School of Science Department of Mathematics 53 Interpolations on Surfaces Supervisor: LEUNG Shing Yu / MATH Student: LI Tao / MATH-GM Course: UROP 1100, Spring UROP 2100, Summer In this paper, we expand on the SPLINE-SIDER-based algorithm introduced in UROP 1100 and further explore its applications. The study is divided into two main parts. The first part provides a detailed explanation of the theoretical foundations and essential ideas behind the algorithm, covering both pose interpolation and location interpolation, and includes a comparative discussion that highlights its advantages over conventional approaches. The second part extends the MATLAB-based camera algorithm from the previous project and evaluates the feasibility and accuracy of different line configurations. Experimental results and visualizations are presented to demonstrate the accuracy, efficiency, and potential uses of the algorithm. Interpolations on Surfaces Supervisor: LEUNG Shing Yu / MATH Student: SOULAT Abdullah Bin / CPEG Course: UROP 1000, Summer We develop two new ideas for integrating curves on the sphere using numerical integration and compare them with existing approaches. The proposed methods utilize surface interpolation techniques, specifically spherical linear interpolation and spherical quadrangle interpolation. The existing methods include the trapezoidal approximation and the Simpson approximation, which integrate the curve in three-dimensional space. Several test integrals are evaluated using all methods, and their orders of convergence and computational times are compared. Interpolations on Surfaces Supervisor: LEUNG Shing Yu / MATH Student: ZHENG Jiayi / MATH-AM Course: UROP 1100, Fall Spherical linear interpolation, known as SLERP, is a standard method for constructing geodesic interpolation on the sphere from points in three-dimensional space. When many points need to be connected, higherorder interpolation schemes become essential. In this project, we investigate the continuity properties of spherical interpolation of order n, abbreviated as SIDER-n, which is constructed through repeated applications of SLERP. The design of SIDER is inspired by the construction of higher-order composite Bezier curves, where a quadratic Bezier curve is applied dynamically to successive triples of points. Our goal is to examine the smoothness of SIDER-n and determine the degree of continuity it achieves.
RkJQdWJsaXNoZXIy NDk5Njg=