AE 脚本进阶：生成大量粒子扩散

Utilities?

正文前面先说个技巧。脚本如何模块化？在前两个案例中，我们写了一些辅助函数（如清空子对象、随机数）。那么，如何在多个脚本中复用呢？

在早期的 js 中，没有模块的概念。事实上，如果有为 html 写 js 的经历，就会发现，几个 js 文件根本不需要设置依赖关系就能互相调用。这个也可以理解为，把所有代码 include 到了全局代码块。

但这个特性在 Adobe ExtendScript（AES）并不适用。直接把公共函数写在同目录其他文件中，符号并不会注入到全局命名空间被找到。AES 提供了一个模块化的方法：

#include "utils.js";

是的，你没看错，这是 js 不是 c……
这种语法就带来了一个问题，会让 prettier 之类的插件需要用的 parser 宕机。

需要注意，AE 执行脚本是有后效性的！一个符号只要通过运行加入了符号表，那么它会在生命周期里一直存在，即使从脚本代码中删掉。

Shape! Shape!

我们需要绘制大量的圆，而圆需要由 Shape 来承载。我们来看看 Shape 的描述：

可以由如下代码创建一个空图形对象。

comp.layers.addShape();

World of Properties

神说，要有圆。

下面我们研究属性。图层是由属性树（Property Tree）描述的，变换（transform）中的位置、旋转都是属性。我们来看看这个圆有哪些属性。

但是得把这个对应到代码上。我们来看看它的属性树长什么样。layer 本身就是一个 PropertyGroup，可以直接获取它的属性。但是我们不知道树的结构！（这个树跟 GUI 上看到的未必一样）

但是，好在我们可以遍历 group 的所有属性，于是有：

function debugProperties(obj) {
  for (var i = 1; i <= obj.numProperties; i++) {
    var prop = obj.property(i);
    prop; // Break here
  }
}

获取完 property 后可以断点查看它的具体数据，尤其是 name 和 matchNamed 域，这有助于我们直接获取该 property。简单测试一下可以得到“内容”的标识名为 ADBE Root Vectors Group。

根据试探，我们可以代码描述出这样的层次……

function test() {
  var layer = app.project.activeItem.layer(1);
  var vectors = layer.property("ADBE Root Vectors Group");
  var eclipse = vectors.property(1); // type: ADBE Vector Group; Name: 椭圆 1
  var transform = eclipse.property("ADBE Vector Transform Group"); // 变换
  debugProperties(transform);
  var content = eclipse.property("ADBE Vectors Group"); // 内容
  var path = content.property("ADBE Vector Shape - Ellipse"); // 椭圆路径 1
}

但是这样依然不够直观，太累了，不能一下子找到想要的属性。可以写个函数，把所有属性的名称和值转字符串。

function reprProperties(obj) {
  var res = "";
  for (var i = 1; i <= obj.numProperties; i++) {
    var prop = obj.property(i);
    res += prop.matchName + ": " + prop.name + " = " + prop.value.toString() + "\n";
  }
  return res;
}

比如椭圆形状的变换属性为：

ADBE Vector Anchor: 锚点 = 0,0
ADBE Vector Position: 位置 = -104.599982776641,14.4000132179264
ADBE Vector Scale: 比例 = 26,26
ADBE Vector Skew: 倾斜 = 0
ADBE Vector Skew Axis: 倾斜轴 = 0
ADBE Vector Rotation: 旋转 = 0
ADBE Vector Group Opacity: 不透明度 = 100

AES 不支持在控制台输出，但在 VSCode 调试时的 VARIABLES 区域可以复制该字符串。

这样还不够！能否一步到位生成树？

function reprPropertyTree(obj) {
  var res = "";
  function impl(pg, indent) {
    for (var i = 0; i < indent; i++) res += "  ";
    var val =
      pg.propertyValueType == PropertyValueType.NO_VALUE || !pg.value
        ? ""
        : " = <" + pg.value.__proto__.constructor.name + "> " + pg.value.toString();
    res += pg.matchName + " [" + pg.name + "]: " + pg.constructor.name + val + "\n";
    for (var i = 1; i <= pg.numProperties; i++) {
      var prop = pg.property(i);
      impl(prop, indent + 1);
    }
  }
  impl(obj, 0);
  return res;
}

这里有几点需要说明。

1. 判断是否有值，如果有值则输出值。值可以用 prop.value 获取，但如果没值却获取了，会 throw，并且提示你需要先用 PropertyValueType.NO_VALUE 来判断是否有值。有值的情况下也可能是空值。
2. 获取属性值的值和类型。在 ES3 没有 Object.getPrototypeOf。而它可能是值类型，没有 constructor。但是可以用 value.__proto__ 来获取原型。于是有：pg.value.__proto__.constructor.name。
3. 获取属性的类型。这一定是 Object 类型，所以可以直接获取它的 constructor.name。其实没啥必要，毕竟叶结点都是 Property，非叶结点和根结点都是 PropertyGroup。

对于这样的一个圆，可能有以下属性（大小和位置后来更改了，与图中不一样）：

ADBE Vector Layer [形状图层 1]: ShapeLayer
  ADBE Marker [标记]: Property = <Object> [object Object]
  ADBE Root Vectors Group [内容]: PropertyGroup
    ADBE Vector Group [椭圆 1]: PropertyGroup
      ADBE Vector Blend Mode [混合模式]: Property = <Number> 1
      ADBE Vectors Group [内容]: PropertyGroup
        ADBE Vector Shape - Ellipse [椭圆路径 1]: PropertyGroup
          ADBE Vector Shape Direction [形状方向]: Property = <Number> 1
          ADBE Vector Ellipse Size [大小]: Property = <Array> 110.400002632141,110.400002632141
          ADBE Vector Ellipse Position [位置]: Property = <Array> 0,0
        ADBE Vector Graphic - Stroke [描边 1]: PropertyGroup
          ADBE Vector Blend Mode [混合模式]: Property = <Number> 1
          ADBE Vector Composite Order [合成]: Property = <Number> 1
          ADBE Vector Stroke Color [颜色]: Property = <Array> 1,1,1,1
          ADBE Vector Stroke Opacity [不透明度]: Property = <Number> 100
          ADBE Vector Stroke Width [描边宽度]: Property = <Number> 10
          ADBE Vector Stroke Line Cap [线段端点]: Property = <Number> 1
          ADBE Vector Stroke Line Join [线段连接]: Property = <Number> 1
          ADBE Vector Stroke Miter Limit [尖角限制]: Property = <Number> 4
          ADBE Vector Stroke Dashes [虚线]: PropertyGroup
            ADBE Vector Stroke Dash 1 [虚线]: Property = <Number> 10
            ADBE Vector Stroke Gap 1 [间隙]: Property = <Number> 10
            ADBE Vector Stroke Dash 2 [虚线 2]: Property = <Number> 10
            ADBE Vector Stroke Gap 2 [间隙 2]: Property = <Number> 10
            ADBE Vector Stroke Dash 3 [虚线 3]: Property = <Number> 10
            ADBE Vector Stroke Gap 3 [间隙 3]: Property = <Number> 10
            ADBE Vector Stroke Offset [偏移]: Property
          ADBE Vector Stroke Taper [锥度]: PropertyGroup
            ADBE Vector Taper Length Units [长度单位]: Property = <Number> 1
            ADBE Vector Taper Start Length [起始长度]: Property
            ADBE Vector Taper End Length [结束长度]: Property
            ADBE Vector Taper StartWidthPx [起始长度]: Property
            ADBE Vector Taper EndWidthPx [结束长度]: Property
            ADBE Vector Taper Start Width [开始宽度]: Property
            ADBE Vector Taper End Width [末端宽度]: Property
            ADBE Vector Taper Start Ease [开始缓和]: Property
            ADBE Vector Taper End Ease [结束缓和]: Property
          ADBE Vector Stroke Wave [波形]: PropertyGroup
            ADBE Vector Taper Wave Amount [数量]: Property
            ADBE Vector Taper Wave Units [单位]: Property = <Number> 1
            ADBE Vector Taper Wavelength [波长]: Property = <Number> 100
            ADBE Vector Taper Wave Cycles [环形]: Property = <Number> 10
            ADBE Vector Taper Wave Phase [相位]: Property
        ADBE Vector Graphic - Fill [填充 1]: PropertyGroup
          ADBE Vector Blend Mode [混合模式]: Property = <Number> 1
          ADBE Vector Composite Order [合成]: Property = <Number> 1
          ADBE Vector Fill Rule [填充规则]: Property = <Number> 1
          ADBE Vector Fill Color [颜色]: Property = <Array> 0.58276235543045,0.75102203967525,0.95137257295496,1
          ADBE Vector Fill Opacity [不透明度]: Property = <Number> 100
      ADBE Vector Transform Group [变换]: PropertyGroup
        ADBE Vector Anchor [锚点]: Property = <Array> 0,0
        ADBE Vector Position [位置]: Property = <Array> -104.599982776641,14.4000132179264
        ADBE Vector Scale [比例]: Property = <Array> 26,26
        ADBE Vector Skew [倾斜]: Property
        ADBE Vector Skew Axis [倾斜轴]: Property
        ADBE Vector Rotation [旋转]: Property
        ADBE Vector Group Opacity [不透明度]: Property = <Number> 100
      ADBE Vector Materials Group [材质选项]: PropertyGroup
        ADBE Vec3D Front RGB [正面颜色]: Property = <Array> 1,0,0,1
        ADBE Vec3D Front Ambient [正面环境]: Property = <Number> 100
        ADBE Vec3D Front Diffuse [正面漫射]: Property = <Number> 50
        ADBE Vec3D Front Specular [正面镜面强度]: Property = <Number> 50
        ADBE Vec3D Front Shininess [正面镜面反光度]: Property = <Number> 5
        ADBE Vec3D Front Metal [正面金属质感]: Property = <Number> 100
        ADBE Vec3D Front Reflection [正面反射强度]: Property
        ADBE Vec3D Front Gloss [正面反射锐度]: Property = <Number> 100
        ADBE Vec3D Front Fresnel [正面反射衰减]: Property
        ADBE Vec3D Front Xparency [正面透明度]: Property
        ADBE Vec3D Front XparRoll [正面透明度衰减]: Property
        ADBE Vec3D Front IOR [正面折射率]: Property = <Number> 1
        ADBE Vec3D Bevel RGB [斜面颜色]: Property = <Array> 1,0,0,1
        ADBE Vec3D Bevel Ambient [斜面环境]: Property = <Number> 100
        ADBE Vec3D Bevel Diffuse [斜面漫射]: Property = <Number> 50
        ADBE Vec3D Bevel Specular [斜面镜面强度]: Property = <Number> 50
        ADBE Vec3D Bevel Shininess [斜面镜面反光度]: Property = <Number> 5
        ADBE Vec3D Bevel Metal [斜面金属质感]: Property = <Number> 100
        ADBE Vec3D Bevel Reflection [斜面反射强度]: Property
        ADBE Vec3D Bevel Gloss [斜面反射锐度]: Property = <Number> 100
        ADBE Vec3D Bevel Fresnel [斜面反射衰减]: Property
        ADBE Vec3D Bevel Xparency [斜面透明度]: Property
        ADBE Vec3D Bevel XparRoll [斜面透明度衰减]: Property
        ADBE Vec3D Bevel IOR [斜面折射率]: Property = <Number> 1
        ADBE Vec3D Side RGB [侧面颜色]: Property = <Array> 1,0,0,1
        ADBE Vec3D Side Ambient [侧面环境]: Property = <Number> 100
        ADBE Vec3D Side Diffuse [侧面漫射]: Property = <Number> 50
        ADBE Vec3D Side Specular [侧面镜面强度]: Property = <Number> 50
        ADBE Vec3D Side Shininess [侧面镜面反光度]: Property = <Number> 5
        ADBE Vec3D Side Metal [侧面金属质感]: Property = <Number> 100
        ADBE Vec3D Side Reflection [侧面反射强度]: Property
        ADBE Vec3D Side Gloss [侧面反射锐度]: Property = <Number> 100
        ADBE Vec3D Side Fresnel [侧面反射衰减]: Property
        ADBE Vec3D Side Xparency [侧面透明度]: Property
        ADBE Vec3D Side XparRoll [侧面透明度衰减]: Property
        ADBE Vec3D Side IOR [侧面折射率]: Property = <Number> 1
        ADBE Vec3D Back RGB [背面颜色]: Property = <Array> 1,0,0,1
        ADBE Vec3D Back Ambient [背面环境]: Property = <Number> 100
        ADBE Vec3D Back Diffuse [背面漫射]: Property = <Number> 50
        ADBE Vec3D Back Specular [背面镜面强度]: Property = <Number> 50
        ADBE Vec3D Back Shininess [背面镜面反光度]: Property = <Number> 5
        ADBE Vec3D Back Metal [背面金属质感]: Property = <Number> 100
        ADBE Vec3D Back Reflection [背面反射强度]: Property
        ADBE Vec3D Back Gloss [背面反射锐度]: Property = <Number> 100
        ADBE Vec3D Back Fresnel [背面反射衰减]: Property
        ADBE Vec3D Back Xparency [背面透明度]: Property
        ADBE Vec3D Back XparRoll [背面透明度衰减]: Property
        ADBE Vec3D Back IOR [背面折射率]: Property = <Number> 1
  ADBE Mask Parade [蒙版]: PropertyGroup
  ADBE Effect Parade [效果]: PropertyGroup
  ADBE Transform Group [变换]: PropertyGroup
    ADBE Anchor Point [锚点]: Property = <Array> 0,0,0
    ADBE Position [位置]: Property = <Array> 960,540,0
    ADBE Position_0 [X 位置]: Property
    ADBE Position_1 [Y 位置]: Property
    ADBE Position_2 [Z 位置]: Property
    ADBE Scale [缩放]: Property = <Array> 203,203,203
    ADBE Orientation [方向]: Property = <Array> 0,0,0
    ADBE Rotate X [X 轴旋转]: Property
    ADBE Rotate Y [Y 轴旋转]: Property
    ADBE Rotate Z [Z 轴旋转]: Property
    ADBE Opacity [不透明度]: Property = <Number> 100
    ADBE Envir Appear in Reflect [在反射中显示]: Property = <Number> 1
  ADBE Layer Styles [图层样式]: PropertyGroup
    ADBE Blend Options Group [混合选项]: PropertyGroup
      ADBE Global Angle2 [全局光角度]: Property = <Number> 120
      ADBE Global Altitude2 [全局光高度]: Property = <Number> 30
      ADBE Adv Blend Group [高级混合]: PropertyGroup
        ADBE Layer Fill Opacity2 [填充不透明度]: Property = <Number> 100
        ADBE R Channel Blend [红色]: Property = <Number> 1
        ADBE G Channel Blend [绿色]: Property = <Number> 1
        ADBE B Channel Blend [蓝色]: Property = <Number> 1
        ADBE Blend Interior [混合内部样式为组]: Property
        ADBE Blend Ranges [使用源的混合范围]: Property = <Number> 1
    dropShadow/enabled [投影]: PropertyGroup
      dropShadow/mode2 [混合模式]: Property = <Number> 5
      dropShadow/color [颜色]: Property = <Array> 0,0,0,1
      dropShadow/opacity [不透明度]: Property = <Number> 75
      dropShadow/useGlobalAngle [使用全局光]: Property
      dropShadow/localLightingAngle [角度]: Property = <Number> 120
      dropShadow/distance [距离]: Property = <Number> 5
      dropShadow/chokeMatte [扩展]: Property
      dropShadow/blur [大小]: Property = <Number> 5
      dropShadow/noise [杂色]: Property
      dropShadow/layerConceals [图层镂空投影]: Property = <Number> 1
    innerShadow/enabled [内阴影]: PropertyGroup
      innerShadow/mode2 [混合模式]: Property = <Number> 5
      innerShadow/color [颜色]: Property = <Array> 0,0,0,1
      innerShadow/opacity [不透明度]: Property = <Number> 75
      innerShadow/useGlobalAngle [使用全局光]: Property
      innerShadow/localLightingAngle [角度]: Property = <Number> 120
      innerShadow/distance [距离]: Property = <Number> 5
      innerShadow/chokeMatte [阻塞]: Property
      innerShadow/blur [大小]: Property = <Number> 5
      innerShadow/noise [杂色]: Property
    outerGlow/enabled [外发光]: PropertyGroup
      outerGlow/mode2 [混合模式]: Property = <Number> 11
      outerGlow/opacity [不透明度]: Property = <Number> 75
      outerGlow/noise [杂色]: Property
      outerGlow/AEColorChoice [颜色类型]: Property = <Number> 1
      outerGlow/color [颜色]: Property = <Array> 1,1,0.74509803921569,1
      outerGlow/gradient [颜色]: Property
      outerGlow/gradientSmoothness [渐变平滑度]: Property = <Number> 100
      outerGlow/glowTechnique [技术]: Property = <Number> 1
      outerGlow/chokeMatte [扩展]: Property
      outerGlow/blur [大小]: Property = <Number> 5
      outerGlow/inputRange [范围]: Property = <Number> 50
      outerGlow/shadingNoise [抖动]: Property
    innerGlow/enabled [内发光]: PropertyGroup
      innerGlow/mode2 [混合模式]: Property = <Number> 11
      innerGlow/opacity [不透明度]: Property = <Number> 75
      innerGlow/noise [杂色]: Property
      innerGlow/AEColorChoice [颜色类型]: Property = <Number> 1
      innerGlow/color [颜色]: Property = <Array> 1,1,0.74509803921569,1
      innerGlow/gradient [颜色]: Property
      innerGlow/gradientSmoothness [渐变平滑度]: Property = <Number> 100
      innerGlow/glowTechnique [技术]: Property = <Number> 1
      innerGlow/innerGlowSource [源]: Property = <Number> 1
      innerGlow/chokeMatte [阻塞]: Property
      innerGlow/blur [大小]: Property = <Number> 5
      innerGlow/inputRange [范围]: Property = <Number> 50
      innerGlow/shadingNoise [抖动]: Property
    bevelEmboss/enabled [斜面和浮雕]: PropertyGroup
      bevelEmboss/bevelStyle [样式]: Property = <Number> 2
      bevelEmboss/bevelTechnique [技术]: Property = <Number> 1
      bevelEmboss/strengthRatio [深度]: Property = <Number> 100
      bevelEmboss/bevelDirection [方向]: Property = <Number> 1
      bevelEmboss/blur [大小]: Property = <Number> 5
      bevelEmboss/softness [柔化]: Property
      bevelEmboss/useGlobalAngle [使用全局光]: Property
      bevelEmboss/localLightingAngle [角度]: Property = <Number> 120
      bevelEmboss/localLightingAltitude [高度]: Property = <Number> 30
      bevelEmboss/highlightMode [高亮模式]: Property = <Number> 11
      bevelEmboss/highlightColor [加亮颜色]: Property = <Array> 1,1,1,1
      bevelEmboss/highlightOpacity [高光不透明度]: Property = <Number> 75
      bevelEmboss/shadowMode [阴影模式]: Property = <Number> 5
      bevelEmboss/shadowColor [阴影颜色]: Property = <Array> 0,0,0,1
      bevelEmboss/shadowOpacity [阴影不透明度]: Property = <Number> 75
    chromeFX/enabled [光泽]: PropertyGroup
      chromeFX/mode2 [混合模式]: Property = <Number> 5
      chromeFX/color [颜色]: Property = <Array> 0,0,0,1
      chromeFX/opacity [不透明度]: Property = <Number> 50
      chromeFX/localLightingAngle [角度]: Property = <Number> 19
      chromeFX/distance [距离]: Property = <Number> 11
      chromeFX/blur [大小]: Property = <Number> 14
      chromeFX/invert [反转]: Property = <Number> 1
    solidFill/enabled [颜色叠加]: PropertyGroup
      solidFill/mode2 [混合模式]: Property = <Number> 1
      solidFill/color [颜色]: Property = <Array> 1,0,0,1
      solidFill/opacity [不透明度]: Property = <Number> 100
    gradientFill/enabled [渐变叠加]: PropertyGroup
      gradientFill/mode2 [混合模式]: Property = <Number> 1
      gradientFill/opacity [不透明度]: Property = <Number> 100
      gradientFill/gradient [颜色]: Property
      gradientFill/gradientSmoothness [渐变平滑度]: Property = <Number> 100
      gradientFill/angle [角度]: Property = <Number> 90
      gradientFill/type [样式]: Property = <Number> 1
      gradientFill/reverse [反向]: Property
      gradientFill/align [与图层对齐]: Property = <Number> 1
      gradientFill/scale [缩放]: Property = <Number> 100
      gradientFill/offset [偏移]: Property = <Array> 0,0
    patternFill/enabled [图案叠加]: PropertyGroup
      patternFill/mode2 [混合模式]: Property = <Number> 1
      patternFill/opacity [不透明度]: Property = <Number> 100
      patternFill/align [与图层链接]: Property = <Number> 1
      patternFill/scale [缩放]: Property = <Number> 100
      patternFill/phase [偏移]: Property = <Array> 0,0
    frameFX/enabled [描边]: PropertyGroup
      frameFX/mode2 [混合模式]: Property = <Number> 1
      frameFX/color [颜色]: Property = <Array> 1,0,0,1
      frameFX/size [大小]: Property = <Number> 3
      frameFX/opacity [不透明度]: Property = <Number> 100
      frameFX/style [位置]: Property = <Number> 1
  ADBE Extrsn Options Group [几何选项]: PropertyGroup
    ADBE Bevel Styles [斜面样式]: Property = <Number> 1
    ADBE Bevel Direction [斜面方向]: Property = <Number> 1
    ADBE Bevel Depth [斜面深度]: Property = <Number> 2
    ADBE Hole Bevel Depth [洞斜面深度]: Property = <Number> 100
    ADBE Extrsn Depth [凸出深度]: Property
  ADBE Material Options Group [材质选项]: PropertyGroup
    ADBE Casts Shadows [投影]: Property
    ADBE Light Transmission [透光率]: Property
    ADBE Accepts Shadows [接受阴影]: Property = <Number> 1
    ADBE Accepts Lights [接受灯光]: Property = <Number> 1
    ADBE Appears in Reflections [在反射中显示]: Property = <Number> 1
    ADBE Ambient Coefficient [环境]: Property = <Number> 100
    ADBE Diffuse Coefficient [漫射]: Property = <Number> 50
    ADBE Specular Coefficient [镜面强度]: Property = <Number> 50
    ADBE Shininess Coefficient [镜面反光度]: Property = <Number> 5
    ADBE Metal Coefficient [金属质感]: Property = <Number> 100
    ADBE Reflection Coefficient [反射强度]: Property
    ADBE Glossiness Coefficient [反射锐度]: Property = <Number> 100
    ADBE Fresnel Coefficient [反射衰减]: Property
    ADBE Transparency Coefficient [透明度]: Property
    ADBE Transp Rolloff [透明度衰减]: Property
    ADBE Index of Refraction [折射率]: Property = <Number> 1
  ADBE Audio Group [音频]: PropertyGroup
    ADBE Audio Levels [音频电平]: Property = <Array> 0,0
  ADBE Layer Sets [集]: PropertyGroup

More Shapes!

如果从头构建属性，会十分麻烦，因此这里采用原型模式。将之前手动创建好的那个椭圆复制多份。由于每个粒子的生命周期是独立的，因此还是分图层比较好。事实上，单个图层里的属性是没法直接复制的，虽然手动创建完整的属性树并不麻烦，比如：

function test() {
  var layer = app.project.activeItem.layer(1);
  var vectors = layer.property("ADBE Root Vectors Group");
  var eclipse2 = vectors.addProperty("ADBE Vector Group");
  eclipse2
    .property("ADBE Vectors Group")
    .addProperty("ADBE Vector Shape - Ellipse");
  eclipse2
    .property("ADBE Vectors Group")
    .addProperty("ADBE Vector Graphic - Fill");
  return;
}

function initLayer(comp, layer) {
  var theta = randf(0, Math.PI * 2);
  const preRange = randf(0, 50);
  const maxTime = 3;
  const speed = 100;
  var dieTime = randf(2, 3);
  var center = [comp.width / 2, comp.height / 2];
  var x0 = Math.cos(theta) * preRange,
    y0 = Math.sin(theta) * preRange;
  var x1 = Math.cos(theta) * speed * maxTime,
    y1 = Math.sin(theta) * speed * maxTime;
  layer.position.setValueAtTime(0, center + [x0, y0]);
  layer.position.setValueAtTime(maxTime, center + [x1, y1]);
  layer.inPoint = 0;
  layer.outPoint = dieTime;
}

在圆多的情况下脚本会执行较长时间，且单帧渲染时间较长。

Electron Cloud

但是上面的太普通了，我们想要粒子模拟出电子云。

众所周知电子云可以用 $n,l,m$ 这 3 个参数来描述。为简化模型，下面我们只考虑基态氢原子 $n=1, l=0, m=0$。

可视化方案有很多，这里采用的是用点密度来表示概率密度。

我们小学二年级时已经学过，单位立体角概率

$$w_{00} = \left| Y_0^0 \right|^2 = 1/4 \pi$$

这也就说，任何方向的概率相同。

径向位置概率密度

$$w_{10}(r) = \chi_{nl}^2(r) = \frac{4}{a_0^3} r^2 \mathrm{e}^{-2r/a_0}$$

现在来根据这个 PDF 概率密度函数来生成随机半径。

积分得 CDF

$$F(r) = 1 - \frac{a_0^2 + 2a_0 r + 2r^2}{a_0^2}\mathrm{e}^{-2r/a_0}, ~r>0$$

取随机数 $R \sim U(0,1)$，令 $R = F(r)$，则反解出 $r = \dots$。行吧，解不了。

下面采取 Acceptance-Rejection 采样法。

设 $f$ 为 PDF，定义域为 $[a, b]$，值域为 $[0, m]$。随机生成 $x \sim U(a,b)$，$y \sim U(0, m)$，若 $y \le f(x)$，则接受采用，否则拒绝并重新生成 $x,y$。

根据函数图像可以设置 $a=0,~b=3,~m=1.023$，当然 $b$ 和 $m$ 设置稍大也可，影响不大。

下面就是生成随机点了，没什么好解释的。包括径向随机点和球面随机点。

function randomR() {
  function f(r) {
    const a = 0.529;
    return (4 / (a * a * a)) * r * r * Math.exp((-2 * r) / a);
  }
  const m = 1.023;
  do {
    var x = randf(0, 3),
      y = randf(0, m);
    if (y <= f(x)) return x;
  } while (true);
}
function randomPoint() {
  var r = randomR();
  var phi = randf(0, 2 * Math.PI);
  var theta = Math.acos(randf(-1, 1));
  return [r, theta, phi];
}
function spherical(r, theta, phi) {
  return (
    r *
    [
      Math.sin(theta) * Math.cos(phi),
      Math.sin(theta) * Math.sin(phi),
      Math.cos(theta),
    ]
  );
}

最后的调用部分。这里将半径放大 40 倍，速度和到原点距离成线性关系，运动轨迹比较随意了。

function initLayer(comp, layer) {
  const maxTime = 1.5;
  var dieTime = randf(1.0, 1.5);
  var center = [comp.width / 2, comp.height / 2];

  var pos = randomPoint();
  var r = pos[0],
    theta = pos[1],
    phi = pos[2];
  var startPos = spherical(r * 40, theta, phi);
  layer.position.setValueAtTime(0, center + startPos);
  layer.position.setValueAtTime(maxTime, center + startPos * 16);
  layer.inPoint = 0;
  layer.outPoint = dieTime;
  layer.position.setInterpolationTypeAtKey(1, KeyframeInterpolationType.BEZIER);
}

上面最后一行设置关键帧插值，开始帧使用二次插值。

运行效果如下。