Two tangential circles in space

As you would do on paper. If they are tangent, they have a common tangent plane in the point where they are tangent.

And the plane is orthogonal to the radius in that point.

So, once you have built one of the spheres (sphere with given center C and radius R https://wiki.geogebra.org/e...) and created the point T on it where you want the other sphere to be tangent, draw CT. The center of the second sphere lies on the line CT.

The problem is how to construct Two tangential circles that are not in the same plane?

draw two tangent spheres, two planes through centers and then intersect each plane with each sphere

There are many ways to construct Awkwardly Osculating Circles depending on how you constrain the location and orientations.

Given:

Point P, the point at which the circles share a tangent

R_{1} and R_{2}, the radii of the first and second circles

F_{1} and F_{2}, points indicating which direction the circle centres are to lie,

P = {-1.5, 2, 2.5}

R_{1} = 1.5

R_{2} = 2.5

find the circle centres and orientations.

First the centres:

f_{1} = Ray(P, F_{1})

s_{1} = Sphere(P, R_{1})

C_{1} = Intersect(s_{1}, f_{1})

f_{2} = Ray(P, F_{2})

s_{2} = Sphere(P, R_{2})

C_{2} = Intersect(s_{2}, f_{2})

The tangent line is normal to both PF_{1} and PF_{2}.

p = Plane(F_{1}, P, F_{2})

t = PerpendicularLine(P, p)

The circles will each lie in their own plane which includes that circle’s centre and the tangent line.

q_{1} = Plane(t, f_{1})

q_{2} = Plane(t, f_{2})

The axis vector for each circle includes that circle’s centre and is perpendicular to its plane.

a_{1} = PependicularLine(C_{1}, q{1})

a_{2} = PependicularLine(C_{2}, q{2})

The radii of the constructed circles may be specified either directly with the given radii or indirectly with the point P.

c_{1} = Circle(a_{1}, P)

c_{2} = Circle(a_{2}, P)

This method goes wobbly if you try to make the circles coplanar. A more stable method might begin with the centre, radius and axis of one circle, the radius of another circle and the plane angle between them.

Thank you all who joined to give solutions....