In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analy...In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analyzes the limitations in them. To overcome these limitations, a novel method for generating 3D fractal sets based on a 3D number system named ternary algebra is proposed. Both theoretical analyses and experimental results demonstrate that the ternary-algebra-based method is superior to any one of the quad-algebra-based methods, including the first two methods presented in this paper, because it is more intuitive, less time consuming and can completely control the geometric structure of the resulting sets. A ray-casting algorithm based on period checking is developed with the goal of obtaining high-quality fractal images and is used to render all the fractal sets generated in our experiments. It is hoped that the investigations conducted in this paper would result in new perspectives for the generalization of 3D Mandelbrot and Julia sets and for the generation of other deterministic 3D fractals as well.展开更多
迈耶·夏皮罗(Meyer Schapiro,1904-1996),美国艺术史家。他的学术成就集中体现在四卷本的选集中。它们是:《罗马式艺术》(Romanesque Art,1977)、《现代艺术:19与20世纪》(Modern Art:19th and20th Centuries,1978)、《古代晚期、...迈耶·夏皮罗(Meyer Schapiro,1904-1996),美国艺术史家。他的学术成就集中体现在四卷本的选集中。它们是:《罗马式艺术》(Romanesque Art,1977)、《现代艺术:19与20世纪》(Modern Art:19th and20th Centuries,1978)、《古代晚期、基督教早期和中世纪艺术》(Late A ntique,Early Christian and Medieval Art,1979),以及《艺术的理论与哲学:风格、艺术家和社会》(Theor y and Philosophy of Art:Style,A展开更多
基金Project supported by the National Basic Research Program (973) of China (Nos. 2004CB719402 and 2002CB312106), the National Natural Science Foundation of China (Nos. 60375020 and 50305033), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20020335112)
文摘In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analyzes the limitations in them. To overcome these limitations, a novel method for generating 3D fractal sets based on a 3D number system named ternary algebra is proposed. Both theoretical analyses and experimental results demonstrate that the ternary-algebra-based method is superior to any one of the quad-algebra-based methods, including the first two methods presented in this paper, because it is more intuitive, less time consuming and can completely control the geometric structure of the resulting sets. A ray-casting algorithm based on period checking is developed with the goal of obtaining high-quality fractal images and is used to render all the fractal sets generated in our experiments. It is hoped that the investigations conducted in this paper would result in new perspectives for the generalization of 3D Mandelbrot and Julia sets and for the generation of other deterministic 3D fractals as well.
文摘迈耶·夏皮罗(Meyer Schapiro,1904-1996),美国艺术史家。他的学术成就集中体现在四卷本的选集中。它们是:《罗马式艺术》(Romanesque Art,1977)、《现代艺术:19与20世纪》(Modern Art:19th and20th Centuries,1978)、《古代晚期、基督教早期和中世纪艺术》(Late A ntique,Early Christian and Medieval Art,1979),以及《艺术的理论与哲学:风格、艺术家和社会》(Theor y and Philosophy of Art:Style,A
