Matlab simulation of drawing circle on sphere

  Derivation description    

For the convenience of formula derivation , In this paper, only when the spherical center is at the origin of coordinates , It should be noted that when the ball center is not at the origin , Just subtract the coordinates of the spherical center from the coordinates of the known conditions , After the solution is completed, add the spherical center coordinates ！

deduction ：

in front Blog , The code of three point circle drawing has been given , The observation function generateCircle You know , To draw a circle , We only need to know the center of the circle , The plane of the circle x Direction vector and y The direction of the vector .

Next , We focus our work on the above three quantities , Suppose the ball is a unit ball whose center is at the origin of coordinates R=1, We need to be on the ball  rs=[0.75,-0.17,0.64] Draw a  15°  The circle of , be

It is known that , The radius of the circle is ： r=R*sin(15/180*pi);

The direction vector of the plane where the circle is located is ：cc=rs=[0.75,-0.17,0.64]

that , The plane equation passing through the origin and parallel to the plane of the circle is ：rs(1)*x+rs(2)*y+rs(3)*z=0

that , Take any vector in this plane as x The direction vector of the axis , Such as ：nx=[1,1,-(rs(1)+rs(2))/rs(3)]

that , According to the cross product rule ,y The direction vector of the axis is ：ny=cross(rs,nx);

Besides , We know , The position of the center of the circle is rs On the straight line connecting with the center of the circle , According to the parameter equation of the ball , The spherical radius of the center of the circle is ：rr=R*cos(15/180*pi)

Code for drawing sphere ：

function drawsphere(a,b,c,R)
%%  Draw sphere 
%  With (a,b,c) For the center of the ball ,R As the radius of 

    %  Generate the data 
    [x,y,z] = sphere(20);

    %  Adjust the radius 
    x = R*x; 
    y = R*y;
    z = R*z;

    %  Adjust the ball center 
    x = x+a;
    y = y+b;
    z = z+c;

    %  Use mesh draw 
%     figure;
%     axis equal;
    mesh(x,y,z);

    %  Use surf draw 
%     figure;
%     axis equal;
%     surf(x,y,z);
end

Generate the code of points on the circle

%%  Generate points on the circle 
% cp  center of a circle ,R  Circle radius ,a Circular plane x Axis direction vector ,b Circular plane y Axis direction vector ,num  Number of interpolation points 
function xyz=generateCircle(cp,R,a,b,num)
    xyz=[];
    i=linspace(0,2*pi,num);
    x=cp(1) + R*cos(i)*a(1) +R*sin(i)*b(1);
    y=cp(2) + R*cos(i)*a(2) +R*sin(i)*b(2);
    z=cp(3) + R*cos(i)*a(3) +R*sin(i)*b(3);
    xyz=[x',y',z'];
end

Transformation from rectangular coordinates to spherical coordinates

function xyz=sp2dikaer(azimuth,elevation,r)
    x = r .* cos(elevation) .* cos(azimuth);
    y = r .* cos(elevation) .* sin(azimuth);
    z = r .* sin(elevation);
    xyz=[x,y,z];
end

Reference from matlab Official website

Test code ：

clc;clear all;close all;
rs=[0.75,-0.17,0.64];
a=0;
b=0;
c=0;
R=1;
flag=0;
drawsphere(a,b,c,R);
hold on;
plot3(rs(1),rs(2),rs(3),'rp');

%%  The radius of the circle 
r=R*sin(15/180*pi);
th=0:0.01:2*pi;
azimuth=atan2(rs(2),rs(1));
elevation=atan2(rs(3),sqrt(rs(2)^2+rs(1)^1));
% xyz=sp2dikaer(azimuth,elevation,R)

nx=[1,1,-(rs(1)+rs(2))/rs(3)];
ny=cross(rs,nx);
nx=nx/norm(nx);
ny=ny/norm(ny);
rs1=sp2dikaer(atan2(rs(2),rs(1)),atan2(rs(3),sqrt(sum(rs(1:2).^2))),R*cos(15/180*pi));
xyz=generateCircle(rs1,r,nx,ny,100);
plot3(xyz(:,1),xyz(:,2),xyz(:,3),'r', 'LineWidth',2);

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