如何计算给定坐标下相机可见的矩形大小？

Question

我制作了一个小的three.js应用程序，它将一堆圆圈从画布底部移到顶部：

let renderer, scene, light, circles, camera;

initialize();
animate();

function initialize() {
  renderer = new THREE.WebGLRenderer({ alpha: true, antialias: true });
  renderer.setSize(window.innerWidth, window.innerHeight);
  document.body.appendChild(renderer.domElement);

  scene = new THREE.Scene();

  light = new THREE.AmbientLight();
  scene.add(light);

  circles = new THREE.Group();
  scene.add(circles);

  camera = new THREE.PerspectiveCamera(45, renderer.domElement.clientWidth / renderer.domElement.clientHeight, 1);
  camera.position.z = circles.position.z + 500;
}


function animate() {
  // Update each circle.
  Array.from(circles.children).forEach(circle => {
    if (circle.position.y < visibleBox(circle.position.z).max.y) {
      circle.position.y += 4;
    } else {
      circles.remove(circle);
    }
  });

  // Create a new circle.
  let circle = new THREE.Mesh();
  circle.geometry = new THREE.CircleGeometry(30, 30);
  circle.material = new THREE.MeshToonMaterial({ color: randomColor(), transparent: true, opacity: 0.5 });
  circle.position.z = _.random(camera.position.z - camera.far, camera.position.z - (camera.far / 10));
  circle.position.x = _.random(visibleBox(circle.position.z).min.x, visibleBox(circle.position.z).max.x);
  circle.position.y = visibleBox(circle.position.z).min.y;
  circles.add(circle);

  // Render the scene.
  renderer.render(scene, camera);
  requestAnimationFrame(animate);
}

function visibleBox(z) {
 return new THREE.Box2(
   new THREE.Vector2(-1000, -1000),
   new THREE.Vector2(1000, 1000)
 );
}

function randomColor() {
  return `#${ _.sampleSize("abcdef0123456789", 6).join("")}`;
}

body {
  width: 100%;
  height: 100%;
  overflow: hidden;
}

<script src="https://cdnjs.cloudflare.com/ajax/libs/three.js/87/three.js">
</script>

<script src="https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.4/lodash.min.js">
</script>

我使用函数visibleBox(z)来确定创建和销毁每个圆圈的位置。我已经硬编码了这个函数的返回值，但是我希望它能够计算在给定深度z处相机可见的矩形的大小。

换句话说，我希望每个圆圈都精确地创建在相机平截头体的底部（上图中红色矩形的底部边缘），并且当它到达平截头体的顶部时（顶部完全被破坏）红色矩形的边缘。）

那么，我如何计算这个矩形？

Answer 1

更改此功能：

function visibleBox(z) {
    var t = Math.tan( THREE.Math.degToRad( camera.fov ) / 2 )
    var h = t * 2 * z;
    var w = h * camera.aspect;
    return new THREE.Box2(new THREE.Vector2(-w, h), new THREE.Vector2(w, -h));
}

并设置圆圈位置：

circle.position.z = _.random(-camera.near, -camera.far);
var visBox = visibleBox(circle.position.z)
circle.position.x = _.random(visBox.min.x, visBox.max.x);
circle.position.y = visBox.min.y;

代码演示：

let renderer, scene, light, circles, camera;

initialize();
animate();

function initialize() {
  renderer = new THREE.WebGLRenderer({ alpha: true, antialias: true });
  renderer.setSize(window.innerWidth, window.innerHeight);
  document.body.appendChild(renderer.domElement);

  scene = new THREE.Scene();

  light = new THREE.AmbientLight();
  scene.add(light);

  circles = new THREE.Group();
  scene.add(circles);

  camera = new THREE.PerspectiveCamera(45, renderer.domElement.clientWidth / renderer.domElement.clientHeight, 1);
  camera.position.z = circles.position.z + 500;
}


function animate() {
  // Update each circle.
  Array.from(circles.children).forEach(circle => {
    if (circle.position.y < visibleBox(circle.position.z).max.y) {
      circle.position.y += 4;
    } else {
      circles.remove(circle);
    }
  });

  // Create a new circle.
  let circle = new THREE.Mesh();
  circle.geometry = new THREE.CircleGeometry(30, 30);
  circle.material = new THREE.MeshToonMaterial({ color: randomColor(), transparent: true, opacity: 0.5 });
  circle.position.z = _.random(-(camera.near+(camera.far-camera.near)/5), -camera.far);
  var visBox = visibleBox(circle.position.z)
  circle.position.x = _.random(visBox.min.x, visBox.max.x);
  circle.position.y = visBox.min.y;
  circles.add(circle);

  // Render the scene.
  renderer.render(scene, camera);
  requestAnimationFrame(animate);
}

function visibleBox(z) {
    var t = Math.tan( THREE.Math.degToRad( camera.fov ) / 2 )
    var h = t * 2 * z;
    var w = h * camera.aspect;
    return new THREE.Box2(new THREE.Vector2(-w, h), new THREE.Vector2(w, -h));
}

function randomColor() {
  return `#${ _.sampleSize("abcdef0123456789", 6).join("")}`;
}

<script src="https://cdnjs.cloudflare.com/ajax/libs/three.js/87/three.js">
</script>

<script src="https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.4/lodash.min.js">
</script>

解释

投影矩阵描述了从场景的3D点到视口的2D点的映射。它从眼睛空间转换到剪辑空间，并且通过用剪辑坐标的w分量进行划分，将剪辑空间中的坐标转换为规范化设备坐标（NDC）。 NDC的范围是（-1，-1，-1）到（1,1,1）。

在透视投影中，深度值与相机的z距离之间的关系不是线性的 透视投影矩阵如下所示：

r = right, l = left, b = bottom, t = top, n = near, f = far

2*n/(r-l)      0              0               0
0              2*n/(t-b)      0               0
(r+l)/(r-l)    (t+b)/(t-b)    -(f+n)/(f-n)    -1    
0              0              -2*f*n/(f-n)    0

由此得出视图空间中z坐标与规范化设备坐标z分量和深度之间的关系。：

z_ndc = ( -z_eye * (f+n)/(f-n) - 2*f*n/(f-n) ) / -z_eye
depth = (z_ndc + 1.0) / 2.0

反向操作如下所示：

n = near, f = far

z_ndc = 2.0 * depth - 1.0;
z_eye = 2.0 * n * f / (f + n - z_ndc * (f - n));

如果透视投影矩阵已知，则可以按如下方式完成：

A = prj_mat[2][2]
B = prj_mat[3][2]
z_eye = B / (A + z_ndc)

请参阅How to render depth linearly in modern OpenGL with gl_FragCoord.z in fragment shader?

视图空间中的投影区域与视图空间的Z坐标之间的实现是线性的。它取决于视角和纵横比。

规范化的装备大小可以转换为视图空间中的大小，如下所示：

aspect = w / h
tanFov = tan( fov_y * 0.5 );

size_x = ndx_size_x * (tanFov * aspect) / z_eye;
size_y = ndx_size_y * tanFov / z_eye;

如果透视投影矩阵是已知的并且投影是对称的（视线位于视口的中心并且视野不会移位），则可以按如下方式进行：

size_x = ndx_size_x * / (prj_mat[0][0] * z_eye);
size_y = ndx_size_y * / (prj_mat[1][1] * z_eye);

请参阅Field of view + Aspect Ratio + View Matrix from Projection Matrix (HMD OST Calibration)

注意，归一化设备坐标中的每个位置都可以通过逆投影矩阵转换为视图空间坐标：

mat4 inversePrjMat = inverse( prjMat );
vec4 viewPosH      = inversePrjMat * vec3( ndc_x, ndc_y, 2.0 * depth - 1.0, 1.0 );
vec3 viewPos       = viewPos.xyz / viewPos.w;

请参阅How to recover view space position given view space depth value and ndc xy

这意味着具有特定深度的未投影矩形可以这样计算：

vec4 viewLowerLeftH  = inversePrjMat * vec3( -1.0, -1.0, 2.0 * depth - 1.0, 1.0 );
vec4 viewUpperRightH = inversePrjMat * vec3(  1.0,  1.0, 2.0 * depth - 1.0, 1.0 );
vec3 viewLowerLeft   = viewLowerLeftH.xyz / viewLowerLeftH.w;
vec3 viewUpperRight  = viewUpperRightH.xyz / viewUpperRightH.w;