MATLAB在低碳的室內裝潢燈光照明的應用

問題描述，一個10*4*3的房間只配備300Watt的照明功率，要如何掛載燈泡使得燈光可以均勻的散布在房間各個角落，假定只是空房間，燈泡的使用個數不限，但功率的總和不超過300Watt。衡量亮度的指標是Watt/㎡

 

使用一個燈，要照亮各個角落當然要掛載天花板的最中間，也就是座標（5，2）

1

可看出一個300Watt的燈泡其燈光分布情況很不均勻，差距很大，所以不是最佳的方案

 

使用兩個燈泡，當然是要放在y=2這條線上了，但最佳座標需要算一下

 

其最暗的角落（0，0）和最亮的地方（5，2）隨這燈泡移動的軌跡如下

可見在1.06和8.9的地方有交叉，則最優的點就是（1.06，2）和（8.9，2）。

 

三個燈泡的時候

曲線：

 

無最優解

 

 

附上代碼：

syms x y;
light1=300/(4*pi*((x-5)^2+(y-2)^2+3^2));
ezcontourf(light1,[0 10 0 4]);
colormap(gray);
axis equal tight;

[x,y]=meshgrid(0:0.1:10,0:0.1:4);
light2=inline('150/(4*pi*((x-d)^2+(y-2)^2+3^2))','x','y','d');
light2=vectorize(light2);
figure;
contourf(light2(x,y,2)+light2(x,y,8),50);
colormap(gray);
axis equal tight;

intensity_corner=[];
for d=0:0.1:10
    intensity_corner=[intensity_corner light2(0,0,d)+light2(0,0,10-d)];
end

intensity_center=[];
for d=0:0.1:10
    intensity_center=[intensity_center light2(5,2,d)+light2(5,2,10-d)];
end
figure;
plot(0:0.1:10,intensity_corner);
hold on;
plot(0:0.1:10,intensity_center,'r-');

optimal=solve('150/(4*pi*((0-d)^2+(0-2)^2+3^2))+150/(4*pi*((0-(10-d))^2+(0-2)^2+3^2))=150/(4*pi*((5-d)^2+(2-2)^2+3^2))+150/(4*pi*((5-(10-d))^2+(2-2)^2+3^2))','d');
optimal=double(optimal);

optimal_intensity=light2(0,0,optimal(1))+light2(0,0,10-optimal(1));

light3=inline('100/(4*pi*((x-d)^2+(y-2)^2+3^2))','x','y','d');
light3=vectorize(light3);
figure;
contourf(light3(x,y,1)+light3(x,y,9)+light3(x,y,5),50);
colormap(gray);
axis equal tight;

test_intensity_corner=light3(0,0,1)+light3(0,0,9)+light3(0,0,5);
test_intensity_center=light3(5,2,1)+light3(5,2,9)+light3(5,2,5);

intensity_corner3=[];
for d=0:0.1:10
    intensity_corner3=[intensity_corner3 light3(0,0,d)+light3(0,0,10-d)+light3(0,0,5)];
end

intensity_center3=[];
for d=0:0.1:10
    intensity_center3=[intensity_center3 light3(5,2,d)+light3(5,2,10-d)+light3(5,2,5)];
end
figure;
plot(0:0.1:10,intensity_corner3);
hold on;
plot(0:0.1:10,intensity_center3,'r-');