Rotational Symmetry and Design

Designing rotational symmetries on surfaces is a necessary task for a wide variety of graphics applications, such as surface parameterization and remeshing, painterly rendering and pen-and-ink sketching, and texture synthesis. In these applications, the topology of a rotational symmetry field such as singularities and separatrices can have a direct impact on the quality of the results. In this paper, we present a design system that provides control over the topology of rotational symmetry fields on surfaces. As the foundation of our system, we provide comprehensive analysis for rotational symmetry fields on surfaces and present efficient algorithms to identify singularities and separatrices. We also describe design operations that allow a rotational symmetry field to be created and modified in an intuitive fashion by using the idea of basis fields and relaxation. In particular, we provide control over the topology of a rotational symmetry field by allowing the user to remove singularities from the field or to move them to more desirable locations. At the core of our analysis and design implementations is the observations that N-way rotational symmetries can be described by symmetric N-th order tensors, which allows an efficient vector-based representation that not only supports coherent definitions of arithmetic operations on rotational symmetries but also enables many analysis and design operations for vector fields to be adapted to rotational symmetry fields. To demonstrate the effectiveness of our approach, we apply our design system to pen-and-ink sketching and geometry remeshing.

Author Index

Designing rotational symmetries on surfaces is a necessary task for a wide variety of graphics applications, such as surface parameterization and remeshing, painterly rendering and pen-and-ink sketching, and texture synthesis. In these applications, the topology of a rotational symmetry field such as singularities and separatrices can have a direct impact on the quality of the results. In this paper, we present a design system that provides control over the topology of rotational symmetry fields on surfaces.

This study focuses on the construction of (25SOR) in Three Dimensions (3D) engaging trigonometric functions. Designing experiments in multiple dimensions is crucial for efficiently exploring complex systems and optimizing various processes. The proposed methodology utilizes trigonometric functions to generate a set of experimental points that exhibit desirable properties, such as rotatability, orthogonality, and uniformity, in the three-dimensional space. By employing trigonometric transformations, a design with twenty-five equally spaced points is constructed, ensuring the ability to conduct thorough investigations across the entire experimental region. The advantages of utilizing trigonometric functions in the design construction process include the flexibility to achieve rotational symmetry and the capability to control the distribution of points systematically. The resulting 25SOR design facilitates comprehensive experimentation and enables researchers to efficiently evaluate response surfaces and identify optimal operating conditions in three-dimensional spaces. This approach holds promise for applications in various fields, including agriculture, where the exploration of multidimensional parameter spaces is essential for enhancing performance and efficiency.