Equation of A Tangent Line To a Sphere

Question

I am curious as to how one would construct the equation of a tangent line to a sphere in $\mathbf{R}^3$. I know that, generally, I'd need a point on the boundary of the sphere and some vector to specify the direction of the line, but the latter part is what I'm having trouble with.

So, all in all, I am confused as to how to describe a line that lies tangent to a sphere that also points in a fixed direction.

If I didn't specify a direction, then it seems one could construct infinitely many tangent lines at a given point on the sphere. Is there a way to describe a tangent line to a sphere that does not have a specific direction?

Feel free to edit my tags to something more appropriate.

Answer 1

At any point $P$ on a sphere we have a tangent plane, that is the plane orthogonal to the radius of the sphere at the point $P$. So there are infinetely many stright line on this plane passing thorough $P$. If we specify a direction on this plane than we have only one line from $P$ with this direction .

Answer 2

You can define the line by a point and a direction. Choose any point on the sphere and any direction perpendicular to the radius vector. Specifically the direction coordinates $(d_x,d_y,d_z)$ need only satisfy

$$(x-x_0)d_x+(y-y_0)d_y+(z-z_0)d_z=0$$

This solution has 3 degrees of freedom: 2 for the point on the sphere and 1 more for the directions perpendicular to the radius vector.

If the direction of the tangent is given then there remains only one degree of freedom for the point, determined by the great circle whose plane is perpendicular to the given direction. Explicitly the equation above (combined with the equation of the sphere itself) can be solved for $(x,y,z)$ to describe all solutions.

Answer 3

N={a,b,c} is a normal vector from the origin to some plane holding the line. d is the distance to the origin. You may not know these when you form the equation. C is the position vector of the center of a sphere and r its radius. This plane tangent to a sphere if dot_product(N,C) - d = r (radius). This is aX+bY+cZ-d=r That is the equation you asked for that DEFINES the tangency. You can use it in a Gaussian elimination or the extended Gaussian that works with geometry. The result of this operation should fill in the numbers. You'll need four equations to determine either the line, or the sphere, or some portion of each if your in a cycle . Tangency is just one.