Fractals offer the ability to generate fascinating geometric shapes with all sorts of unique characteristics (for instance, fractal geometry provides a basis for modelling infinite detail found in nature). While fractals are non-euclidean mathematical objects which possess an assortment of properties (e.g., attractivity and symmetry), they are also able to be scaled down, rotated, skewed and replicated in embedded contexts. Hence, many different types of fractals have come into limelight since their origin discovery. One particularly popular method for generating fractal geometry is using Julia sets. Julia sets provide a straightforward and innovative method for generating fractal geometry using an iterative computational modelling algorithm. In this paper, we present a method that combines Julia sets with dual-quaternion algebra. Dual-quaternions are an alluring principal with a whole range interesting mathematical possibilities. Extending fractal Julia sets to encompass dual-quaternions algebra provides us with a novel visualize solution. We explain the method of fractals using the dual-quaternions in combination with Julia sets. Our prototype implementation demonstrate an efficient methods for rendering fractal geometry using dual-quaternion Julia sets based upon an uncomplicated ray tracing algorithm. We show a number of different experimental isosurface examples to demonstrate the viability of our approach.

翻译：分形能够生成具有各种独特性质的迷人几何形状（例如，分形几何为模拟自然界中无限细节提供了基础）。分形作为非欧几何数学对象，具备吸引性、对称性等多种属性，同时能在嵌入场景中进行缩放、旋转、扭曲和复制。因此，自发现以来，多种分形类型备受关注。生成分形几何的一种特别流行方法是使用朱莉娅集。朱莉娅集提供了一种通过迭代计算建模算法生成分形几何的简洁创新方法。本文提出一种将朱莉娅集与对偶四元数代数相结合的方法。对偶四元数是一个极具吸引力的数学原理，蕴含一系列有趣的数学可能性。将分形朱莉娅集扩展至对偶四元数代数，为我们提供了新颖的可视化方案。我们阐述了结合对偶四元数与朱莉娅集的分形生成方法，并通过原型实现展示了基于简单光线追踪算法、使用对偶四元数朱莉娅集高效渲染分形几何的技术。文中给出多个实验性等值面示例，以验证本方法的可行性。