3D数学基础

第四章 矢量数学与几何两种含义数学含义 矢量是一组数字。计算机含义，矢量是数组。包括多少数字是矢量的维度。几何含义 矢量是线段的长度和指向magnitude & directiondisplacement 位移矢量velocity 速率矢量distance 距离标量speed 速度标量矢量没有位置，相对存在标量无方向矢量中的数字在各维度，标识带符号的位移。Vectors are specified by giving the signed displacement in each dimension.点 指示位置(与矢量不同)矢量标识出相对位移，在大框架之下找到位置。

实际城市中存在多层次的权限划分，每层次有自己的独立坐标空间，又要参考上下层次的空间。特殊领域内容需要一些不同的坐标空间信息+世界空间(east, west)Latitude 33度01' NorthLongitude 96度59' Westworld spaceglobal spaceuniversal coordinate space[Questions] 物体位置朝向; 镜头位置朝向; 景物位置外观; 物体移动规律;- what is the position and orientation of each object?- what is the position and orientation of the camera?- what is the terrain like in each position in the world?(hills,mountains,lakes,buildings)- how does each object get from where it is to where it wants to be?(motion planning for npc)+物体空间(left, forward)object spacemodeling spacebody space[Questions] 附近物体识别; 物体相对方位;- is there another object near me that I need to interact with?(Do I need to kill it?)- in what direction is it? is it front of me? Slightly to my left? To my right? (So I can shoot at it or rum in the opposite direction)+镜头空间camera spacea coordinate with an observer[Questions] 镜头内可见; 镜头的平截头体内相对位置; 整体局部可见或不可见; 物体间遮挡;- is a given point in 3D space in front of the camera?- is a given point in 3D space on screen, or is it off to the left, right, top, or bottom edges of the camera frustum? 镜头的平截头体- is an object completely on screen, partially on screen, or completely off screen?- which of the two objects is in front of the other?(This is called occlusion information) 3D to 2D method : projection process. +惯性空间? 物体空间经过旋转达到惯性空间，而由惯性空间经过偏移到达世界空间；用惯性空间分割了rotate和transform两个动作可以简化问题 Clearly, the robot thinks that her y-axis points from her feet to her head and that her x-axis points to her left. The robot’s inertial space is obtained from her object space by rotating her object axes about their origin until the axes are parallel with the world axes (Figure 3.4). Finally, inertial space can be transformed to world space by moving the origin of inertial space to the origin of world spaceto be precise, we must specify the location and orientation of the object's axes in world coordinates. 需要用世界坐标解决物体相对位置positive 正向的 positive x-axes;negative 负向的嵌套的坐标系统世界系统 -> 物体系统不在镜头内的物体,仅记录"物体系统"对于"世界系统"的位置+朝向因物体在世界里溜达，世界空间做父，物体空间做子很方便。物体打开成组件也可方便的独立动画 羊可边前进(世界空间)，边前后晃头(羊空间)，边上下摆耳朵(头部空间)物体内部形成垂直体系，各个组件按次序依存在层次结构中，根据物体的动作复杂程度，划分多层次，进行相对运动计算。子坐标系统嵌套(nested)在父坐标系统中。世界坐标系统是树的根，虚拟世界的树结构可以在生命周期里不断的动态改变。[coordinate space transformation] 我们需把世界空间的位置 转换 到 物体空间 或反之总结：1. 绘制物体树结构2. 物体与世界空间向连接的媒体，旋转和移动的相对性角度是相反的。

三维数学用于度量位置，距离，角度。2.1 1DCartesion coordinate system 笛卡尔坐标系René Descartes 1596-1650 French philosopher, physicist, physiologist, mathematician自然数 natural number, 常叫做counting number, 几千年前发明的 (! 手指也许十进制的来历)整数 integer number, consist of the natural numbers and their negative counterparts分数 fractional number, consisting of one integer divided by another有理数 rational number, fit in the number line in the obvious places between integers小数记法 decimal notation, 分数冗长乏味PI ratio of the circumference of a circle to its diameter.实数 real number, 包括有理数，小数。harmless delusion离散数学 discrete mathematics连续数学 continuous mathematicsThe First Law of Computer Graphics: If it looks right, it is right.三角学 trigonometrybecause Descartes notwithstanding, we told you so, because it would be nice, and because it makes sense.2.2 2D假想城市 The Hypothetical City of Cartesia二维坐标系 “等价”, 8种指向方向能完全转换，四种旋转，四种反向观察。而三维坐标系不等价。2.3 3D垂直正交 perpendicularIn 3D, points are specified using three numbers, x, y, and z, which give the signed distance to the yz, xz, and xy planes, respectively.The distance is measured along a line parallel to the axis.left-handed coordinate system3D Studio Max. the default orientation of the axes is for +x to point right, +y point forward, and +z to point up, Is this a left- or right-handed coordinate space?

发现了和http://book.douban.com/annotation/32032123/ 一样的问题正确的向量投影，应该参考： https://en.wikipedia.org/wiki/Vector_projection推理过程总结如下：假设 ($\vec v$) 在 ($n$) 上的投影，平行于 ($\vec n$) 的称为 ($\vec v_1$)，垂直于 ($\vec n$) 的称为 ($\vec v_2$)。那么\begin{eqnarray*} \vec v_1 & = & \hat n \|\vec v_1\| & = & \hat n\| \vec v\|\cos \theta & = & \vec n \frac{\vec v \cdot \vec n}{\|\vec n\|^2} \end{eqnarray*}($$ \vec v_2 = \vec v - \vec v_1 = \vec v - \vec n \frac{\vec v \cdot \vec n}{\|\vec n\|^2} $$)PS: 豆瓣的公式保持了一贯的风格，字都那么小，唉。