【GAMES101】透视投影

原文链接：【GAMES101】透视投影

GAMES101作业1中编程实现一个三角形旋转投影的任务，期间配置环境、找Bug费了好几天的功夫。本期，推送一些注意的地方。

1 3D Perspective Projection

How to do perspective projection

• First “squish” the frustum into a cuboid $n\to n,f\to f$ ($M_{persp\to ortho}$)

• Do orthographic projection ($M_{ortho}$, already known!)

就是将Frustum拉成一个长方体，然后再利用正交投影进行变换。

注意：在右手系中，相机正对$-z$方向。这是一个坑，在编程时，应该$n=-zNear$。

由$△OAC\sim △OBD$，有
$$\{\begin{array}{r}{y}^{\prime }=\frac{n}{z}y\\ x^{\prime }=\frac{n}{z}x\end{array}。$$
在齐次坐标系中，，有
$$\left(\begin{array}{c}x\\ y\\ z\\ 1\end{array}\right)⟹\left(\begin{array}{c}\frac{n}{z}x\\ \frac{n}{z}y\\ \text{unknown}\\ 1\end{array}\right){⟹}^{\times z}\left(\begin{array}{c}nx\\ ny\\ \text{still unknown}\\ z\end{array}\right)$$
因此
KaTeX parse error: No such environment: equation* at position 8: \begin{̲e̲q̲u̲a̲t̲i̲o̲n̲*̲}̲M_{persp\righta…
$M_{persp\to ortho}$如下
$$M_{persp\to ortho}=\left(\begin{array}{cccc}n& 0& 0& 0\\ 0& n& 0& 0\\ ？& ？& ？& ？\\ 0& 0& 1& 0\end{array}\right)$$

Observation: the third row is responsible for $z’$

• Any point on the near plane will not change
• Any point’s $z$ on the far plane will not change

• Near plane

$$M_{persp\to ortho}\left(\begin{array}{c}x\\ y\\ n\\ 1\end{array}\right)=\left(\begin{array}{c}x\\ y\\ n\\ 1\end{array}\right)$$

所以$M_{persp\to ortho}$第三行为$(0,0,A,B)$

• Far plane

$$M_{persp\to ortho}\left(\begin{array}{c}0\\ 0\\ f\\ 1\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ f\\ 1\end{array}\right)$$

综上
$$\{\begin{array}{r}A{n}^2+B=n\\ A{f}^2+B=f\end{array}。$$
解得
$$\{\begin{array}{rl}A& =n+f\\ B& =-n\times f\end{array}。$$

2 Orthographic Projection

In general

• We want to map a cuboid $[l,r]\times [b,t]\times [f,n]$ to the canonical cube $[-1,1{]}^3$.

需要两部操作：

• Center cuboid by translating
• Scale into “canonical” cube

因此
$$M_{ortho}=\left(\begin{array}{cccc}\frac{2}{r-l}& 0& 0& 0\\ 0& \frac{2}{t-b}& 0& 0\\ 0& 0& \frac{2}{n-f})& 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{cccc}0& 0& 0& -\frac{r+l}{2}\\ 0& 0& 0& -\frac{t+b}{2}\\ 0& 0& 0& -\frac{n+f}{2}\\ 0& 0& 0& 1\end{array}\right)$$

3. Filed of View

Filed of View如下

How to convert from $fovY$ and aspect to $l,r,b,t$?

易得
$$\{\begin{array}{rlr}t& =\tan (\frac{fovY}{2})|n|,& b=-t\\ r& =aspect\times t,& l=-r\end{array}。$$
综上所述，透视投影矩阵
$$\begin{array}{rl}{M}_{persp}=& M_{ortho}\times M_{persp\to ortho}\\ =& \left(\begin{array}{cccc}\frac{2}{r-l}& 0& 0& 0\\ 0& \frac{2}{t-b}& 0& 0\\ 0& 0& \frac{2}{n-f})& 0\\ 0& 0& 0& 1\end{array}\right)\\ & \left(\begin{array}{cccc}0& 0& 0& -\frac{r+l}{2}\\ 0& 0& 0& -\frac{t+b}{2}\\ 0& 0& 0& -\frac{n+f}{2}\\ 0& 0& 0& 1\end{array}\right)\\ & \left(\begin{array}{cccc}n& 0& 0& 0\\ 0& n& 0& 0\\ 0& 0& n+f& -n\times f\\ 0& 0& 1& 0\end{array}\right)\end{array}$$

Eigen::Matrix4f get_model_matrix(float rotation_angle)
{
    Eigen::Matrix4f model = Eigen::Matrix4f::Identity();

    // TODO: Implement this function
    // Create the model matrix for rotating the triangle around the Z axis.
    // Then return it.

	double rotation_radian = rotation_angle * MY_PI / 180;

    model << std::cos(rotation_radian), -std::sin(rotation_radian), 0, 0,
             std::sin(rotation_radian), std::cos(rotation_radian), 0, 0,
             0, 0, 1, 0,
             0, 0, 0, 1;   
    return model;
}

Eigen::Matrix4f get_projection_matrix(float eye_fov, float aspect_ratio,
                                      float zNear, float zFar)
{
    // Students will implement this function

    Eigen::Matrix4f projection = Eigen::Matrix4f::Identity();

    // TODO: Implement this function
    // Create the projection matrix for the given parameters.
    // Then return it.

    float n = -zNear;
    float f = -zFar;
    float t = std::abs(n) * std::tan(.5f * eye_fov * MY_PI / 180);
    float b = -t;
    float r = aspect_ratio * t;
    float l = -r;
    

    Eigen::Matrix4f perspective_to_orthogonal = Eigen::Matrix4f::Identity();
    perspective_to_orthogonal << n, 0, 0, 0,
                                 0, n, 0, 0,
                                 0, 0, n + f, - n * f,
                                 0, 0, 1, 0;

    Eigen::Matrix4f orthogonal_projection = Eigen::Matrix4f::Identity();
    orthogonal_projection << 2.f/(r-l), 0, 0, 0,
                             0, 2.f/(t-b), 0, 0,
                             0, 0, 2.f/(n-f), 0,
                             0, 0, 0, 1;

    projection = orthogonal_projection * perspective_to_orthogonal;
    return projection;
}