MATLAB_Scattered_Interpolation_Surface

Dini的表面由以下参数方程给出： x = acos(u)sin(v);y = asin(u)sin(v);z = a(cos(v) + log(tan(0.5v))) + bu; 这些方程使用mesh命令绘制为Surface Plot，可以通过MATLAB代码生成Dini表面的可视化效果。更多细节和背景请参考维基百科链接：Dini's Surface。

该工具箱包含了各种绘制NURBS曲线和NURBS曲面的程序。对于从事曲面设计或相关领域的工作者来说，非常有用；同时，也为学习MATLAB编程提供了强大的支持，帮助用户更好地掌握该语言的应用。

在高维插值中，我们面临“维数灾难”：当我们增加维数时，样本数呈指数增长。减少这种影响的一种方法是使用稀疏网格。当梯度信息可用时，例如来自伴随求解器，梯度增强稀疏网格提供了进一步减少样本数量的可能性。

插值动画：本项目探讨了动画多项式近似的递增顺序，提升动画表现力。当前版本的剪辑可观看：点击这里

使用stlread加载矩阵时，利用PatchSlim可以有效减小顶点矩阵的大小。表面网格通常包含重复顶点。此函数会解析点列表，查找并移除v（顶点）矩阵中的所有重复条目，并相应更新f（面）矩阵的索引。通常，这会显著减少补丁矩阵的大小。请注意，在非常大的网格上运行可能会需要较长时间。在加载STL文件后，我使用此工具来优化表面网格尺寸。用法：[v, f]=patchslim(v, f)。更多信息请访问： STL File Reader和Esmonde-White's Site

Quaternions are an extension of the complex number concept, playing a crucial role in 3D space rotation. To better understand the relationship between quaternions and 3D rotations, we begin with the fundamentals of complex numbers, as there are notable parallels between the properties of complex numbers and their role in 2D rotation. Complex numbers consist of real and imaginary components, where the imaginary unit (i) is defined as ( \sqrt{-1} ), so any complex number (z) can be represented as ( z = a + bi ), with (a) as the real part and (b) as the imaginary part, where (i^2 = -1). Complex numbers can be viewed as points or vectors in the complex plane, where the real part corresponds to the x-axis and the imaginary part to the y-axis. Complex addition and subtraction are straightforward, performed by adding or subtracting the real and imaginary parts, respectively. Multiplication of complex numbers, however, is slightly more complex, requiring expansion and simplification using (i^2 = -1). A significant feature of complex multiplication is its connection to matrix multiplication. Specifically, every complex number can be associated with a particular 2x2 real matrix. For example, the complex number ( z = a + bi ) can be represented by the matrix: [ \begin{bmatrix} a & -b \\ b & a \end{bmatrix} ] This matrix form allows us to use matrix multiplication to perform complex number multiplication. During this process, we observe that complex number multiplication effectively corresponds to a rotation and scaling transformation in 2D space. For instance, the unit complex number 1 corresponds to the identity matrix, leaving points in the complex plane unchanged, while the imaginary unit (i) corresponds to a 90-degree rotation matrix, meaning that multiplying a complex number by (i) rotates it 90 degrees clockwise. After grasping these complex fundamentals, we move to quaternions. Quaternions are a further extension of complex numbers into four-dimensional space. Each quaternion consists of a real part and three imaginary parts, generally written as ( q = w + xi + yj + zk ), where (w) is the real part and (x), (y), and (z) are the imaginary parts. The units (i), (j), and (k) follow distinct multiplication rules in four-dimensional space, where quaternion multiplication is analogous to complex multiplication yet complies with four-dimensional rules. Quaternions are highly beneficial in 3D rotation operations because they avoid gimbal lock, a common issue with Euler angle representation. This makes quaternions a stable and continuous method for rotation interpolation, essential in computer graphics for 3D animation and simulation. Through quaternions, we can use linear algebra and basic geometry principles to intuitively derive and comprehend 3D spatial rotation. Quaternions provide direct representation of the rotation axis and rotation angle, while interpolating quaternions enables smooth transitions between two rotations. This is extremely helpful in animation and real-time rendering for creating fluid movement. For example, in game development and flight simulation, quaternions are often used to achieve coherent flight paths, allowing engineers to implement smooth motion.

PM_Solver_Matlab is an FEM solver developed using MATLAB for synchronous motors, with a focus on surface-mounted permanent magnet motors. Some of the code is based on or modified from Smeklib().

FSSF2是一款用于在分散的3D数据上拟合2-D傅立叶级数表面的工具，支持系列配方优化拟合。该工具具备直观的GUI界面。

本模型用于捕捉球落在非理想表面的轨迹，具有一定的恢复系数。用户可以调整该系数，以查看不同情况下的运动结果。