3f:null 40:["$","title",null,{"children":"透视投影(Perspective Projection) One | o_O | Observable"}] 46:[{"id":"7efcf74e97e4cc086cc27f1f335dba991f2fa1c810adb322b359f134df895376e34c71625b463c55d580c3724f41ba7939aa7ab7ae01892df701a30bff92914e","href":"https://static.observableusercontent.com/files/7efcf74e97e4cc086cc27f1f335dba991f2fa1c810adb322b359f134df895376e34c71625b463c55d580c3724f41ba7939aa7ab7ae01892df701a30bff92914e","name":"image@1.png","type":"image/png","lastModified":1656220232185.655,"size":8922},{"id":"0c58802ecf2a4c7937778c628f360740741b9ac696354ec1da2208eda4db3cc0b3f7c0699f6f1793c5bbad768c8e28deb1de4d47f26bac0f3af46b142280ed73","href":"https://static.observableusercontent.com/files/0c58802ecf2a4c7937778c628f360740741b9ac696354ec1da2208eda4db3cc0b3f7c0699f6f1793c5bbad768c8e28deb1de4d47f26bac0f3af46b142280ed73","name":"image@2.png","type":"image/png","lastModified":1656220247046.656,"size":6585}] 44:{"id":"b7216f04fff017b8","slug":"perspective-projection-one","title":"透视投影(Perspective Projection) One","thumbnail":"0c3a272d258c97b83fc26609962d26d82fafb62ac6a93cd0350cedd4ec5fe944","access_level":"public","create_time":"$D2022-06-26T04:20:36.841Z","update_time":"$D2022-06-26T07:37:54.903Z","view_time":null,"first_public_time":"$D2022-06-26T08:10:31.381Z","has_unpublished_edits":null,"fork":false,"stars":1,"starred":false,"owner":{"type":"team","id":"32165dab00feb5f5","login":"musixnotmusic","name":"o_O","avatar_url":"https://avatars-next.observableusercontent.com/00e1f2e2d9054519261eedf61fb2d59cda7f6eedf972722679f250270ac68638","tier":"starter_2024","bio":"","following":false},"version":400,"publish_version":400,"role":null,"authored":false,"authors":[{"type":"individual","id":"65e905bbe8e390e3","login":"musixnotmusic","name":"o_O","avatar_url":"https://avatars-next.observableusercontent.com/00e1f2e2d9054519261eedf61fb2d59cda7f6eedf972722679f250270ac68638","tier":null,"bio":"","following":false}],"members":[{"type":"individual","id":"65e905bbe8e390e3","login":"musixnotmusic","name":"o_O","avatar_url":"https://avatars-next.observableusercontent.com/00e1f2e2d9054519261eedf61fb2d59cda7f6eedf972722679f250270ac68638","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"透视投影(Perspective Projection) One","cells":[{"id":0,"value":"# [透视投影(Perspective Projection)](https://www.wgqing.com/%E9%80%8F%E8%A7%86%E6%8A%95%E5%BD%B1%E5%8F%98%E6%8D%A2%E6%8E%A8%E5%AF%BC/) One","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":12,"value":"### 1.齐次方程","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":15,"value":"md`透视投影变换是在***齐次坐标***下进行的，而***齐次坐标***本身就是一个令人迷惑的概念，这里我们先把它理解清楚。 \n简短的解释：***齐次坐标***就是将一个原本是***n维的向量***用一个***n+1维向量***来表示。`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":98,"value":"“齐次坐标表示是计算机图形学的重要手段之一，它既能够用来明确区分向量和点，同时也更易用于进行仿射（线性）几何变换。”—— F.S. Hill, JR","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":31,"value":"### 2.线性插值","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":34,"value":"md`这是在图形学中普遍使用的基本技巧，我们在很多地方都会用到，比如2D位图的放大、缩小，Tweening变换，以及我们即将看到的透视投影变换等等。基本思想是：给一个x属于[a, b]，找到y属于[c, d]，使得x与a的距离比上ab长度所得到的比例，等于y与c的距离比上cd长度所得到的比例，用数学表达式描述很容易理解：`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":37,"value":"tex.block`\\frac{x - a}{b - a} = \\frac{y - c}{d- c}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":44,"value":"md`这样，从a到b的每一个点都与c到d上的唯一一个点对应。有一个x，就可以求得一个y。\n此外，如果x不在[a, b]内，比如x < a或者x > b，则得到的y也是符合y < c或者y > d，比例仍然不变，插值同样适用。`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":56,"value":"## 投影变换","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":58,"value":"md`好，有了上面两个理论知识，我们开始分析这次的主角——透视投影变换。这里我们选择OpenGL的透视投影变换进行分析，其他的APIs会存在一些差异，但主体思想是相似的，可以类似地推导。经过相机矩阵的变换，顶点被变换到了相机空间。这个时候的多边形也许会被视锥体裁剪，但在这个不规则的体中进行裁剪并非那么容易的事情，所以经过图形学前辈们的精心分析，裁剪被安排到规则观察体(Canonical View Volume, CVV)中进行，CVV是一个正方体，x, y, z的范围都是[-1，1]，多边形裁剪就是用这个规则体完成的。所以，事实上是透视投影变换由两步组成：`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":63,"value":"md`1）用透视变换矩阵把顶点从视锥体中变换到裁剪空间的CVV中。 \n2） CVV裁剪完成后进行透视除法（一会进行解释）`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":50,"value":"![image@1.png](${await FileAttachment(\"image@1.png\").url()})","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":53,"value":"image2 = FileAttachment(\"image@2.png\").image()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":73,"value":"上图是右手坐标系中顶点在相机空间中的情形。设***P(x,z)***是经过相机变换之后的点，视锥体由***eye(眼睛位置)***，***np(近裁剪平面)***，***fp(远裁剪平面)***组成。***N***是眼睛到近裁剪平面的距离，***F***是眼睛到远裁剪平面的距离。投影面可以选择任何平行于近裁剪平面的平面，这里我们选择近裁剪平面作为投影平面。设***P'(x',z')***是投影之后的点，则有z' = -***N***。通过相似三角形性质，我们有关系:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":110,"value":"tex.block`\\frac{x}{x'} = \\frac{z}{z'} = \\frac{z}{-N}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":113,"value":"tex.block`x' = -N\\frac{x}{z}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":125,"value":"同理:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":118,"value":"tex.block`y' = -N\\frac{y}{z}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":122,"value":"tex.block`z' = -N`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":130,"value":"***P(x, y, z)*** 映射到 ***P'(x', y', z')***:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":136,"value":"tex.block`p' = (-N\\frac{x}{z}, -N\\frac{y}{z}, -N)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":139,"value":"从上面可以看出，投影的结果z’始终等于***-N***，在投影面上。实际上，z’对于投影后的P’已经没有意义了，这个信息点已经没用了。但对于3D图形管线来说，为了便于进行后面的片元操作，例如z缓冲消隐算法，有必要把投影之前的z保存下来，方便后面使用。因此，我们利用这个没用的信息点存储z，处理成：","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":145,"value":"tex.block`p' = (-N\\frac{x}{z}, -N\\frac{y}{z}, z)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":150,"value":"这个形式最大化地使用了3个信息点，达到了最原始的投影变换的目的，但是它太直白了，有一点蛮干的意味，我感觉我们最终的结果不应该是它，你说呢？我们开始结合CVV进行思考，把它写得在数学上更优雅一致，更易于程序处理。假入能够把上面写成这个形式：","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":148,"value":"tex.block`p' = (-N\\frac{x}{z}, -N\\frac{y}{z}, -\\frac{az + b}{z})`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":156,"value":"那么我们就可以非常方便的用矩阵以及齐次坐标理论来表达投影变换：","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":158,"value":"tex.block`\\begin{bmatrix}\n N & 0 & 0 & 0 \\\\\n 0 & N & 0 & 0 \\\\\n 0 & 0 & a & b \\\\\n 0 & 0 & -1 & 0\n\\end{bmatrix}\n\\begin{bmatrix}\n x \\\\\n y \\\\\n z \\\\\n 1\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n Nx \\\\\n Ny \\\\\n az + b \\\\\n -z\n\\end{bmatrix}\n\\xrightarrow{齐次坐标转换}\n\\normalsize \\begin{bmatrix}\n -Nx/z \\\\\n -Ny/z \\\\\n -(az + b)/z \\\\\n 1\n\\end{bmatrix}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":195,"value":"影矩阵的第一个版本。你一定会问为什么要把z写成","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":197,"value":"tex.block`-\\frac{az + b}{z}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":215,"value":"md`1.后面投影之后的光栅化阶段，要通过x’和y’对z进行线性插值，以求出三角形内部片元的z，进行z缓冲深度测试。在数学上，投影后的x’和y’，与z不是线性关系，与1/z才是线性关系。而正***-(az + b) / z *** 是***1/z***的线性关系，即***-a+b/z***。用这个***1/z***的线性组合值和x’、y’进行插值才是正确的。 \n\n2.P’的3个代数分量统一地除以分母-z，易于使用齐次坐标变为普通坐标来完成，使得处理更加一致、高效。 \n\n3.后面的CVV是一个x,y,z的范围都为[-1，1]的规则体，便于进行多边形裁剪。而我们可以适当的选择系数a和b，使得***-(az + b) / z ***这个式子在z = -***N***的时候值。\n为-1，而在z = -***F***的时候值为1，从而在z方向上构建CVV。`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":238,"value":"**上面是原始作者列举的原因，但我认为主要是一个原因：由于在光栅化的过程中，要进行 Z 坐标的倒数的插值,参考透视校正插值。**","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":244,"value":"**接下来我们就求出a和b:**","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":248,"value":"tex.block`-\\frac{az + b}{z} = \\begin{cases}\n -1 &\\text{if } z = -N \\\\\n 1 &\\text{if } z = -F\n\\end{cases}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":271,"value":"tex.block`a = -\\frac{F + N}{F - N}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":274,"value":"tex.block`b = -\\frac{2FN}{F - N}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":277,"value":"这样我们就得到了透视投影矩阵的第一个版本：","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":280,"value":"tex.block`\\begin{bmatrix}\n N & 0 & 0 & 0 \\\\\n 0 & N & 0 & 0 \\\\\n 0 & 0 & a & b \\\\\n 0 & 0 & -1 & 0\n\\end{bmatrix}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":298,"value":"tex.block`a = -\\frac{F + N}{F - N}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":300,"value":"tex.block`b = -\\frac{2FN}{F - N}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":304,"value":"**使用这个版本的透视投影矩阵可以从z方向上构建CVV，但是x和y方向仍然没有限制在[-1,1]中，我们的透视投影矩阵的下一个版本就要解决这个问题。**","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":309,"value":"为了能在x和y方向把顶点从Frustum情形变成CVV情形，我们开始对x和y进行处理。先来观察我们目前得到的最终变换结果：","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":311,"value":"tex.block`\\begin{bmatrix}\n -Nx/z \\\\\n -Ny/z \\\\\n -(az + b)/z \\\\\n 1\n\\end{bmatrix}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":314,"value":"**我们知道-Nx / z的有效范围是投影平面的左边界值（记为left）和右边界值（记为right），即[left, right]，-Ny / z则为[bottom, top]。而现在我们想把-Nx / z属于[left, right]映射到x属于[-1, 1]中，-Ny / z属于[bottom, top]映射到y属于[-1, 1]中。你想到了什么？哈，就是我们简单的线性插值，你都已经掌握了！我们解决掉它：**","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":316,"value":"tex.block`\\begin{cases}\n \\frac{\\frac{-Nx}{z} - left}{right - left} = \\frac{x' - (-1)}{1 - (-1)}\\\\\n \\frac{\\frac{-Ny}{z} - bottom}{top - bottom} = \\frac{y' - (-1)}{1 - (-1)}\n\\end{cases}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":323,"value":"tex.block`\\begin{cases}\n x' = \\frac{2Nx/-z}{right - left} - \\frac{right + left}{right - left}\\\\\n y' = \\frac{2Ny/-z}{top - bottom} - \\frac{top + bottom}{top - bottom}\n\\end{cases}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":343,"value":"***最终投影点：***","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":346,"value":"tex.block`P' = \\begin{bmatrix}\n \\frac{2Nx/-z}{right - left} - \\frac{right + left}{right - left} \\\\\n \\frac{2Ny/-z}{top - bottom} - \\frac{top + bottom}{top - bottom} \\\\\n -(az + b)/z \\\\\n 1\n\\end{bmatrix}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":354,"value":"下面要做的就是从这个新形式出发反推出下一个版本的透视投影矩阵。注意到","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":360,"value":"tex.block`\\begin{bmatrix}\n -Nx/z \\\\\n -Ny/z \\\\\n -(az + b)/z \\\\\n 1\n\\end{bmatrix} \n\\rightarrow\n\\begin{bmatrix}\n Nx \\\\\n Ny \\\\\n (az + b) \\\\\n -z\n\\end{bmatrix}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":375,"value":"经过透视除法的形式，而P’只变化了x和y分量的形式，az+b和-z是不变的，则我们做透视除法的逆处理——给P’每个分量乘上-z，得到","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":369,"value":"tex.block`\\begin{bmatrix}\n \\frac{2Nx/-z}{right - left} - \\frac{right + left}{right - left} \\\\\n \\frac{2Ny/-z}{top - bottom} - \\frac{top + bottom}{top - bottom} \\\\\n -(az + b)/z \\\\\n 1\n\\end{bmatrix}\n\\rightarrow\n\\begin{bmatrix}\n \\frac{2Nx}{right - left} + \\frac{right + left}{right - left}z \\\\\n \\frac{2Ny}{top - bottom} + \\frac{top + bottom}{top - bottom}z \\\\\n az + b \\\\\n -z\n\\end{bmatrix}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":386,"value":"***最终投影矩阵:***","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":389,"value":"tex.block`\\begin{cases}\n M' = \\begin{bmatrix}\n \\frac{2N}{right - left} & 0 & \\frac{right + left}{right - left} & 0 \\\\\n 0 & \\frac{2N}{top - bottom} & \\frac{top + bottom}{top - bottom} & 0 \\\\\n 0 & 0 & a & b \\\\\n 0 & 0 & -1 & 0\n \\end{bmatrix}\\\\\n a = -\\frac{F + N}{F - N} \\\\\n b = -\\frac{2FN}{F - N}\n\\end{cases}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":399,"value":"**M就是最终的透视变换矩阵。相机空间中的顶点，如果在视锥体中，则变换后就在CVV中。如果在视锥体外，变换后就在CVV外。而CVV本身的规则性对于多边形的裁剪很有利。OpenGL在构建透视投影矩阵的时候就使用了M的形式。注意到M的最后一行不是(0 0 0 1)而是(0 0 -1 0)，因此可以看出透视变换不是一种仿射变换，它是非线性的。另外一点你可能已经想到，对于投影面来说，它的宽和高大多数情况下不同，即宽高比不为1，比如640/480。而CVV的宽高是相同的，即宽高比永远是1。这就造成了多边形的失真现象，比如一个投影面上的正方形在CVV的面上可能变成了一个长方形。解决这个问题的方法就是在对多变形进行透视变换、裁剪、透视除法之后，在归一化的设备坐标(Normalized Device Coordinates)上进行的视口(viewport)变换中进行校正，它会把归一化的顶点之间按照和投影面上相同的比例变换到视口中，从而解除透视投影变换带来的失真现象。进行校正前提就是要使投影平面的宽高比和视口的宽高比相同。**","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"}],"files":"$W46","stdlib":"1"}} 45:"$44:body"