基于不变式论的Regular Polytope艺术图案可视化

Regular Polytopes (RP) are the generalization of 2-D regular polygons and 3-D regular polyhedra in high dimensional Euclidean space that possess high abstract and perfect symmetry. Since the existing visualization of RP (VRP) couldn't consider both metric information and symmetries and the low dimensional visualization of aesthetic patterns (VAP) couldn't overcome the huge order bottleneck of RP's symmetry group, the VAP of RP (RP-VSP) has obvious difficulty. Based on invariant theory and fundamental region, this project plans to investigate the following problems. First, to compensate the defect of the existing VRP lacking metric information, establish special homomorphic mapping which can realize the equivalent symmetrical decomposition of RP and provide metric information. Second, by establishing the elementary symmetrical polynomial, use invariant theory to overcome the order bottleneck appeared in the traditional method for treating RP-VAP. At last, to treat the circumstances that many types and complex structure of RP, explore the self-adaption rendering scheme of RP-VAP. The project not only regards the scientific meaning of investigated problems, but also emphasizes the application of theoretical results. It could promote the development and communion of group representation and invariant theory of RP and present new method for realizing RP-VAP.

Regular Polytope(RP)是正多边形、正多面体在高维欧氏空间的推广，它具有高度抽象性和完美对称性。现有RP可视化方法不能兼顾度量与对称性，且低维艺术图案可视化(VAP)方法难以克服RP对称群的巨大阶数瓶颈，因此，RP艺术图案可视化(RP-VAP)存在显著困难。本项目拟以基础域原理和不变式论为基础，阐明RP空间任意点映射到基础域的既约型对称变换，通过建立RP几何空间和对称群间具有对称剖分意义的特定同态映射，实现RP空间等价对称剖分，弥补现有可视化方法丢失度量信息的不足；探讨RP对称群的基本对称多项式，借助不变式论克服传统方法实现RP-VAP所面临的阶数瓶颈；建立RP自适应渲染算法，藉此应对RP结构复杂且族类众多的特征。本项目既注重研究问题的科学意义，又强调研究成果的应用价值，不仅能促进群表示论和RP不变式论的发展与交融，还有望为RP-VAP提供新思路。

Regular Polytope(RP)是正多边形、正多面体在高维欧氏空间的推广，它具有高度抽象性和完美对称性。现有RP 可视化方法不能兼顾度量与对称性，且低维艺术图案可视化(VAP)方法难以克服RP 对称群的巨大阶数瓶颈，因此，RP 艺术图案可视化(RP-VAP)存在显著困难。本项目拟以基础域原理和不变式论为基础，阐明RP 空间任意点映射到基础域的既约型对称变换，通过建立RP 几何空间和对称群间具有对称剖分意义的特定同态映射，实现RP 空间等价对称剖分，弥补现有可视化方法丢失度量信息的不足；探讨RP 对称群的基本对称多项式，借助不变式论克服传统方法实现RP-VAP 所面临的阶数瓶颈；建立RP 自适应渲染算法，藉此应对RP 结构复杂且族类众多的特征。本项目既注重研究问题的科学意义，又强调研究成果的应用价值，不仅能促进群表示论和RP 不变式论的发展与交融，还有望为RP-VAP 提供新思路。..针对项目涉及的相关问题，我们在RP剖分、艺术图案可视化、动力系统、保形变换和分形等领域进行相应研究，共发表12篇SCI、2篇EI，获授权数项软件著作权专利，一项发明专利在发文实质审核阶段。上述工作，较为系统地解决项目涉及核心科学，建立简单快捷的RP相关问题解决策略。