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I need to know how to find a tangent vector to a point on the surface of a sphere if I am given the point P and the normal vector at that point N.

I know that there are many possible tangent vectors to choose from, but I was wondering also if it is possible so that regardless of which point on the surface I choose, the tangent vectors are all "oriented" the same way.

Lastly, does the nature of the surface matter? For instance, would the method used to find the tangent vector differ at all if I was dealing with a flat surface/plane instead of a sphere?

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    $\begingroup$ If by "oriented the same way" you mean what I think you mean -- is it possible to choose a unit tangent vector at every point on a sphere such that the field is continuous? -- then no, there is no way to do so. $\endgroup$ – Rahul Mar 4 '14 at 0:10
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If you know the normal vector $\vec N=(n_1,n_2,n_3)$ of a surface at a point $P=(x_0,y_0,z_0)$ then the plane tangent to the surface at $P$ is given by $n_1(x-x_0)+n_2(y-y_0)+n_3(z-z_0)=0$. This is just the equation of a plane given a point and it's normal vector (you should look into that). Any vector in this plane is tangent to the surface. This holds for any surface as long as you know the normal vector at $P$. If you don't know what the normal vector is then you have to use partial derivatives.

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Check out this paper that presents an analytical way to calculate tangent surface vectors of an implicit surface.

"D.S. Lopes et al., Tangent vectors to a 3-D surface normal: A geometric tool to find orthogonal vectors based on the Householder transformation, Computer-Aided Design, 2013, 45:683 - 694"

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