Tangent point of sphere and circle

Question

I would like to know parametric equation of the tangent point of a sphere and a circle. Circle center point is $(a,b,c)$ and its radius is $K$. Sphere center point is $(d,e,f)$ and its radius is $L$.

I assume that specified circle and a sphere intersect on one tangent point. I would like to know how I can determine the tangent point $(x,y,z)$.

If there is a misinformation please define for me.

Thanks in advance.

Answer 1

In the plane passing through the centers $A,B$ and the tangent point $C$, you get a triangle $ABC$ of which you know the three sides.
So you can determine every parameter of it, and in particular the height $h$ from $C$ to the side $AB$.

When reported in $3$D, the triangle is free to rotate around the axis $AB$ and the point C can be whichever along a circle on the sphere of radius $h$ from the $AB$ axis.

You need one more constraint to fix the circle.