【光线追迹】基于龙格库塔法实现光线追迹附Matlab代码

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本文详细介绍了如何在光线追踪中使用龙格-库塔法来求解光线路径的微分方程，强调了这种方法的高精度、稳定性和效率，同时也讨论了其在计算成本和自适应步长上的挑战。

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🔥 内容介绍

光线追踪是一种计算机图形技术，它模拟光线在场景中的传播，以生成逼真的图像。它通过计算光线从光源发出，与场景中的物体交互，最终到达观察者的路径来实现。

龙格-库塔法

龙格-库塔法是一种求解常微分方程的数值方法。它通过使用一系列中间步骤来逼近方程的解，从而提高了计算的精度和效率。

光线追踪中的龙格-库塔法

在光线追踪中，光线路径可以用常微分方程来表示：

dx/dt = v(x, t)

其中：

x 是光线的位置
t 是时间
v 是光线的速度

使用龙格-库塔法，我们可以将该方程离散化为一系列步骤：

k1 = v(x, t) k2 = v(x + h * k1 / 2, t + h / 2) k3 = v(x + h * k2 / 2, t + h / 2) k4 = v(x + h * k3, t + h) x = x + h * (k1 + 2 * k2 + 2 * k3 + k4) / 6

其中：

h 是时间步长

实现

我们可以使用以下步骤来实现基于龙格-库塔法的【光线追踪】：

初始化光线位置和方向
循环迭代，直到光线到达场景中的某个物体或超过最大步数
在每个迭代中，使用龙格-库塔法更新光线位置
检查光线是否与场景中的任何物体相交
如果相交，计算光线与物体的交互，并根据材料属性更新光线方向
重复步骤 3-5，直到光线到达观察者或超过最大步数

优点

使用龙格-库塔法进行【光线追踪】具有以下优点：

**高精度：**龙格-库塔法是一种高阶方法，可以提供比低阶方法更高的精度。
**稳定性：**龙格-库塔法是一种稳定的方法，这意味着即使时间步长较大，它也能产生合理的解。
**效率：**龙格-库塔法是一种高效的方法，因为它只需要少量中间步骤即可获得高精度。

局限性

龙格-库塔法也有一些局限性，包括：

**计算成本：**龙格-库塔法需要比低阶方法更多的计算量。
**自适应步长：**龙格-库塔法不能自动调整时间步长，因此可能需要手动调整以获得最佳性能。

结论

基于龙格-库塔法的【光线追踪】是一种功能强大且准确的技术，可用于生成逼真的图像。它的高精度、稳定性和效率使其成为各种图形应用程序的理想选择。然而，需要考虑其计算成本和自适应步长的局限性。

📣 部分代码

clear;TIR = 0;in = [1, 0, 1]; %Direction vector for incident rayix = [0; 0; 0]; %Position vector for ray on first planen = [0, 0, -1]; %Unit normal of first planeray_energy = 1; %Preset energy of the incident ray%The medium variable "med_var" keeps track of which medium the ray is%currently travelling through. When med_var = 1, the ray is currently in%medium 1 whereas when med_var = -1 the ray is currently travelling in%medium 2.med_var = 1;%The direction variable "dir_var" stores the z-direction in which the ray %is travelling. When dir_var = 1, the ray being followed is travelling in%the positive z-direction otherwise it is travelling in the negative%z-direction. dir_var = 1;%The lambda parameter controls the slope of the exponential%distribution from which the distances between successive interfaces%are randomly chosenlambda = 1;r_z = [0, 0, 1]; %General ray along the z-axis%Refractive indices of the two median1 = 1;n2 = 1.1;%Calculating the critical angletheta_c = asind(n1/n2);%Normalising the incident ray direction vectorin_unit = in/norm(in);i = 0;while ix(3)>=0        i = i+1;        ix_m(i,:) = ix;        if i == 1                r_z = in_unit;            else                r_z = [0,0,dir_var];            end        th_m1 = acosd(dot(r_z, -sign(r_z(3))*n)); %Calculating the incident angle        %Outer if statement checks which media the refracted ray is travelling    %through wih each iteration.    %The inner if statements check for total internal reflection by checking    %whether the refracted ray returned is complex.    %NB: Total internal reflection can only occur when the light ray is    %travelling from a more dense medium to a less dense medium        rec_v = 0;        if med_var == -1                rf = refract(r_z, sign(r_z(3))*n, n2, n1);                if(isreal(rf) == 0 && th_m1>theta_c)                        rf = [0,0,0];                        TIR = i;                        rec_v = 1;                    end                    else                rf = refract(r_z, sign(r_z(3))*n, n1, n2);            end        rl = reflect(r_z, sign(r_z(3))*n);        if rec_v == 1                th_m2 = -acosd(dot(rf,-sign(rf(3))*n));            else                th_m2 = acosd(dot(rf,-sign(rf(3))*n)); %Calculating the refraction angle            end        %Calculating the reflection coefficient at each interface    rs = ((n1*cosd(th_m1) - n2*cosd(th_m2))/(n1*cosd(th_m1) + n2*cosd(th_m2)))^2;    rp = ((n1*cosd(th_m2) - n2*cosd(th_m1))/(n1*cosd(th_m2) + n2*cosd(th_m1)))^2;    R_coef = (rp+rs)/2;        ray_energy = ray_energy*R_coef;        %Randomising the distance between successive interfaces over an exponential    %distribution    sy = rand(1);        s = (-log(sy))/lambda;        if i == 1                %Calculating the rotation matrix        R_mat = vecRotMat(rf,r_z);                ix = ix + s*rf';                rf_m(1,:) = rf;                med_var = -med_var;            else                        %By comparing a randomly generated number to the reflection coefficent at        %each interface a decision is made as to whether the reflected or the        %refracted ray willl be followed.        if ((rand(1) <= R_coef) || (rec_v == 1))                        rl_r = R_mat'*rl';                        rl = rl_r';                        %Using the reflected vector            ix = ix + s*rl';                        R_mat = vecRotMat(rl,r_z);                        dir_var = sign(rl(3));                    else            med_var = -med_var;                        rf_r = R_mat'*rf';                        rf = rf_r';                        %Using the refracted vector            ix = ix + s*rf';                        R_mat = vecRotMat(rf,r_z);                        dir_var = sign(rf(3));                    end            end        %Using the George Marsaglia method to generate random normal vectors    %distributed evenly over the surface of a half-sphere    S_n = 2;        while S_n>1                n_3 = -1 + rand(1,1);        n_1 = -1 + 2*rand(1,1);                S_n = n_3^2 + n_1^2;        n(2) = 1 - 2*S_n;            end        n(3) = (2*n_3)*sqrt((1-S_n));    n(1) = (2*n_1)*sqrt((1-S_n));    n(2) = 1 - 2*S_n;        endstr3 = sprintf('The ray exited the material after %d interfaces', i);disp(str3);ix_m(i+1,:) = [0,0,0];ix_m2 = zeros(i+1,3);%NB: ix_m2 is a matrix which stores the coordinates at which the light rays%intercept each plane. The columns have been re-ordered to comply with the%altered optic coordinate system.ix_m2(:,1) = ix_m(:,3);ix_m2(:,2) = ix_m(:,1);ix_m2(:,3) = ix_m(:,2);ix2(1) = ix(3);ix2(2) = ix(1);ix2(3) = ix(2);%Outer if statement checks whether TIR has occurred at interfcae 2 in which%case no plot is generatedif any(ix_m(2,:))    figure();    for k = 1:i        if any(ix_m2(k+1,:))                        %Displaying each of the rays between interfaces            vectarrow(ix_m2(k,:),ix_m2(k+1,:));                        hold on;                    else                        hold on;                        vectarrow(ix_m2(k,:),ix2);                    end            end    hold on;    scatter3(0,0,0,'filled','g');    hold on;    scatter3(ix(3),ix(1),ix(2),'filled','r');        xlabel('z')    ylabel('x')    zlabel('y')        end%Printing all the data to a text filefid = fopen('R13Data.txt','w');fprintf(fid,'Data for test run of Ray_Trace12\r\n\r\nFor this run the ray exited the material after %d interfaces\r\n\r\n',i);fprintf(fid,'Preset Parameters:\r\n\r\n');fprintf(fid,'n1 = %.2f\n n2 = %.2f\r\n',n1,n2);fprintf(fid, 'lambda = %.2f\r\n\r\n',lambda);fprintf(fid, 'Intercept Points:\r\n\r\nix_m =\r\n\r\n');for j = 1:(i)    fprintf(fid, '%.5f  %.5f  %.5f\r\n',ix_m(j,:));endfclose(fid);