透視投影

行列

$ M = \begin{bmatrix} \frac{1}{aspect \cdot \tan(fov/2)} & 0 & 0 & 0 \\ 0 & \frac{1}{\tan(fov/2)} & 0 & 0 \\ 0 & 0 & -\frac{far + near}{far - near} & \frac{-2 \cdot far \cdot near}{far - near} \\ 0 & 0 & -1 & 0 \end{bmatrix}

$ M \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix} = \begin{bmatrix} \frac{1}{aspect \cdot \tan(fov/2)} x \\ \frac{1}{\tan(fov/2)} y \\ -\frac{far + near}{far - near} z - \frac{2 \cdot far \cdot near}{far - near} \\ -z \end{bmatrix}

code:glsl

float aspect = resolution.x / resolution.y;

float sy = 1.0 / tan(fov / 2.0);

mat4 projectionMatrix = mat4(

sy / aspect, 0, 0, 0,

0, sy, 0, 0,

0, 0, -(far + near) / (far - near), -1,

0, 0, -2.0 * far * near / (far - near), 0

);

ベクトルで計算する

行列用意するのが面倒ならベクトルで計算してもいいよ

code:glsl

vec4 perspective(vec3 pos) {

float sy = 1.0 / tan(fov / 2.0);

float aspect = uResolution.x / uResolution.y;

return vec4(pos, 1.0) * vec4(

sy / aspect,

sy,

-(far + near) / (far - near),

0.0

) + vec4(

0.0,

0.0,

-2.0 * far * near / (far - near),

-pos.z

);

}

Depthどうなんの

$ depth = \frac{z'}{w'} = \frac{(far + near)z + 2 \cdot far \cdot near}{(far - near)z}

はぇ

far = 0.1, near = 10.0 のとき

$ depth = \frac{10.1 z + 2}{9.9 z}

わかんねー　直感的じゃねー

たすけてdesmos～

https://gyazo.com/c380d156d09ab51bf89d52a02c8a9e7d

あーーーーー

完全理解した

Depth逆変換

$ depth (far - near) z = (far + near)z + 2 \cdot far \cdot near

$ z = \frac{2 \cdot far \cdot near}{depth(far - near) - (far + near)}

オッケー

Screen Coordで1