss and threatens the survival of numerous species, someof which may become extinct if their habitats are destroyed.Economically, trees are valuable resources that provide various goods and services. Timber harvested from forestsis used in construction, furniture manufacturing, and paper production, among other industries. Additionally, many communities rely on forests for livelihoods throughactivities such as logging, eco-tourism, and the harvesting of non-timber forest products like fruits, nuts, and medicinal plants. Moreover, trees contribute to soil stabilization, erosion control, and watershed protection, which are essential for agriculture and water resource management.In terms of human well-being, trees have numerous benefits. Urban green spaces, such as parks and gardens,contribute to mental and physical health by providing opportunities for recreation, relaxation, and exercise. Studies have shown that exposure to nature reduces stress, anxiety, and depression while promoting overall well-being. Furthermore, trees in urban areas help mitigate the urban heat island effect by providing shade and cooling the surrounding environment, which is particularly important in densely populated cities.Additionally, trees have cultural and spiritual significance for many societies around the world. They feature prominently in folklore, mythology, and religious beliefs, symbolizing strength, wisdom, and the cycle of life. Trees are often revered as sacred entities and are central to various cultural practices and ceremonies. Preserving forests and protecting trees is not only essential for ecological reasons but also for maintaining cultural heritage and identity.In conclusion, trees are invaluable assets to our planet, providing a wide range of environmental, economic, and social benefits. From mitigating climate change andpreserving biodiversity to supporting human well-being and cultural traditions, the importance of trees cannot be overstated. It is imperative that we recognize the significance of trees and take concerted efforts to conserve and protect them for future generations.。
应对区爱护学校专题一阅读理解第一讲细节理解类真题验证题组一A(2017·全国Ⅱ,A) 47404000 In the coming months, we are bringing together artists from all over the globe, to enjoy speaking Shakespeare's plays in their own language, in our Globe, within the architecture Shakespeare wrote for. Please come and join us.National Theatre of China Beijing|ChineseThis great occasion(盛会)will be the National Theatre of China's first visit to the UK. The company's productions show the new face of 21st century Chinese theatre. This production of Shakespeare's RichardⅢ will be directed by the National's Associate Director, Wang Xiaoying.Date & Time:Saturday 28 April, 2.30pm & Sunday 29 April, 1.30pm & 6.30pmMarjanishvili Theatre Tbilisi|GeorgianOne of the most famous theatres in Georgia, the Marjanishvili, founded in 1928, appears regularly at theatre festivals all over the world. This new production of As You Like It is helmed (指导)by the company's Artistic Director Levan Tsuladze.Date & Time:Friday 18 May, 2.30pm & Saturday 19 May, 7.30pmDeafinitely Theatre London I British Sign Language(BSL)By translating the rich and humourous text of Love's Labour's Lost into the physical language of BSL, Deafinitely Theatre creates a new interpretation of Shakespeare's comedy and aims to build a bridge between deaf and hearing worlds by performing to both groups as one audience.Date & Time: Tuesday 22 May, 2.30pm & Wednesday 23 May, 7.30pmHabima National Theatre Tel Aviv|HebrewThe Habima is the centre of Hebrew-language theatre worldwide. Founded in Moscow after the 1905 revolution, the company eventually settled in Tel Aviv in the late 1920s. Since 1958, they have been recognised as the national theatre of Israel. This production of Shakespeare's TheMerchant of Venice marks their first visit to the UK.Date & Time:Monday 28 May, 7.30pm & Tuesday 29 May, 7.30pm文章大意:本文是一篇应用文,主要介绍了全球的艺术家将聚集到一起用他们自己的语言表演莎士比亚的戏剧的相关内容。
如何维持人与自然之间的动态平衡英语作文How to Keep Humans and Nature in BalanceHi there! My name is Sam, and I'm going to tell you all about how we can keep a good balance between humans and nature. It's really important that we take care of the environment and the plants and animals that live in it. If we mess things up too much, it can lead to big problems like pollution, climate change, and even animals going extinct! But don't worry, there are lots of easy things we can do to help.First off, let's talk about why nature is so awesome. Have you ever gone for a hike in the woods or played outside in a park? Isn't it so much fun? Nature gives us beautiful places to explore, fresh air to breathe, and all sorts of cool plants and critters to discover. Without healthy forests, oceans, and all the other parts of nature, we wouldn't have most of the food we eat or even the wood we use to build our homes and furniture. Nature does so much for us!But sadly, a lot of human activities like burning fossil fuels, cutting down too many trees, and littering end up damaging the environment. The smoke from factories and cars creates smog that makes it hard to breathe. Cutting down too many trees getsrid of animal habitats and can lead to soil erosion. And litter like plastic bags and bottles ends up clogging waterways and hurting marine life. Yuck!So how can we fix these problems and have a good relationship with nature? Well, there are actually a ton of simple things we can all do in our daily lives:Reduce, reuse, recycle! This means trying to use less stuff, finding ways to reuse things instead of throwing them out, and recycling whatever we can. Things like bottles, cans, paper and cardboard can all go into the recycling bin instead of the trash.Turn off lights and electronics when you're not using them. This saves energy so we don't have to burn as much coal or gas to make electricity.Walk, bike, or take public transportation when you can instead of having someone drive you everywhere. Cars create a lot of pollution.Don't litter! Always put trash in the garbage can instead of throwing it on the ground. And pick up any litter you see too.Plant trees, flowers, and gardens. More plants means cleaner air, less soil erosion, and habitats for animals. Gardens can provide fresh fruits and veggies too.Conserve water by taking shorter showers, turning off the faucet when you're brushing your teeth, and not wasting water when you're washing dishes or playing outside.Buy local and organic foods when possible. Foods that are grown locally don't have to travel as far to get to you, which saves energy. And organic foods don't use as many chemical pesticides that can pollute the environment.Support organizations that protect nature like the World Wildlife Fund, Greenpeace, and local land trusts andstate/national parks. They work hard to conserve habitats and endangered species.Those are just a few ideas, but there are so many other ways both kids and adults can live a more eco-friendly lifestyle. The most important thing is just being aware of how our everyday actions impact the planet and making an effort to tread a little lighter on the Earth.We need nature, but nature also needs us to take care of it! Plants and animals can't speak up for themselves, so we have to be their voice. If every single person does a few small, simple things, it can add up to huge positive changes for the environment. We're all in this together with Mother Nature.So next time you're outside, look around and appreciate how amazing the natural world is. Breathe in that fresh air, listen to the birds singing, and admire the beauty of the flowers, trees, and clouds. Don't we want to make sure it stays healthy and beautiful forever? I know I do! By finding a nice balance between humans and nature, we can go on enjoying all the wonderful things the outdoors has to offer for many, many years to come. The Earth is pretty awesome, so let's take good care of it!。
幼儿园观察记录身体协调情况为了了解幼儿身体协调发展情况,并为他们提供相应支持和指导,我进行了一次观察记录。
以下是我在观察中所得到的观察结果。
1. 幼儿园背景介绍:这个幼儿园是一所私立幼儿园,位于市中心地带,拥有宽敞明亮的教室和安全舒适的户外场地。
幼儿园采用了玩乐学的教育理念,注重于培养幼儿的身体协调能力。
2. 观察对象:我观察了一群3-4岁的幼儿,他们一共有12人,男女比例基本平衡。
3. 观察时间和地点:观察进行了一周的时间,每天持续观察两小时,观察地点为幼儿园的室内和室外活动场所。
4. 观察方法和记录方式:我通过实地观察和记录表的方式来记录幼儿的身体协调情况。
观察包括站立、行走、跑跳、抓握等行为,记录表包括了幼儿姓名、观察日期、观察内容等项目。
观察结果如下:1) 站立:大部分幼儿能够稳定地站立,但少数幼儿站立时出现脚步不稳、姿势不端正等现象。
这些幼儿需要额外的指导和练习,提高他们的平衡性和站立姿势。
2) 行走:幼儿们能够熟练地行走,表现出较好的身体协调和平衡感。
他们能够在不摔倒的情况下自如地转弯、提高步伐速度等。
然而,有少数幼儿在行走中脚步不稳、踉跄或者行走速度过快。
针对这些情况,我们需要提供更多的实践机会和引导,帮助他们提高身体协调能力。
3) 跑跳:大多数幼儿能够自如地跑跳,保持良好的平衡感。
他们能够连续跑跳或者跳跃过障碍物。
但是,个别幼儿在跑跳时姿势不稳,缺乏平衡感,需要进行相关的训练和指导。
4) 抓握:幼儿们的手部协调能力较好,能够进行简单的握持动作,如握笔、用餐、穿扣子等。
但仍有少数幼儿握力不足,手部协调能力相对较弱,需要有针对性的练习集中发展。
通过以上观察,我发现了幼儿园幼儿在身体协调方面的整体情况。
对于身体协调能力较弱的幼儿,我们需要针对性地开展相关的干预措施。
例如,我们可以在日常活动中加入一些促进平衡感和身体协调的游戏、活动,提供更多的机会让幼儿锻炼身体协调能力。
同时,可以与家长沟通,提供家庭指导,鼓励家长和幼儿一起进行户外活动,增强幼儿的身体素质。
树木写作英文作文高级句型Trees are an integral part of our natural world, providing us with oxygen, shade, and beauty. From towering redwoods to delicate cherry blossoms, trees come in all shapes and sizes, each with its own unique personality.One of the most striking features of trees is their ability to adapt to their environment. Whether they grow in a dry desert or a wet rainforest, trees are able to adjust their growth patterns to survive and thrive. This adaptability is what makes trees such an important part of our ecosystem, as they help to maintain the balance of nature.In addition to their ecological importance, trees also have a cultural significance. Many cultures around the world have stories and myths about trees, and they are often used as symbols of strength, wisdom, and longevity. Trees are also an important part of many religious traditions, such as the Christmas tree in Christianity andthe Bodhi tree in Buddhism.Despite their many benefits, trees are facing a number of threats in today's world. Deforestation, climate change, and pollution are all taking a toll on our forests, and many species of trees are at risk of extinction. It is up to us to take action to protect these vital resources, by planting new trees, supporting conservation efforts, and reducing our impact on the environment.In conclusion, trees are a vital part of our natural world, providing us with countless benefits and serving as an important symbol of our connection to nature. By working to protect and preserve our forests, we can ensure that future generations will be able to enjoy the beauty and benefits of these magnificent plants.。
树木大小分化英语作文Title: Disparity in Tree Sizes: A Cause for Concern。
In recent years, there has been a noticeable trend in the size differentiation among trees in various ecosystems. This phenomenon, characterized by a significant gap between small and large trees within a given area, poses several ecological implications. In this essay, we will explore the factors contributing to this divergence and its potential consequences.One prominent factor contributing to the divergence in tree sizes is human-induced disturbances to natural ecosystems. Deforestation, logging, urbanization, and land-use changes have all disrupted the natural balance of forested areas, leading to the removal of large trees and the proliferation of smaller, opportunistic species. This alteration in the composition of tree size classes can have cascading effects on ecosystem structure and function.Furthermore, climate change is exacerbating the size differentiation among trees. Shifts in temperature and precipitation patterns influence the distribution and abundance of tree species, favoring those adapted to new environmental conditions. As a result, certain tree species may thrive and dominate, while others struggle to survive, leading to an imbalance in tree sizes within ecosystems.In addition to external factors, internal mechanisms within ecosystems also play a role in driving tree size differentiation. Competition for resources such as light, water, and nutrients can favor the growth of larger, more competitive trees at the expense of smaller ones. Over time, this competitive exclusion can lead to a pronounceddisparity in tree sizes within a given area.The consequences of size differentiation among treesare multifaceted and extend beyond the realm of ecology. Ecologically, the loss of large trees can disrupt important ecosystem processes such as carbon sequestration, nutrient cycling, and habitat provision. Large trees often provide critical habitat for a myriad of species, and their declinecan have cascading effects on biodiversity.Moreover, the aesthetic and cultural value of large trees cannot be overstated. They serve as landmarks, symbols of longevity, and sources of inspiration for artists, poets, and nature enthusiasts. The disappearance of large trees from landscapes can diminish the cultural heritage and intrinsic value associated with natural environments.Addressing the issue of size differentiation among trees requires a multifaceted approach that addresses both the underlying causes and the symptoms of this phenomenon. Conservation efforts aimed at protecting remaining old-growth forests and restoring degraded ecosystems are essential for preserving the integrity of natural habitats and promoting biodiversity.Furthermore, sustainable land management practices, such as selective logging and reforestation, can help mitigate the impacts of human-induced disturbances on tree size differentiation. By maintaining a diverse range oftree sizes within ecosystems, we can enhance resilience to environmental change and safeguard the numerous ecological, cultural, and aesthetic benefits associated with large trees.In conclusion, the divergence in tree sizes observed in many ecosystems is a cause for concern due to its ecological, cultural, and aesthetic implications. Addressing this issue requires concerted efforts to address the underlying drivers of size differentiation and implement strategies to promote the conservation and restoration of diverse forest ecosystems. By doing so, we can ensure the continued health and vitality of our natural environments for future generations.。
以树木形态对比的作文英文回答:As we compare the forms of trees, we can observe the diversity and uniqueness that each tree possesses. Trees come in various shapes, sizes, and structures, reflecting their adaptation to different environments and their growth patterns. Some trees have tall, straight trunks that reach for the sky, while others have branches that spread out in all directions. The differences in tree forms can be attributed to factors such as species, climate, and available resources.For example, the oak tree is known for its strong and sturdy form. Its trunk is thick and its branches are wide-spreading, creating a broad canopy that provides shade and shelter. On the other hand, the willow tree has a slender and graceful form. Its branches hang down, giving it a weeping appearance. These contrasting forms reflect the different growth habits and characteristics of these twotree species.Another example is the palm tree, which has a unique and recognizable form. Its trunk is tall and slender, topped with a cluster of large leaves that fan out in a circular shape. This form allows the palm tree to withstand strong winds and conserve water in arid environments. In contrast, the pine tree has a conical shape, with astraight trunk and branches that extend horizontally. This form helps the pine tree shed snow and withstand harsh winter conditions.中文回答:当我们比较树木的形态时,我们可以观察到每棵树所具有的多样性和独特性。
节约树木的方法作文英语Title: Methods to Conserve Trees。
Trees are vital components of our ecosystem, providing oxygen, purifying the air, and supporting diverse habitats for various organisms. However, deforestation and indiscriminate logging have led to a decline in global tree cover, posing severe environmental challenges. To address this pressing issue, several methods can be implemented to conserve trees effectively.Firstly, promoting sustainable forestry practices is paramount in tree conservation efforts. Sustainableforestry involves managing forest resources in a mannerthat ensures their long-term viability while meeting the needs of current and future generations. This can be achieved through practices such as selective logging, where only mature trees are harvested, and reforestation initiatives, which involve planting new trees to replace those that have been cut down. By adopting sustainableforestry practices, we can minimize the negative impact of logging on tree populations and maintain healthy forest ecosystems.Secondly, raising awareness about the importance of trees and the consequences of deforestation is essential in garnering public support for tree conservation. Education campaigns, workshops, and outreach programs can be organized to inform individuals about the ecological, economic, and social benefits of trees. Additionally, highlighting the detrimental effects of deforestation, such as loss of biodiversity, soil erosion, and climate change, can motivate people to take action to protect trees and forests. By fostering a sense of environmental stewardship and responsibility, we can inspire collective efforts towards tree conservation.Furthermore, implementing stringent laws and regulations to regulate deforestation and protect tree species is crucial. Governments can enact legislation to restrict the conversion of forests into agricultural land, enforce bans on illegal logging activities, and establishprotected areas to safeguard critical habitats. Additionally, incentivizing companies to adopt sustainable practices through tax breaks, subsidies, and certification schemes can encourage responsible forestry management. By enforcing strict regulations and promoting accountability within the forestry sector, we can prevent the wanton destruction of trees and promote their conservation.Moreover, embracing alternative materials and technologies that reduce the demand for wood products can help alleviate pressure on tree populations. Innovations such as engineered wood products, bamboo, and recycled materials offer viable alternatives to traditional timber and can help mitigate the environmental impact of deforestation. Additionally, promoting sustainable land-use practices, such as agroforestry and urban forestry, can integrate tree planting into agricultural and urban landscapes, enhancing biodiversity and ecosystem services while providing economic benefits to communities.In addition to these measures, fostering international cooperation and collaboration is essential in addressingglobal deforestation challenges. Countries can worktogether to develop and implement strategies forsustainable forest management, share best practices, and provide support to regions facing deforestation pressures. International agreements and initiatives, such as theUnited Nations Forum on Forests and the REDD+ (Reducing Emissions from Deforestation and Forest Degradation) program, play a crucial role in coordinating global efforts to conserve trees and forests.In conclusion, conserving trees requires a multifaceted approach that encompasses sustainable forestry practices, public awareness and education, legal protections, technological innovation, and international cooperation. By implementing these methods collectively, we can safeguard trees and forests for future generations and ensure a sustainable and healthy environment for all living beings.It is imperative that individuals, communities, governments, and organizations come together to prioritize tree conservation and take decisive action to protect ourplanet's precious natural resources.。
大树测量比的知识Trees are one of nature's most majestic creations, standing tall and proud, providing us with much more than just shade and beauty. 大树是大自然最雄伟的创造之一,高大挺拔,不仅给我们提供阴凉和美丽,还给我们带来了更多的好处。
One of the most important aspects of trees is their ability to absorb carbon dioxide and release oxygen through photosynthesis. 树木最重要的一个方面是它们通过光合作用吸收二氧化碳并释放氧气的能力。
In addition to their environmental benefits, trees also provide countless cultural and spiritual significance to many societies around the world. 除了环境上的益处之外,树木在世界各地的许多社会中也提供了无数的文化和精神意义。
Furthermore, trees play a crucial role in maintaining biodiversity by providing habitats for a wide variety of animal species. 此外,树木通过为各种动物物种提供栖息地,对维持生物多样性起着至关重要的作用。
Measuring the height of trees has always been a fascinating challenge for researchers and nature enthusiasts alike. 测量树木的高度一直是研究人员和自然爱好者们非常有趣的挑战。
农艺师告诉果农作文英文回答:As an agronomist, I would like to provide some advice and suggestions to fruit farmers. It is important to note that different fruits have different requirements and needs. Therefore, I will focus on a few common fruits and discuss their cultivation techniques.Firstly, let's talk about apple trees. Apple trees require well-drained soil and full sunlight for optimal growth. It is crucial to prune the trees regularly to promote airflow and prevent diseases. Additionally, apple trees need proper fertilization to ensure healthy growthand high-quality fruit. For example, I recommend using a balanced fertilizer with a ratio of 10-10-10. Furthermore,it is essential to monitor pests and diseases, such asapple scab and codling moth. Implementing integrated pest management strategies can help reduce the use of chemical pesticides and protect the environment.Moving on to citrus fruits, such as oranges and lemons. Citrus trees prefer slightly acidic soil with good drainage. They also require regular watering, especially during dry periods. Pruning is necessary to maintain the shape of the tree and remove dead or diseased branches. Fertilizingcitrus trees with a specialized citrus fertilizer can enhance fruit production and quality. Additionally, protecting citrus trees from frost during winter is crucial, as frost can damage the fruit and leaves. Using frost blankets or installing a heating system can help prevent frost damage.Now, let's discuss grapes. Grapes thrive in well-drained soil and full sunlight. Pruning is essential to control the growth and ensure proper air circulation. Grapevines need support systems, such as trellises or fences, to grow properly. Proper irrigation is crucial, especially during the growing season, to prevent waterstress and maintain fruit quality. It is also important to protect grapes from common pests, such as birds and grapevine moths. Netting or scare devices can be used todeter birds, while pheromone traps can help monitor and control grapevine moths.中文回答:作为一名农艺师,我想给果农们提供一些建议和建议。
写作文常绿树常绿树的对比手法开头英文回答:When it comes to the comparison between evergreen trees and deciduous trees, there are several aspects that can be considered. Firstly, evergreen trees, as the name suggests, retain their leaves throughout the year, while deciduous trees shed their leaves during certain seasons. This fundamental difference in leaf retention is adistinguishing feature between the two types of trees.Another difference lies in the appearance of the trees. Evergreen trees, with their constant foliage, provide a year-round greenery that can be quite pleasing to the eye. They bring a sense of permanence and stability to the landscape. On the other hand, deciduous trees go through a cycle of changing colors, from vibrant greens in the spring and summer to various shades of red, orange, and yellow in the fall. This seasonal transformation adds a dynamic and visually appealing element to the surroundings.Furthermore, evergreen trees are known for their resilience and ability to withstand harsh weather conditions. Their thick, waxy leaves and sturdy branches provide protection against strong winds and heavy snowfall. In contrast, deciduous trees, with their leafless branches during the winter months, are more vulnerable to the elements. However, this vulnerability also allows for a unique phenomenon the growth of delicate, blossoming flowers in the spring, which can be seen in trees such as cherry blossoms and magnolias.In terms of ecological benefits, both types of trees play important roles. Evergreen trees provide year-round shelter and food for wildlife, as their leaves offer a constant source of nourishment. Deciduous trees, on the other hand, provide habitats for a variety of species during the warmer months and contribute to the nutrient cycle by shedding their leaves and enriching the soil. This diversity in ecological functions highlights the importance of having a mix of both evergreen and deciduous trees in an ecosystem.中文回答:谈到常绿树和落叶树的对比,有几个方面可以考虑。
初二关于比例英语作文The Magic of Proportions in Daily Life.Proportions, a fundamental concept in mathematics, play a pivotal role in our daily lives, often going unnoticed. From the smallest details in our surroundings to the grandest architectures, proportions are the silent architects that give shape and meaning to the world we live in.In the realm of art, proportions are what give paintings and sculptures their harmonious feel. The ratio of lengths, widths, and heights determine whether a work of art appears balanced and pleasing to the eye. The same goes for our clothes. The proportion of a shirt's sleeves to its body, or the length of pants relative to the wearer's height, all contribute to the overall aesthetic appeal.Architecture is another field where the importance of proportions is evident. From the towering skyscrapers tothe humble homes, the balance of windows, doors, and other elements is crucial. A building with disproportionate features would look awkward and out of place. The ancient Greeks and Romans understood this well, as their temples and palaces still stand as testament to their mastery of proportions.Even in nature, we can see the beauty of proportions. The spiral shells of snails, the symmetric wings of butterflies, and the branches of trees all exhibit a remarkable sense of proportion. These natural phenomena are not just random occurrences; they are the result ofmillions of years of evolution, perfected through the lens of proportion.In the realm of cooking, proportions are equally important. The balance of ingredients in a recipe determines whether a dish will taste delicious or not. Too much salt can ruin a soup, while too little sugar can leave a dessert tasting flat. It is the careful consideration of these proportions that turns a simple meal into a culinary masterpiece.In the workplace, too, proportions play a vital role. The allocation of resources, the balance of workload, and the proportion of decision-making power all contribute to the efficiency and harmony of an organization. When these proportions are well-managed, a company can thrive and prosper.Moreover, proportions have a profound impact on our psychology. The proportion of positive to negative experiences in our lives shapes our overall happiness and well-being. A life filled with more positive moments, in proportion to the negative ones, is more likely to lead to a content and satisfied individual.In conclusion, proportions are a ubiquitous force in our lives, shaping every aspect of our world from the smallest detail to the grandest vision. They are the invisible hand that guides us towards harmony, balance, and beauty. By understanding and appreciating the magic of proportions, we can gain a deeper understanding of theworld we live in and create a more harmonious existence for ourselves and those around us.。
自然平衡英语演讲稿范文
英文回答,Natural balance is essential for the well-being of our planet. When we talk about natural balance, we are referring to the delicate equilibrium that exists in nature, where all living organisms and their environment coexist in harmony. For example, in a healthy ecosystem, predators keep the population of prey in check, preventing overpopulation and ensuring the balance of the food chain. This balance is crucial for the survival of all species.
中文回答,自然平衡对于地球的健康至关重要。
当我们谈论自然平衡时,我们指的是自然界中存在的微妙平衡,所有生物和它们的环境和谐共存。
例如,在一个健康的生态系统中,捕食者控制着猎物的数量,防止过度繁殖,保证了食物链的平衡。
这种平衡对于所有物种的生存都至关重要。
英语作文如何平衡人与自然Balancing human needs with the preservation of nature is a crucial aspect of sustainable development. Here are some strategies to achieve this equilibrium:Firstly, raising awareness is key. Educatingindividuals about the importance of environmental conservation and the interconnectedness of all living beings fosters a sense of responsibility towards nature. Through educational campaigns, people can learn about the impact of their actions on the environment and how they can contribute to its preservation.Secondly, implementing strict environmental regulations is essential. Governments and regulatory bodies play a crucial role in ensuring that industries and businesses operate in an environmentally sustainable manner. By enforcing laws that limit pollution, protect natural habitats, and promote the use of renewable resources, we can mitigate the negative impact of human activities on theenvironment.Furthermore, promoting sustainable practices in various sectors is imperative. In agriculture, for instance, embracing organic farming methods and reducing reliance on chemical pesticides and fertilizers can help minimize harm to ecosystems. Similarly, adopting energy-efficient technologies and investing in renewable energy sources like solar and wind power can reduce our carbon footprint and combat climate change.Moreover, fostering a closer connection with nature can cultivate a deeper appreciation for the environment. Encouraging outdoor activities such as hiking, camping, and wildlife observation not only promotes physical and mental well-being but also instills a sense of reverence for the natural world. By experiencing firsthand the beauty and complexity of nature, individuals are more likely to become advocates for its protection.In addition, fostering collaboration between communities, businesses, and conservation organizations isessential. By working together towards common goals, we can leverage collective expertise and resources to address environmental challenges more effectively. Initiatives such as community clean-up events, reforestation projects, and wildlife conservation efforts are just some examples of how collaboration can yield positive outcomes for both people and nature.Ultimately, achieving harmony between humans and nature requires a multifaceted approach that encompasses education, regulation, innovation, appreciation, and collaboration. By embracing sustainable practices and prioritizing the well-being of our planet, we can create a more balanced and resilient world for future generations.。
Balanced Aspect Ratio Trees and Their Usefor Drawing Large GraphsChristian A.DuncanMax-Planck-Institut f¨u r InformatikSaarbr¨u cken,Germanyhttp://www.mpi-sb.mpg.de/~duncanchristian.duncan@mpi-sb.mpg.deMichael T.Goodrich Stephen G.KobourovCenter for Geometric ComputingThe Johns Hopkins UniversityBaltimore,MD21218/labs/cgc/goodrich@ kobourov@AbstractWe describe a new approach for cluster-based drawing of large graphs,which obtains clusters by using binary space partition(BSP)trees.Wealso introduce a novel BSP-type decomposition,called the balanced aspectratio(BAR)tree,which guarantees that the cells produced are convex andhave bounded aspect ratios.In addition,the tree depth is O(log n),andits construction takes O(n log n)time,where n is the number of points.We show that the BAR tree can be used to recursively divide a graphembedded in the plane into subgraphs of roughly equal size,such thatthe drawing of each subgraph has a balanced aspect ratio.As a result,weobtain a representation of a graph as a collection of O(log n)layers,whereeach succeeding layer represents the graph in an increasing level of detail.The overall running time of the algorithm is O(n log n+m+D0(G)),wheren and m are the number of vertices and edges of the graph G,and D0(G)is the time it takes to obtain an initial embedding of G in the plane.Inparticular,if the graph is planar each layer is a graph drawn with straightlines and without crossings on the n×n grid and the running time reducesto O(n log n).Communicated by G.Liotta and S.H.Whitesides:submitted November1998;revised November1999.Duncan,Goodrich,and Kobourov,BAR Trees,JGAA,4(3)19–46(2000)201IntroductionIn the past decade hundreds of graph drawing algorithms have been developed (e.g.,see[7,8]),and research in methods for visually representing graphical information is now a thriving area with several different emphases.One general emphasis in graph drawing research is directed at algorithms that display an entire graph,with each vertex and edge explicitly depicted.Such drawings have the advantage of showing the global structure of the graph.A disadvantage, however,is that they can be cluttered for drawings of large graphs,where details are typically hard to discern.For example,such drawings are inappropriate for display on a computer screen any time the number of vertices is more than the number of pixels on the screen.For this reason,there is a growing emphasis in graph drawing research on algorithms that do not draw an entire graph, but instead partially draw a graph,either by showing high-level structures and allowing users to“zoom in”on areas of interest,or by showing substructures of the graph and allowing users to“scroll”from one area of the graph to another. Such approaches are well suited for displaying large graphs,such as significant portions of the world wide web graph,where every web page is a vertex and every hyper-link is an edge.A common technique used for scrolling viewpoints is thefish-eye view[16, 18,27],which shows an area of interest quite large and detailed(such as nodes representing a user’s web pages)and shows other areas successively smaller and in less detail(such as nodes representing a user’s department and organization web pages).Fish-eye views allow a user to understand the structure of a graph near a specific set of nodes,but they often do not display global structures.An alternate technique displays the global structure present in a graph by clustering smaller subgraphs and drawing these subgraphs as single nodes or filled-in regions.By grouping vertices together into clusters,we can recursively divide a given graph into layers of increasing detail.These layers can then be viewed in a top-down fashion or even infish-eye view by following a single path in a cluster-based recursion tree.If clusters of a graph are given as input along with the graph itself,then several authors give various algorithms for displaying these clusters in two or three dimensions[10,11,13,14,24,31].If,as will often be the case,clusters of a graph are not given a priori,then various heuristics can be applied forfinding clusters using properties such as connectivity,cluster size, geometric proximity,or statistical variation[1,17,23,25].Once a clustering has been determined,we can generate the layers in a hierarchical drawing of the graph,with the layer depth(i.e.,number of layers)being determined by the depth of the recursive clustering hierarchy.This approach allows the graph to be represented by a sequence of drawings of increasing detail.As illustrated by Eades and Feng[10],this hierarchical approach to drawing large graphs can be very effective.Thus,our interest in this paper is to further the study of methods for producing good graph clusterings that can be used for graph drawing purposes.We feel that a good clustering algorithm and its associated drawing method should come as close as possible to achieving the following goals:Duncan,Goodrich,and Kobourov,BAR Trees,JGAA,4(3)19–46(2000)211.Balanced clustering:in each level of the hierarchy the size of the clustersshould be about the same.2.Small cluster depth:there should be a small number of layers in the re-cursive decomposition.3.Convex cluster drawings:the drawing of each cluster shouldfit in a simpleconvex region,which we call the cluster region for that subgraph.4.Balanced aspect ratio:cluster regions should not be too“skinny”.5.Efficiency:computing the clustering and its associated drawing shouldnot take too long.In this paper we study how well we can achieve these goals for large graph drawings using clustering.Previous algorithms optimize one or more of the above criteria at the expense of some of the rest.Our goal is to simultaneously satisfy all of them.Our approach relies on creating the clusters using binary space partition(BSP)trees,defined by recursively cutting regions with straight lines.1.1BSP Tree Based Clustered Graph DrawingThe main idea behind the use of a BSP tree in I R2to define clusters is very simple.Given a graph G=(V,E),where n=|V|and m=|E|,we can use any existing method to embed it in the plane,provided that method places vertices at distinct points in the plane(e.g.,see[7,20,32]).For example,if G is planar we can use any existing method for embedding G in the plane such that vertices are at grid points,and edges of the graph are straight lines that do not cross[6,12,28,30,33].Once the graph drawing is defined,we build a binary space partition tree on the vertices of this drawing.Each node v in this tree corresponds to a convex region R of the plane,and associated with v is a line that separates R into two regions,each of which are associated with a child of v.Thus,any such BSP tree defined on the points corresponding to vertices of G naturally defines a hierarchical clustering of the nodes of G. Such a clustering could then be used,for example,with an algorithm like that of Eades and Feng[10],who present a technique for drawing a3-dimensional representation of a clustered graph.The main problem with using BSP trees to define clusters for a graph drawing algorithm is that previous methods for constructing BSP trees do not give rise to clustered drawings that achieve the design goals listed above.For example, the standard k-d tree and its variants(e.g.,see[15,26]),which use axis-parallel lines to recursively divide the number of points in a region in half,maintain every criteria but the balanced aspect ratio.Likewise,quad-trees and fair-split trees(e.g.,see[4,26]),which always split by a line parallel to a coordinate axis to recursively divide the area of a region more or less in half,maintain balanced aspect ratio but can have a depth that isΘ(n).In graph drawing,aesthetics are very important,and while“fat”regions appear rounder,a series of skinny regions can be distracting.But depth is alsoDuncan,Goodrich,and Kobourov,BAR Trees,JGAA,4(3)19–46(2000)22 important,for a deep hierarchy of clusterings would be computationally expen-sive to traverse and would not provide very balanced clusters.The balanced box-decomposition tree of Arya et al.[3,2]has O(log n)depth and has regions with good aspect ratio,but it sacrifices convexity by introducing holes into the middle of regions,which makes this data structure less attractive for use in clustering for graph drawing applications.Indeed,to our knowledge,there is no previous BSP-type hierarchical decomposition tree that achieves all of the above design goals.1.2The Balanced Aspect Ratio(BAR)TreeIn this paper we present a new type of binary space partition tree that is bet-ter suited for the application of defining clusters in a large graph.Our data structure,which we call the balanced aspect ratio(BAR)tree,is a BSP-type decomposition tree that has O(log n)depth and creates convex regions with bounded aspect ratio(also called“fat”regions).In this paper we present the BAR tree in I R2.The generalized BAR tree in I R d is presented in[9].The construction of the BAR tree is very similar to that of a k-d tree,except for two important differences:1.In addition to axis-aligned cuts,the BAR tree allows for one more cutdirection:a45◦-angled cut.2.Rather than insisting that the number of points in a region be cut in halfat every level,the BAR tree guarantees that the number of points is cut roughly in half every two levels,which is something that does not seem possible to do with either a k-d tree or a quadtree(or even a hybrid of the two)while guaranteeing regions with bounded aspect ratios.In short,the BAR tree is an O(log n)-depth BSP-type data structure that creates fat,convex regions.Thus,the BAR tree is“balanced”in two ways:on the one hand,clusters on the same level have roughly the same number of points,and, on the other hand,each cluster region has a bounded aspect ratio.We show that a BAR tree achieves this combined set of goals by proving the existence of a cut,which we call a two-cut.A two-cut might not reduce the point size by any amount but maintains balanced aspect ratio and ensures the existence of a subsequent cut,which we call a one-cut,that both maintains good aspect ratio and reduces the point size by at least two-thirds.In Section 3,we formally define one-and two-cuts and describe how to construct a BAR tree.1.3Our Results for Cluster-Based Graph DrawingIn Section4,we show how to use the BAR tree in a cluster-based graph drawing algorithm.The Large Graph Drawing(LGD)algorithm runs in O(n log n+m+ D0(G))time,where n and m are the number of vertices and edges in the graph G and D0(G)is the time to embed G in the plane.If the graph is planar,Duncan,Goodrich,and Kobourov,BAR Trees,JGAA,4(3)19–46(2000)23Figure1:A clustered graph C=(G,T).The underlying graph G is at the lowest level on the right.The clustering of G on the right is obtained from the BSP cuts on the left. Each cluster is represented by a single node.Edges between layers on the right are edges of the tree T.the algorithm introduces no edge crossings and the running time reduces to O(n log n).The algorithm creates a hierarchical cluster representation of a graph,with balanced clusters at each layer and with cluster depth O(log n).Each cluster region has a balanced aspect ratio,guaranteed by the BAR tree data structure. In the actual display of the clustered graph we represent the clusters either by their convex hulls,or by a larger region defined by the BSP tree,or simply by a single node,see Figure1.2Using a BSP Tree for Cluster DrawingLet G=(V,E)be the graph that we want to draw,where|V|=n and|E|= m.Note that graph G is given combinatorially,i.e.,defined by the order of the neighbors around each vertex.An embedding of G also assigns distinct coordinates in I R2for every vertex v∈V(G).The edges of the graph are drawn as straight lines.For the rest of this paper,we assume that the vertices of G have integer coordinates,that is,the graph is embedded on the integer grid.The goal of our LGD algorithm is to produce a representation of the graph G given a BSP tree T,see Figure1.Similar to[10]we define the clustered graph C=(G,T)to be the graph G,and the BSP tree T,such that the vertices of G coincide with the leaves of T.An internal node of T represents a cluster,whichDuncan,Goodrich,and Kobourov,BAR Trees,JGAA,4(3)19–46(2000)24Figure2:A2-dimensional representation of a clustered graph C=(G,T).The under-lying graph G and the clustering are the same as in Figure1.a simple closed curve. consists of all the vertices in its subtree.All the nodes of T at a given depth i represent the clusters of that level.A view at level i,G i=(V i,E i),consists of the nodes of depth i in T and a set of representative edges,for0≤i≤depth(T).An edge(u,v)belongs to E i if there is an edge between a and b in G,where a is in the subtree of u and b is in the subtree of v.In addition,each node u∈T has an associated region,corresponding to the partition given by T.In Figure1we show an example of a3-dimensional representation of a graph G and in Figure2we show a2-dimensional representation of the same graph.We create the graphs G i in a bottom-up fashion,starting with G k and going all the way up to G0,where k=depth(T).Define the combinatorial graph H=(V(H),E(H)),where initially V(H)={u∈T:depth(u)=k}and E(H)=E(G).Notice that H is well defined since the leaves of T are exactly the vertices of G.At each new level i we perform a shrinking of H.Suppose u,v∈V(H),and parent(u)=parent(v).We replace the pair by their parent and remove the edge(u,v)if it exists.We also remove any multiple edges that this operation may have created and maintain for each surviving edge a pointer to the original edge in G.Thus a shrinking of the graph H consists of all such operations, necessary to transform H into a representation of G at one higher level in the tree T.At each level G i is a subgraph of G with certain edges removed.Since we are producing a representation of G in3-dimensions,every vertex must have three coordinates.Thefirst two coordinates correspond to the location of the vertex on the integer grid.The third coordinate of a vertex v∈V i is equal to i,that is,all the vertices in G i are embedded in the plane given by z=i.To obtain G i from G i+1,for i=0,...,k−1,we use the combinatorial graph H from level i+1.Initially E i=E i+1.We then perform a shrinking of H and while removing an edge from H we remove its associated edge from E i.Thus the algorithm on Figure3runs in O(n·depth(T)+m)ing any of the previous known types of BSP trees,we can maintain most but never all of the desired properties.For example,if T is a k-d tree the cluster regions do not have balanced aspect ratios.We next describe how to construct a BSP tree which satisfies all of our goal criteria.Duncan,Goodrich,and Kobourov,BAR Trees,JGAA,4(3)19–46(2000)25create graph(T,G)H←Gk←depth(T)for i=k downto0obtain G i from Hshrink Hreturn CFigure3:Given graph G embedded in the plane and BSP tree T create clustered graph C.Here H is a combinatorial graph initially the same as G.The operations of obtaining G i from H and shrinking of H are defined in Section2.3The BAR treeLet us now discuss in detail the definition of our particular BSP-type decom-position tree,the BAR tree,and its construction.We begin with some general definitions.Definition1The following terms relate to various potential cuts:•A canonical cut direction is any of the following three vectors:v x=(1,0), v y=(0,1), v z=(1,−1).•A canonical cut is any line whose normal is a canonical cut direction.For example,the line x−y=3has normal v z.•A canonical region is any convex polygon such that each side is a segment of a canonical cut.Since there are three cut directions1,a canonical region can have at most six sides.For convenience,we define six labels representing the six sides of the polygon.Notice that some of these sides may have zero length.For a canonical region R,we let x l and x r represent the corresponding left and right sides of R with normal v x.Similarly,we define y l,y r,z l,and z r,see Figure4.Definition2For a canonical region R,let diam i(R)be the L m metric distance between the two sides of R with normal v i.For a side l in R,we define|l|to be the length of the line segment l measured in the L m metric.For simplicity in our arguments and notation,we use the L∞metric although any of the standard L m metrics is acceptable.In the L∞metric the distance between two lines normal to v z and the length of a line segment normal to v z areDuncan,Goodrich,and Kobourov,BAR Trees ,JGAA ,4(3)19–46(2000)26114523z r x lz l x r y ly r Figure 4:A labelling of the various sides of a canonical region R .defined differently than in the L 2metric.In particular,for a canonical region R with sides z l and z r ,the length |z l |(or |z r |)is the vertical distance between the two endpoints.The distance between the lines associated with z l and z r is one half the vertical distance between the two lines.Definition 3The aspect ratio of a canonical region R isar (R )=max(diam i (R ))/min(diam j (R )),∀i,j ∈{x,y,z }.Given an aspect ratio parameter α,a region R is α-balanced if ar (R )≤α.This definition is valid only for canonical regions.Since all of the regions that appear in this section are canonical regions,whenever we refer to any region we mean a canonical region.When the term αis understood,we refer to α-balanced regions as simply balanced regions and refer to non-α-balanced regions as unbalanced regions.Throughout the paper,we also call balanced and unbalanced regions,respectively,fat and skinny regions.To understand the various notions of a canonical region,let us look at one specific canonical region R in Figure 4.Here we see the various sides of R ,x l ,x r ,y l ,y r ,z l ,z r .In particular,although not actually a true side of R ,we still represent the side z r .It is tangent to R and has zero length.From the figure,we see the various lengths of each side:|x l |=2,|y l |=5,|z l |=1,|x r |=3,|y r |=4,|z r |=0.Since we are using the L ∞metric,the length of z l is 1rather than √Duncan,Goodrich,and Kobourov,BAR Trees,JGAA,4(3)19–46(2000)273.1Constructing the BAR treeWe now introduce the BAR tree data structure.Suppose we are given a point set S in the plane,|S|=n,and an initially square region R containing S.We construct a BAR tree T on S recursively dividing R into cells such that the following properties are guaranteed:•Every cell in the tree is convex.•Every cell in the tree has balanced aspect ratio.•Every leaf cell contains at most a constant number of points of S.•The tree has O(n)nodes.•The depth of the tree is O(log n).The structure is straightforward and reminiscent of the original k-d tree. Recall that in a k-d tree,every node u in the tree represents a cell region u.region and an axis-parallel cut u.cut partitioning that region into two sub-regions,u.left and u.right.The leaves of the tree are cells with a constant number of points.In general,each cut divides the region into two roughly equal halves,and thus the tree has O(log n)depth and uses O(n)space.However,if the vast majority of the points is concentrated close to any particular corner of the region,no constant number of axis-parallel cuts can effectively reduce the size of the point set and maintain good aspect ratio.This is a serious concern for many applications and for ours in particular.As a result,an extensive amount of research has been dedicated to improving and analyzing the performance of k-d trees and its derivatives,often concentrating on trying to maintain some form of balanced aspect ratio[5,19,29].We now show how to construct a BAR tree T from a point set S using an aspect ratio parameterαand a balance parameterβ.We prove that anyα-balanced region can be divided by a sequence of one or two cuts into at most three subregions.We also guarantee that each subregion isα-balanced and the number of points in each of the three subregions is less thanβtimes the number of points in the original region.We begin by defining the notions of a one-cut and a two-cut.Definition4Let R be anα-balanced canonical region containing n points.Let βbe a given balance parameter.A one-cut is any canonical cut dividing R into two subregions R1and R2such that:1.R1and R2are bothα-balanced canonical regions.2.R1and R2contain at mostβn points.If there exists a one-cut for R,we say R is one-cuttable.Definition5Let R be anα-balanced canonical region containing n points.Let βbe a given balance parameter.A two-cut is any canonical cut dividing R into two subregions R1and R2such that:Duncan,Goodrich,and Kobourov,BAR Trees,JGAA,4(3)19–46(2000)28 create tree(R,α,β)create node uu.region←Rif number of points in R≤c,return uif an(α,β)-balanced one-cut s,exists in Ru.cut←s(R1,R2)←s(R)else let s be an(α,β)-balanced two-cut in Ru.cut←s(R1,R2)←s(R)u.left←create tree(R1,α,β)u.right←create tree(R2,α,β)return uFigure5:Creating the BAR tree.The recursion stops when a cell has a constant number of points,c≥1.1.R1and R2are bothα-balanced canonical regions.2.R2contains at mostβn points.3.R1is one-cuttable.If there exists a two-cut for R,we say R is two-cuttable.For anα-balanced region R which is two-cuttable,let s represent the two-cut dividing R into two regions R1and R2,and let s′represent the one-cut dividing R1.In other words,the sequence of two cuts,s and s′,results in three α-balanced regions each containing at mostβn points.To make it clear thatαandβare parameters,we often refer to one-cuts(resp.two-cuts)of a region R as(α,β)-balanced one-cuts(resp.two-cuts).Figure5shows the pseudo-code for the construction of a BAR tree.Here we use the notation(R1,R2)←s(R)as a shorthand for cutting the region R with a cut s resulting in subregions R1and R2.We prove in the next section that everyα-balanced region is either one-cuttable or two-cuttable for sufficiently large constant values ofαandβ.Since the algorithm only uses one-cuts and two-cuts,the regions produced are allα-balanced regions.The algorithm stops the recursion when a leaf cell has a constant number of points from S.Because at least every other cut used is a one-cut,the depth of the tree is O(log1/βn) and the size is O(n).Therefore,the algorithm correctly creates a tree which satisfies the properties for a BAR tree.Duncan,Goodrich,and Kobourov,BAR Trees,JGAA,4(3)19–46(2000)29i l be the line containing i l and let P be the region between i r and i l towards i r until either P is empty or just before P becomes unbalanced.We call thisfinal region R i,r=P maximized in the direction from i l.Similarly,we calli l as defined above,if R i,r is not empty and we continue sweeping in the same direction,the region betweenDuncan,Goodrich,and Kobourov,BAR Trees ,JGAA ,4(3)19–46(2000)30x rz,rR x,r x,l x l y ry l Figure 7:The labels on the sides of a general canonical region and the maximizing cuts from the respective directions.Corollary 1For an α-balanced region R ,if the region R i,r is maximized in the direction from i l ,then min {diam x (R i,r ),diam y (R i,r ),diam z (R i,r )}=diam i (R i,r ).Corollary 2For an α-balanced region R and direction v i ,if R i,l ∩R i,r =∅,then any cut i m with a normal v i and lying between i r produces twoα-balanced subregions R 1and R 2.Lemma 2Suppose we are given a region R with n points,a balance parameter β≥1/2and two parallel lines c l and c r .Without loss of generality,let us orient these lines so that c l lies to the left of c r .Then one of the following must be true:•The number of points from R to the left of c l (i.e.,away from c r )is more than βn ;•The number of points from R to the right of c r (i.e.,away from c l )is more than βn ;•There exists a line c ′parallel and between c l and c r dividing R into two subregions R 1and R 2such that the number of points in either subregion is less than βn .Proof:Assume the first two conditions do not hold.Thus,we only need to prove that the last condition must hold.Let n 1be the number of points to the left of c l and let n 2be the number of points to the left of c r .We know then that n 1>βn ≥n/2.Similarly,we know that (n −n 2)>βn ≥n/2.It followsDuncan,Goodrich,and Kobourov,BAR Trees,JGAA,4(3)19–46(2000)31 then that n2<n/2.Sweep a line c′from c l to c r letting n3be the number of points to the left of c′.Since the sweep is continuous,n3varies from n1>n/2 to n2<n/2.In particular,there is a point where n3=n/2.This cut divides R into two subregions each with less than n/2points.2Corollary3For anα-balanced region R with n points,a direction v i,and β≥1/2,either R is one-cuttable or R i contains more thanβn points.Proof:If the two subregions R i,l and R i,r intersect each other,then by defini-tion R i=R and thus contains n points.If R is one-cuttable,then the statement is trivially true.Otherwise,we have two cuts i l associated with R i,l and R i,r respectively.From Lemma2,either R i,l or R i,r contains more thanβn points or there exists a line c′parallel and between i l dividing R into two subregions R1and R2such that the number of points in either subregion is less thanβn.However,this implies that R is one-cuttable.2 The above corollary is quite useful in proving that certain regions are one-cuttable.For instance,let R be anα-balanced region such that,for some canonical direction v i,both R i,l and R i,r are empty.Since neither of these two subregions can contain any points,R must be one-cuttable.In fact,this notion can be extended to include multiple canonical directions.Lemma3Let R be anα-balanced region R with n points andβ≥2/3.If R x∩R y∩R z=∅,then R is one-cuttable.Proof:This is a standard extension from set theory.For a set of points S,it is impossible to have three subsets of S each contain more than2/3of S without their intersection containing at least one point.2 If we can prove that there exist regions such that no possible assignment for the R i’s allows for a non-empty intersection,then the region R is always one-cuttable.Do there exist regions which are guaranteed to be one-cuttable? We describe two such regions which we will use to argue that everyα-balanced region is inevitably two-cuttable.Definition8For a given aspect ratio parameterαwe define two special canon-ical regions with aspect ratioαas follows:•Canonical isosceles trapezoidal(CIT)regions are trapezoids which have z l and z r as the two opposing parallel base sides,see Figure8a.•Canonical right-angle trapezoidal(CRT)regions are trapezoids which have their two opposing parallel base sides normal to either v x or v y,see Fig-ure8b.Lemma4Forα>4andβ≥2/3,canonical isosceles trapezoidal(CIT)regions are one-cuttable.Duncan,Goodrich,and Kobourov,BAR Trees ,JGAA ,4(3)19–46(2000)32xx rd r(a)d (b)Figure 8:Examples of (a)CIT and (b)CRT regions.Proof:Without loss of generality,we can analyze the region R in Figure 8a,since the other possible CIT regions are symmetrical.Let d i =diam i (R )for i ∈{x,y,z }.Define δ=|z r |=d x −|x r |.Since the trapezoid’s two parallel sides are z l and z r ,we know that d x =d y and |x r |=|y l |.Recall that in the L ∞metric,d z =(|x l |+|y l |)/2=|y l |/2.Similarly,we get d z =|x r |/2.Since the region has aspect ratio α,we have ar (R )=α=d x /d z .It follows thatd x =αd z =α|x r |/2=α(d x −δ)/2=αδ/(α−2)(1)Let us examine the possible intersections of R x ∩R y ∩R z .Since R x,l is empty,we know that R x =R x,r .Since by definition,R x,r is maximized from x l ,we know that diam x (R x )≤d y /α=d x /α.From Equation 1and from α>4,it follows that diam x (R x )<δ/2.Similarly,we know that R y =R y,l and diam y (R y )<δ/2.This implies that R x ∩R y =∅.From Lemma 3,R must be one-cuttable.2Lemma 5For α>4and β≥1/2,canonical right-angle trapezoidal (CRT)regions are one-cuttable.Proof:Without loss of generality,we can again analyze the region R in Fig-ure 8b,since the other possible CRT regions are symmetrical.Let d i =diam i (R )for i ∈{x,y,z }.We know that max i ∈{x,y,z }(d i )=d x and min i ∈{x,y,z }(d i )=d y from the definition of the region.Therefore,we know that ar (R )=α=d x /d y .Observing that |y r |=d x −d y ,we obtain:d y =d x −|y r |。