School of Science Department of Mathematics 52 Interpolations on Surfaces Supervisor: LEUNG Shing Yu / MATH Student: CHAN Sze Yu / MATH-CS Course: UROP 1100, Fall UROP 2100, Spring We propose a novel framework for interpolation on general genus zero surfaces, a problem for which only a small number of existing methods have been explored. The core idea is to apply a conformal map from the surface to the sphere, perform the desired interpolation using established spherical techniques, and then transfer the resulting curve back to the original geometry. We demonstrate the applicability of the approach through an example on an ellipsoid. Interpolations on Surfaces Supervisor: LEUNG Shing Yu / MATH Student: CHENG Yiyang / MATH-AM Course: UROP 1100, Spring Quaternions form a number system with many remarkable algebraic properties. Their importance reaches far beyond pure mathematics, as they serve as a powerful tool in applied mathematics. In particular, quaternions offer an efficient way to represent orientations and rotations of objects in three-dimensional space. Building on this notation, researchers have developed a wide range of interpolation methods on surfaces such as the unit sphere. These techniques play an important role in computer graphics and scientific computing. This project introduces the basic ideas of quaternions and then examines several interpolation methods, with a focus on understanding their structure and analytical properties. Interpolations on Surfaces Supervisor: LEUNG Shing Yu / MATH Student: FU Ming Hon Harvey / MATH-PMA Course: UROP 1100, Summer This project investigates several interpolation schemes for rotations in SO(3). Methods such as SLERP, SENO, and SQUAD are examined and compared to identify their respective strengths and limitations. A custom Python plugin for the three-dimensional modeling software Blender is used to visualize these schemes. The study is based on two main visualizations. In the first one, the rotation is applied to a camera, and the resulting camera motion after interpolation is analyzed. In the second one, the rotation is applied to the x, y, and z axes, and the transformed axes are displayed on three separate spheres. The findings from this project provide guidance for selecting suitable interpolation methods for rotations in practical applications.
RkJQdWJsaXNoZXIy NDk5Njg=