Plot cross tangent of two circle

I have center coordinate of two circles and radius of two circles. How to draw this figure joining the cross tangent points?

[Adam Danz]

I don't have a solution off the top of my head but wouldn't there be 4 possible tangent lines that are tangential to both circles?

[John D'Errico]

Yes. As long as a circle does not contain the other. If one circle is wholly inside the other, then there are no mutually tangent lines. There may also be a problem if the two circles intersect, in which case it would appear there will usually be exactly two tangent lines. If one circle is entirely inside the other, but there is one point of intersection, then you will have exactly one tangent line. And of course, if the two circles have the same center AND the same radius, then there are infinitely many multual tangent lines.

So the number of mutual tangent lines can be any of 0,1, 2, 4, inf.

Answer 1

[Bruno Luong]

% Generate two random circles
boxsize = 10;
% center circle 1
x1 = boxsize*rand;
y1 = boxsize*rand;
% center circle 2
x2 = boxsize*rand;
y2 = boxsize*rand;
% their radius
r1 = 1+rand;
r2 = 1+rand;

% Compute cross tangent s-t
xy1 = [x1; y1];
xy2 = [x2; y2];
v = xy2-xy1;
d = norm(v);
rs = r1+r2;
if rs >= d
    error('not possible try to generate again');
end
v = v/d;
s = 1; % sign of s allows to select which of two cross tangents
Q = [v, sign(s)*[-v(2); v(1)]]; 
c = rs/d;
Qx = Q*[c; sqrt(1-c^2)];
s = xy1 + Qx*r1;
t = xy2 - Qx*r2;

% need to plot circle
phi = linspace(0,2*pi,361);
cs = cos(phi);
ss = sin(phi);
plotcirclefun = @(xy,r) plot(xy(1)+r*cs, xy(2)+r*ss, 'k', 'linewidth', 3);

% Graphical output
figure(1)
clf
hold on
plotcirclefun(xy1, r1);
plotcirclefun(xy2, r2);
plot([s(1) t(1)], [s(2) t(2)], 'k', 'linewidth', 3);
axis equal