I. Introduction

//URP pipeline used;

This part of atmospheric scattering still involves a lot of knowledge points, so I decided to write a separate article to summarize the relevant knowledge points of atmospheric scattering.

First of all, there are a few Amway articles, which are really good. I personally feel that reading these articles is enough hehe:

Realize a complete set of single atmospheric scattering from scratch

https://zhuanlan.zhihu.com/p/238048524

There are four articles in total. After reading them, you should have a good understanding of atmospheric scattering.

Since what we want to achieve here is cartoon-like rendering, we made two modifications:

Only the effect of Mie scattering is calculated;

In the optical distance part, only D(PA¯) is calculated, and the calculation of D(CP¯) is omitted;

Regarding the implementation of atmospheric scattering for cartoon rendering, we can also refer to this article:

https://blog.csdn.net/qq_41835314/article/details/128409989

2. Some analysis of ideas

2.1 (General idea) It is mainly to look at the sky from our perspective, which is the process of intersection between the human eye and the sky. If the ray is blocked by an object, it will be returned as black . If it is not blocked, a single atmospheric scattering will be returned. value ;

2.2 When calculating dpa and dpc (here we ignore the calculation of dpc), we will take multiple points between AB (see the picture below), value each point separately and accumulate them to get the final result. dpa value;

2.3

Let’s start the analysis of the code

3. Code framework

The code part mainly revolves around the following two equations:

3.1 Construct a coordinate system

First we will construct the coordinate system, as shown below. Since we use the top center of the earth as our (0,0,0) position, we do not need to adjust and convert the world coordinates. We directly use Unity's world coordinates. Correct results can be obtained;

For the explanation of the above figure, the human eye is at point A in the figure. Our ray will intersect with the atmospheric sphere at point B. AB is the direction of the ray;

Before calculating the main function, we will first find the atmospheric scattering coefficient function, phase function at different positions, and the atmospheric density coefficient function at different heights:

3.2 Intersection of rays and large balloons

The first is to calculate the intersection of the ray and the sphere. The main idea is:

1.联立视线与球体的方程，得到一个二次函数的表达式，如果这一个二次函数的表达式有解，则说明存在交点，若不存在则返回-1;

2.这里我们返回的是射线方程中的+-t的值，之后我们用rayOrigin + t * rayDir(射线方程)就可以得到两个交点的三维坐标

关于数学的分析可以参考以下的文章:

射线与球体/三角面片求交、重心坐标、插值

https://blog.csdn.net/xiji333/article/details/108581804#:~:text=%E7%90%83%E4%BD%93%E6%B1%82%E4%BA%A4%20%E4%B8%8D%E5%A6%A8%E8%AE%BE%E7%90%83%E4%BD%93%E7%9A%84%E6%96%B9%E7%A8%8B%E4%B8%BA%20%28X%20%E2%88%92%20C%292%20%3D%20R2%20%EF%BC%8C%E5%85%B6%E4%B8%AD,%E2%88%97P%20%2BP%202%20%2BC%202%20%E2%88%92R2%20%3D%200

代码如下：

//--RaySphereIntersection --首先是两个点的方向t值的计算;
//返回的是±t，rayOrigin+t*rayDir就能得到两个交点,这两个交点就是光线从与大气相交的位置传入到我们的眼睛位置的两个点;
            float2 RaySphereIntersection(float3 rayOrigin,float3 rayDir,float3 sphereCenter,float sphereRadius)
            {
                rayOrigin -= sphereCenter;//因为一开始传入的是一个点,因此这里我们减去原点的坐标得到一个向量;
                float a = dot(rayDir,rayDir);//这里是二次方程的a;
                float b = 2.0 * dot(rayOrigin,rayDir);//这里是二次方程的b;
                float c = dot(rayOrigin,rayOrigin)-(sphereRadius * sphereRadius);
                //下面是二次函数的求解;
                float d = b*b -4*a*c;
                if(d<0)
                {
                    return -1;//如果b^2 - 4ac小于01则返回,此时无解;
                }
                else
                {
                    d = sqrt(d);
                    return float2(-b-d,-b +d)/(2*a);
                    //此处最后返回的的+-t,之后我们仅需要rayOrigin+t*rayDir就能得到两个交点;
                }
            }

接下来是大气散射的密度系数函数以及相位函数的方程;

3.3大气密度函数

主要是根据公式exp(-height / H);//其中H为海平面的大气密度,height为高度为height处的大气密度，这里返回的是一个比值;

代码如下:

//1.首先我们先计算大气密度--GetAtmosphereDensity--;
              //思路:这里我们会根据position以及地表中心距离减去地球的半径来求得高度,最后我们会输出dpc以及dpa,这里我们省去了dpc;
            void GetAtmosphereDensity(float3 position,float3 plantCenter,float3 lightDir,out float dpa,out float dpc)
            {
                //计算公式为dpa = exp{-h/H};
                float height = length(position - plantCenter) - _PlanetRadius;//获得高度;
                dpa = exp(-height / _DensityScaleHeight);//求得此处的大气密度;
                dpc = 0;//dpc我们暂时省略;
            }

3.4相位函数

接下来是相位函数的计算;

相位函数代表的是在所有的散射方向中，某一个方向的比例,由于我们这里仅会计算Mie散射，因此先给出Mie散射的相位函数公式:

代码部分我们仅需要求出这个公式的值并与传入的scattering相乘即为得出最终在此方向上的sacttering；

//2.下面是相位函数的计算;--ApplyPhaseFunction--;
            //这里我们仅引入Mie相位函数!!;
            void AppltPhaseFunction(inout float scatterMie,float cosAngle)
            {
                float g = _MieG;//注意这里的g在传入的时候要将值设定为0.76,这是一个固定的值!;
                float phase = (1.0 / (4.0 * PI)) * ((3.0 * (1.0 - g*g)) / (2.0 * (2.0 + g*g))) * ((1 + cosAngle * cosAngle) / (pow((1 + g*g - 2 * g * cosAngle), 3.0 / 2.0)));
                scatterMie *= phase;
            }

3.5大气散射主函数

到此处，前置的一些方程都已经完成，接下来就是主函数的计算，在这一个函数中我们会返回最中对应的颜色值;

代码如下:

//3.到这里先前工作都完成,接下来我们要进行的是大气散射的主函数!!!!--IntegrateInscattering--;
            //此函数计算返回值就是在我们的A点得到的总能量大小!!(用返回的颜色来进行表示);
            //rayStart为摄像机的起点,也就是理论图中的A点;//rayDir为射线的方向;//raylength为AB的长度;
            half4 IntegrateInscattering(float3 rayStart, float3 rayDir,float rayLength,float3 plantCenter,float3 lightDir,float samplerCount)
            {
                //参考Mie散射的光的波长来进行赋值; 
                float4 MieSct = float4(2.0f, 2.0f, 2.0f, 0.0f) * 0.00001f;
                //光波对应的颜色值，与散射系数的乘积，这里会在最后进行的I与散射系数的相乘中用到;
                _ExtinctionM = MieSct * MieExtinctionCoef;//Mie对应波长乘以散射系数;
                _ScatteringM = MieSct * MieScatterCoef;
                
                float3 stepVector = rayDir * (rayLength / samplerCount);
                float stepSize = length(stepVector);//这里会根据samplerCount分成几个点,此处是每两个点之间的长度;

                //下面是一些代传入参数的创建;
                float prevDPA =0;//在循环前算的的Density大气密度对应值;
                float prevTransmittance = 0;//透射率方程的参数,用于存储循环前的最后投射率值;
                float densityPA = 0;//PA的大气密度;
                float densityCP = 0;//CP的大气密度;
                float localDPA = 0;//地平线处的大气密度;
                float scatterMie = 0;
                
                //下面先求T(CP) * T(PA)的部分,具体看原理,主要就是先求出两个光学深度的和再乘上地平线处的大气密度;
                GetAtmosphereDensity(rayStart,plantCenter,lightDir,localDPA,densityCP);//最后我们得到的是摄像机位置的大气密度localDensity;

                //下面是PA的大气密度,这里进行加和;
                densityPA +=stepSize * localDPA;
                prevDPA = localDPA;

                //下面求T(CP) * T(PA)透射率方程的部分;
                float Transmittance = exp(-(densityPA + densityCP)*_ExtinctionM) * localDPA;//这里获取到透射率方程部分_ExtinctionM用于调节;
                prevTransmittance = Transmittance;//获取循环计算前的透射率比例值;

                //下面进行主函数的计算;求出的是BA上无数个(samplerCount个)点经过一次散射之后的光线能量,依次加入到最后的结果;
                for(int i = 0;i<samplerCount;i+=1.0)
                {
                    float p = rayStart + stepVector * i;//获取到每一个分点的位置;
                    GetAtmosphereDensity(p,plantCenter,lightDir,localDPA,densityCP);//这里求出该点位置的大气密度;
                    densityPA +=(localDPA + prevDPA) *stepSize/2.0;//加上两点间的密度值的累加,这里取平均;
                    Transmittance = exp(-(densityPA+densityCP)* _ExtinctionM) * localDPA;
                    scatterMie +=  (prevTransmittance +Transmittance) * stepSize /2.0;//同样进行一个累加,表示该点到A的透射率;
                     //准备下一次的循环;
                    prevTransmittance = Transmittance;
                    prevDPA = densityPA;
                }
                //下面结束循环,还差三个量，相位值，散射系数，以及I;I与散射系数的乘积已经表示出来了;;
                AppltPhaseFunction(scatterMie,dot(rayDir,-lightDir.xyz));//角度为摄像机方向和光照反方向间的夹角;
                half3 lightInscatter = _ScatteringM * scatterMie;//乘上散射值与光强I的乘积，这一部分在前面有提及;;
                
                return half4(lightInscatter,1);//最后算出颜色值;
            }

关于累加项的dpa计算，我们可以在AB上取不同个数的相等距离的点p，之后对PA,PB分别求dpa dpb再相加，然后跳到下一个点上进行处理;

如图：

3.6片元着色器

最后我们在片元着色器里面传入参数即可(原文是通过cpp传入参数，这里参数我都直接放在Shader里里面)

 //Mie scattering
                float3 scatteringColor = 0;
                float3 rayStart = float3(0,10,0);//人眼观察的位置;
                rayStart.y = saturate(rayStart.y);
                float3 rayDir = normalize(i.uv.xyz);
                float3 planetCenter = float3(0, -_PlanetRadius, 0);//地球的中心;
                float2 intersection = RaySphereIntersection(rayStart,rayDir,planetCenter,_PlanetRadius + _AtmosphereHeight);//获取两个t值;
                float rayLength = intersection.y;//获取正的t,也就是rayLength;
                intersection = RaySphereIntersection(rayStart, rayDir, planetCenter, _PlanetRadius);
                //这里判断遮挡是利用射线的方向与地球来进行相交，得到两个大于0的t值;
if (intersection.x > 0)
                {
                     //没有被遮挡则可以进行光线长度的计算;
                    rayLength = min(rayLength, intersection.x*100);
                }
               //下面传入参数进行散射的计算;
                float4 inscattering = IntegrateInscattering(rayStart, rayDir, rayLength,planetCenter, -main_light.direction.xyz, 16);
                scatteringColor = _MieColor*_MieStrength * inscattering;
          //加入到最终的颜色里面;
                finalSkyColor +=scatteringColor;