-*- coding: utf-8 -*-

Solution of a line in parametric formula:
({p1, p2, p3} - {n1, n2, n3})t + {n1, n2, n3}
and the sphere x^2+y^2+(z-r)^2-r^2==0.

------------

{

{
 n1 + ((-n1 + p1)*(2*n1^2 + 2*n2^2 + 2*n3^2 - 2*n1*p1 - 2*n2*p2 - 2*n3*p3 - 2*n3*r + 2*p3*r -  Sqrt[-4*(n1^2 + n2^2 + n3^2 - 2*n1*p1 + p1^2 - 2*n2*p2 + p2^2 - 2*n3*p3 + p3^2)*(n1^2 + n2^2 + n3^2 - 2*n3*r) +  (-2*n1^2 - 2*n2^2 - 2*n3^2 + 2*n1*p1 + 2*n2*p2 + 2*n3*p3 + 2*n3*r - 2*p3*r)^2]))/(2*(n1^2 + n2^2 + n3^2 - 2*n1*p1 + p1^2 - 2*n2*p2 + p2^2 - 2*n3*p3 + p3^2))
,

 n2 + ((-n2 + p2)*(2*n1^2 + 2*n2^2 + 2*n3^2 - 2*n1*p1 - 2*n2*p2 - 2*n3*p3 - 2*n3*r + 2*p3*r -  Sqrt[-4*(n1^2 + n2^2 + n3^2 - 2*n1*p1 + p1^2 - 2*n2*p2 + p2^2 - 2*n3*p3 + p3^2)*(n1^2 + n2^2 + n3^2 - 2*n3*r) +  (-2*n1^2 - 2*n2^2 - 2*n3^2 + 2*n1*p1 + 2*n2*p2 + 2*n3*p3 + 2*n3*r - 2*p3*r)^2]))/(2*(n1^2 + n2^2 + n3^2 - 2*n1*p1 + p1^2 - 2*n2*p2 + p2^2 - 2*n3*p3 + p3^2))
,

 n3 + ((-n3 + p3)*(2*n1^2 + 2*n2^2 + 2*n3^2 - 2*n1*p1 - 2*n2*p2 - 2*n3*p3 - 2*n3*r + 2*p3*r -  Sqrt[-4*(n1^2 + n2^2 + n3^2 - 2*n1*p1 + p1^2 - 2*n2*p2 + p2^2 - 2*n3*p3 + p3^2)*(n1^2 + n2^2 + n3^2 - 2*n3*r) +  (-2*n1^2 - 2*n2^2 - 2*n3^2 + 2*n1*p1 + 2*n2*p2 + 2*n3*p3 + 2*n3*r - 2*p3*r)^2]))/(2*(n1^2 + n2^2 + n3^2 - 2*n1*p1 + p1^2 - 2*n2*p2 + p2^2 - 2*n3*p3 + p3^2))
}, 

{
 n1 + ((-n1 + p1)*(2*n1^2 + 2*n2^2 + 2*n3^2 - 2*n1*p1 - 2*n2*p2 - 2*n3*p3 - 2*n3*r + 2*p3*r +  Sqrt[-4*(n1^2 + n2^2 + n3^2 - 2*n1*p1 + p1^2 - 2*n2*p2 + p2^2 - 2*n3*p3 + p3^2)*(n1^2 + n2^2 + n3^2 - 2*n3*r) +  (-2*n1^2 - 2*n2^2 - 2*n3^2 + 2*n1*p1 + 2*n2*p2 + 2*n3*p3 + 2*n3*r - 2*p3*r)^2]))/(2*(n1^2 + n2^2 + n3^2 - 2*n1*p1 + p1^2 - 2*n2*p2 + p2^2 - 2*n3*p3 + p3^2)), 

 n2 + ((-n2 + p2)*(2*n1^2 + 2*n2^2 + 2*n3^2 - 2*n1*p1 - 2*n2*p2 - 2*n3*p3 - 2*n3*r + 2*p3*r +  Sqrt[-4*(n1^2 + n2^2 + n3^2 - 2*n1*p1 + p1^2 - 2*n2*p2 + p2^2 - 2*n3*p3 + p3^2)*(n1^2 + n2^2 + n3^2 - 2*n3*r) +  (-2*n1^2 - 2*n2^2 - 2*n3^2 + 2*n1*p1 + 2*n2*p2 + 2*n3*p3 + 2*n3*r - 2*p3*r)^2]))/(2*(n1^2 + n2^2 + n3^2 - 2*n1*p1 + p1^2 - 2*n2*p2 + p2^2 - 2*n3*p3 + p3^2)), 

 n3 + ((-n3 + p3)*(2*n1^2 + 2*n2^2 + 2*n3^2 - 2*n1*p1 - 2*n2*p2 - 2*n3*p3 - 2*n3*r + 2*p3*r +  Sqrt[-4*(n1^2 + n2^2 + n3^2 - 2*n1*p1 + p1^2 - 2*n2*p2 + p2^2 - 2*n3*p3 + p3^2)*(n1^2 + n2^2 + n3^2 - 2*n3*r) +  (-2*n1^2 - 2*n2^2 - 2*n3^2 + 2*n1*p1 + 2*n2*p2 + 2*n3*p3 + 2*n3*r - 2*p3*r)^2]))/(2*(n1^2 + n2^2 + n3^2 - 2*n1*p1 + p1^2 - 2*n2*p2 + p2^2 - 2*n3*p3 + p3^2))
}


}