# Tangent point of sphere and circle

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.

• Are you wondering about the determination of a circle tangent to a sphere given its respective centers, the plane of the circle and the ray of the sphere? Aug 7, 2019 at 7:24
• This is equivalent to finding the intersection of two spheres.
– amd
Aug 7, 2019 at 8:36
• I exactly want to find the cartesian coordinate of the tangent point regarding to given center point and radius of sphere and circle.
– Onur
Aug 7, 2019 at 9:45
• A circle outside a sphere and touching it could be rotated about the axis between their centers and trace a circle on the sphere orthogonal to that axis. Is that what you want? As @amd says, this is the circle of intersection of 2 spheres.
– Paul
Aug 7, 2019 at 11:23
• Actually, I think that the question can be asked for intersection point of two spheres. Because circle and sphere have same equations. But I would like to know one specific point that they are tangent. Because if the spheres intersect with each other, it will have two intersection point. But I need only one tangent point.
– Onur
Aug 8, 2019 at 4:50

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.
• @onur: if the plane of the circle is known, then the plane $ABC$ is also known as being normal to the first. Then the problem is much easier: please specify exactly what is known and what is to be determined. Aug 8, 2019 at 14:20