How do you find the point touching the tangent line on a unit circle?

Question

For a unit circle and a normalised vector representing the tangent (could be pointing in either direction), of which, would be touching the circle's perimeter, how do you find the point on the circle for it ?

Answer 1

For spheres the normal pointing outwards in a point $p$ on the boundary is given by the vector $p$. The tangent vectors are orthogonal to the normal.

So in your case, finding the point where, given a tangent direction, the tangent can meet the circle is just a matter of rotating the direction by 90 degrees. However the question and this solution don’t specify a unique such point, but rather two antipodal points. But this can be easily modified to result in a unique solution, at least for 2d spheres: We fix a vector field of tangent directions, say counterclockwise. Then rotating the tangent directions by 90 degrees in clockwise direction will give you normals pointing outwards, which are unique and give you the corresponding touching point.