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问题： 如何使 $1+2+4+\cdots+2^n+\cdots=-1$ ？ 全新定义距离函数，同时保持这一函数的平移不变性： $$ \operatorname{dist}(A, B)=\operatorname{dist}(A+x, B+x) \quad \forall x $$ 首先假设 $0$ 与 $2^n$ 距离为 $2^{-n}$，之后通过平移不变性确定...

问题： 假设一位音乐天才，对任何频率之比为有理数的一对音符都觉得悦耳，那么他是否会觉得所有的音符都是悦耳的？ 详细说明：当一对音符的频率比为有理...

Linux 的基础知识 Linux 简介 Linux 分为内核版本和发行版本。发行版本在内核基础上增加各种功能。在服务器领域使用 redhat，ubuntu 具有图形界面偏向个人...

The course notes give detailed code implementation and mathematical derivation. Newton Method $$ \begin{gather*} f(x)\approx f(x_n)+f'(x_n)\Delta x+\frac{1}{2}f''(x_n)\Delta x^2 \\ 0=\frac{d}{d\Delta x}\left(f(x_n)+f'(x_n)\Delta x+\frac{1}{2}f''(x_n)\Delta x^2\right)=f'(x_{n})+f''(x_{n})\Delta x \end{gather*} $$ so $$ x_{n+1}=x_n-H^{-1}\nabla f(x_n) $$ where $H=\nabla^2f$ is the Hessian matrix Euler Method $$ \mathbf{x}(t_0 + h) = \mathbf{x}_0+h \dot{\mathbf{x}}(t_0) $$ Euler Method is the simplest numerical method and achieve $O(h^2)$ accuracy. But it has many disadvantages. It will produce outward drift when evaluating an orbiting movement so it is not accurate....

Paper Reading Note Implicit Euler Solver $$ \begin{gather*} \mathbf{q}_{n+1}=\mathbf{q}_n+h\mathbf{v}_{n+1} \\ \mathbf{v}_{n+1}=\mathbf{v}_n+h\mathbf{M}^{-1}(\mathbf{f}_{\text{int}}(\mathbf{q}_{n+1})+\mathbf{f}_{\text{ext}}) \end{gather*} $$ This system can be converted to an optimization problem: $$ \require{mathtools} \DeclarePairedDelimiters\norm{\lVert}{\rVert} \min_{\mathbf{q}_{n+1}} \dfrac{1}{2h^2} \norm*{\mathbf{M}^{\frac{1}{2}} (\mathbf{q}_{n+1} - \mathbf{s}_n)}_F^2 + \sum_i W_i(\mathbf{q}_{n+1}) $$ where $\mathbf{s}_n = \mathbf{q}_n + h\mathbf{v}_n + h^2\mathbf{M}^{-1}\mathbf{f}_{\text{ext}}$ and $W_i(\mathbf{q})$ is a scalar potential energy function. Intuitively, this minimization problem describes the compromise between the momentum potential $$ \begin{equation}\label{potential} \dfrac{1}{2h^2} \norm*{\mathbf{M}^{\frac{1}{2}} (\mathbf{q}_{n+1} - \mathbf{s}_n)}_F^2 \end{equation} $$ which states that the solution should follow its momentum (plus external forces), and the elastic potential, that requires the solution to minimize the elastic deformation....

This paper introduces position-based methods and its differences from other methods. Moreover, it presents the algorithm and several solvers. Specific constraints are also discussed in detail. Finally, Implementation and application are briefly presented. Constraints Constraints are kinematic restrictions in the form of equations and inequalities that constrain the relative motion of bodies. Equality and inequality constraints are referred to as bilateral and unilateral constraints, respectively. Bilateral constraint $$ C(\mathbf{x}_{i_1}, \mathbf{q}_{i_1}, \cdots , \mathbf{x}_{i_{n_j}}, \mathbf{q}_{i_{n_j}}) = 0 $$ Unilateral constraint $$ C(\mathbf{x}_{i_1}, \mathbf{q}_{i_1}, \cdots , \mathbf{x}_{i_{n_j}}, \mathbf{q}_{i_{n_j}}) \ge 0 $$ where $\{i_1, \cdots, i_{n_j}\}$ is a set of body indices and $n_j$ is the cardinality of the constraint....

This paper presents the most important methods used in computer graphics to produce realistic animations and simulations: finite element/difference/volume methods, mass-spring systems, meshfree methods, coupled particle systems and reduced deformable models based on modal analysis. Also, it discusses the benefits and drawbacks of each method, reports the improvement that made in other papers. Time Integration Time integration schemes are evaluated by two main criteria, their stability and their accuracy. In the field of physically based animation in Computer Graphics, stability is often much more important than accuracy....

基于 Hexo 和 NexT 生成 blog ...

博客写作时的一些规范和特殊用法参考 ...