{"title":"Diagonal decompositions of shapes and their algebras","authors":"Djordje Kristic","doi":"10.1017/S0890060421000238","DOIUrl":null,"url":null,"abstract":"Abstract The formal approach to shapes and their algebras, as it appears in shape grammar theory, has been reviewed. It starts with geometric elements and their partial algebras, continues to shapes, their algebras, and boundaries, as well as algebras that calculate with shapes and their boundaries. There is a number of new concepts introduced along the way. These include diagonal decompositions and their algebras which simplify calculations with shapes, b-paired diagonal decompositions which extend calculations with shapes and their boundaries from diagonal shapes only to all shapes, and m-order boundaries which extend the concept of shape boundaries and allow for calculations with multiple representations of shapes. It also shows that algebras of shapes are infinite direct sums of diagonal algebras.","PeriodicalId":50951,"journal":{"name":"Ai Edam-Artificial Intelligence for Engineering Design Analysis and Manufacturing","volume":" ","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2022-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ai Edam-Artificial Intelligence for Engineering Design Analysis and Manufacturing","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1017/S0890060421000238","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE","Score":null,"Total":0}
引用次数: 1
Abstract
Abstract The formal approach to shapes and their algebras, as it appears in shape grammar theory, has been reviewed. It starts with geometric elements and their partial algebras, continues to shapes, their algebras, and boundaries, as well as algebras that calculate with shapes and their boundaries. There is a number of new concepts introduced along the way. These include diagonal decompositions and their algebras which simplify calculations with shapes, b-paired diagonal decompositions which extend calculations with shapes and their boundaries from diagonal shapes only to all shapes, and m-order boundaries which extend the concept of shape boundaries and allow for calculations with multiple representations of shapes. It also shows that algebras of shapes are infinite direct sums of diagonal algebras.
期刊介绍：
The journal publishes original articles about significant AI theory and applications based on the most up-to-date research in all branches and phases of engineering. Suitable topics include: analysis and evaluation; selection; configuration and design; manufacturing and assembly; and concurrent engineering. Specifically, the journal is interested in the use of AI in planning, design, analysis, simulation, qualitative reasoning, spatial reasoning and graphics, manufacturing, assembly, process planning, scheduling, numerical analysis, optimization, distributed systems, multi-agent applications, cooperation, cognitive modeling, learning and creativity. AI EDAM is also interested in original, major applications of state-of-the-art knowledge-based techniques to important engineering problems.
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