MATLAB 还只会画爱心？，教你画水晶簇以及水晶爱心

创意来自于初中时做的硫酸铜爱心，硫酸铜是三斜晶型，这里为了方便绘制并没有采用同样的结构。

效果：
果然还是自然生长出的晶体更好看呜呜呜


坐标变换
想要画出水晶新要先会画晶体簇，画晶体簇要先会画一个晶体，画晶体我们需要这样一个结构：

我们首先需要一个到直线距离一定且垂线落点为cutPnt的点，这个非常好找，例如cutPnt=[x0,y0,z0];中轴线方向为V=[x1,y1,z1];到直线距离为L则如下点显然符合条件：

v2=[z1,z1,-x1-y1];
v2=v2./norm(v2).*L;
pnt=cutPnt+v2;

但是广找到一个点还不够，我们要找到四个而且每个点都是之前的点绕直线旋转pi/2得到的，这就要请出我们的点绕直线旋转变换矩阵：

。。。。可以说是超级复杂了：

rotateMat=[u^2+(v^2+w^2)*cos(theta)       ,  u*v*(1-cos(theta))-w*sin(theta),  u*w*(1-cos(theta))+v*sin(theta),  (a*(v^2+w^2)-u*(b*v+c*w))*(1-cos(theta))+(b*w-c*v)*sin(theta);
		   u*v*(1-cos(theta))+w*sin(theta),  v^2+(u^2+w^2)*cos(theta)       ,  v*w*(1-cos(theta))-u*sin(theta),  (b*(u^2+w^2)-v*(a*u+c*w))*(1-cos(theta))+(c*u-a*w)*sin(theta);
           u*w*(1-cos(theta))-v*sin(theta),  v*w*(1-cos(theta))+u*sin(theta),  w^2+(u^2+v^2)*cos(theta)       ,  (c*(u^2+v^2)-w*(a*u+b*v))*(1-cos(theta))+(a*v-b*u)*sin(theta);
           0                              ,  0                              ,  0                              ,  1];
        

其中[u,v,w]是方向单位向量，[a,b,c]是轴初始点坐标

水晶簇绘制完整代码

function crystall

hold on
for i=1:50
    len=rand(1)*8+5;
    tempV=rand(1,3)-0.5;
    tempV(3)=abs(tempV(3));
    tempV=tempV./norm(tempV).*len;
    tempEpnt=tempV;
    drawCrystal([0 0 0],tempEpnt,pi/6,0.8,0.1,rand(1).*0.2+0.2)
    disp(i)
end
ax=gca;
ax.XLim=[-15,15];
ax.YLim=[-15,15];
ax.ZLim=[-2,15];
grid on
ax.GridLineStyle='--';
ax.LineWidth=1.2;
ax.XColor=[1,1,1].*0.4;
ax.YColor=[1,1,1].*0.4;
ax.ZColor=[1,1,1].*0.4;
ax.DataAspectRatio=[1,1,1];
ax.DataAspectRatioMode='manual';
ax.CameraPosition=[-67.6287 -204.5276   82.7879];

    function drawCrystal(Spnt,Epnt,theta,cl,w,alpha)
       %plot3([Spnt(1),Epnt(1)],[Spnt(2),Epnt(2)],[Spnt(3),Epnt(3)])
       mainV=Epnt-Spnt;
       cutPnt=cl.*(mainV)+Spnt;
       cutV=[mainV(3),mainV(3),-mainV(1)-mainV(2)];
       cutV=cutV./norm(cutV).*w.*norm(mainV);
       cornerPnt=cutPnt+cutV;
       cornerPnt=rotateAxis(Spnt,Epnt,cornerPnt,theta);
       cornerPntSet(1,:)=cornerPnt';
       for ii=1:3
           cornerPnt=rotateAxis(Spnt,Epnt,cornerPnt,pi/2);
           cornerPntSet(ii+1,:)=cornerPnt';
       end
       for ii=1:4
           jj=mod(ii,4)+1;
           fill33(Spnt,cornerPntSet(ii,:),cornerPntSet(jj,:),alpha)
           fill33(Epnt,cornerPntSet(ii,:),cornerPntSet(jj,:),alpha)
       end
    end

    function fill33(p1,p2,p3,alpha)
        fill3([p1(1),p2(1),p3(1)],[p1(2),p2(2),p3(2)],[p1(3),p2(3),p3(3)],...
            [0 71 177]./255,'FaceAlpha',alpha,'EdgeColor',[0 71 177]./255.*0.8,'EdgeAlpha',0.6,...
            'EdgeLighting','gouraud','SpecularStrength',0.3)
        
    end

    function newPnt=rotateAxis(Spnt,Epnt,cornerPnt,theta)
        V=Epnt-Spnt;V=V./norm(V);
        u=V(1);v=V(2);w=V(3);
        a=Spnt(1);b=Spnt(2);c=Spnt(3);
        cornerPnt=[cornerPnt(:);1];
        
        rotateMat=[u^2+(v^2+w^2)*cos(theta)       ,  u*v*(1-cos(theta))-w*sin(theta),  u*w*(1-cos(theta))+v*sin(theta),  (a*(v^2+w^2)-u*(b*v+c*w))*(1-cos(theta))+(b*w-c*v)*sin(theta);
                   u*v*(1-cos(theta))+w*sin(theta),  v^2+(u^2+w^2)*cos(theta)       ,  v*w*(1-cos(theta))-u*sin(theta),  (b*(u^2+w^2)-v*(a*u+c*w))*(1-cos(theta))+(c*u-a*w)*sin(theta);
                   u*w*(1-cos(theta))-v*sin(theta),  v*w*(1-cos(theta))+u*sin(theta),  w^2+(u^2+v^2)*cos(theta)       ,  (c*(u^2+v^2)-w*(a*u+b*v))*(1-cos(theta))+(a*v-b*u)*sin(theta);
                   0                              ,  0                              ,  0                              ,  1];
        
        newPnt=rotateMat*cornerPnt;
        newPnt(4)=[];
    end

end

水晶心绘制完整代码

function crystalHeart
clc;clear;close all
hold on
% drawCrystal([1,1,1],[3,3,3],pi/6,0.8,0.14)
sep=pi/8;
t=[0:0.2:sep,sep:0.02:pi-sep,pi-sep:0.2:pi+sep,pi+sep:0.02:2*pi-sep,2*pi-sep:0.2:2*pi];
x=16*sin(t).^3;
y=13*cos(t)-5*cos(2*t)-2*cos(3*t)-cos(4*t);
z=zeros(size(t));
plot3(x,y,z,'Color',[186,110,64]./255,'LineWidth',1)
for i=1:length(t)
    for j=1:6
        len=rand(1)*2+2;
        tempV=rand(1,3)-0.5;
        tempV=tempV./norm(tempV).*len;
        tempSpnt=[x(i),y(i),z(i)];
        tempEpnt=tempV+tempSpnt;
        drawCrystal(tempSpnt,tempEpnt,pi/6,0.8,0.14)
        disp([i,j])
    end
end
ax=gca;
ax.XLim=[-22,22];
ax.YLim=[-20,20];
ax.ZLim=[-10,10];
grid on
ax.GridLineStyle='--';
ax.LineWidth=1.2;
ax.XColor=[1,1,1].*0.4;
ax.YColor=[1,1,1].*0.4;
ax.ZColor=[1,1,1].*0.4;
ax.DataAspectRatio=[1,1,1];
ax.DataAspectRatioMode='manual';

    function drawCrystal(Spnt,Epnt,theta,cl,w)
       %plot3([Spnt(1),Epnt(1)],[Spnt(2),Epnt(2)],[Spnt(3),Epnt(3)])
       mainV=Epnt-Spnt;
       cutPnt=cl.*(mainV)+Spnt;
       cutV=[mainV(3),mainV(3),-mainV(1)-mainV(2)];
       cutV=cutV./norm(cutV).*w.*norm(mainV);
       cornerPnt=cutPnt+cutV;
       cornerPnt=rotateAxis(Spnt,Epnt,cornerPnt,theta);
       cornerPntSet(1,:)=cornerPnt';
       for ii=1:3
           cornerPnt=rotateAxis(Spnt,Epnt,cornerPnt,pi/2);
           cornerPntSet(ii+1,:)=cornerPnt';
       end
       for ii=1:4
           jj=mod(ii,4)+1;
           fill33(Spnt,cornerPntSet(ii,:),cornerPntSet(jj,:))
           fill33(Epnt,cornerPntSet(ii,:),cornerPntSet(jj,:))
       end
    end

    function fill33(p1,p2,p3)
        fill3([p1(1),p2(1),p3(1)],[p1(2),p2(2),p3(2)],[p1(3),p2(3),p3(3)],[0 71 177]./255.*1.03,...
            'FaceAlpha',0.2,'EdgeColor',[0 71 177]./255.*0.9,'EdgeAlpha',0.25,'LineWidth',0.5,...
            'EdgeLighting','gouraud','SpecularStrength',0.3)
        
    end

    function newPnt=rotateAxis(Spnt,Epnt,cornerPnt,theta)
        V=Epnt-Spnt;V=V./norm(V);
        u=V(1);v=V(2);w=V(3);
        a=Spnt(1);b=Spnt(2);c=Spnt(3);
        cornerPnt=[cornerPnt(:);1];
        
        rotateMat=[u^2+(v^2+w^2)*cos(theta)       ,  u*v*(1-cos(theta))-w*sin(theta),  u*w*(1-cos(theta))+v*sin(theta),  (a*(v^2+w^2)-u*(b*v+c*w))*(1-cos(theta))+(b*w-c*v)*sin(theta);
                   u*v*(1-cos(theta))+w*sin(theta),  v^2+(u^2+w^2)*cos(theta)       ,  v*w*(1-cos(theta))-u*sin(theta),  (b*(u^2+w^2)-v*(a*u+c*w))*(1-cos(theta))+(c*u-a*w)*sin(theta);
                   u*w*(1-cos(theta))-v*sin(theta),  v*w*(1-cos(theta))+u*sin(theta),  w^2+(u^2+v^2)*cos(theta)       ,  (c*(u^2+v^2)-w*(a*u+b*v))*(1-cos(theta))+(a*v-b*u)*sin(theta);
                   0                              ,  0                              ,  0                              ,  1];
        
        newPnt=rotateMat*cornerPnt;
        newPnt(4)=[];
    end

end