# Finding the tangent point on a sphere, knowing the angle of the tangent plane to the x and y axes

If I have a sphere of known radius, R, centred at the origin and a plane with a known angle to the x and y axes (roll, φ, and pitch, θ), how can I find the coordinates of the points that this plane could be a tangent to the sphere assuming the plane can be translated.

I have done some research into spherical coordinates, however, I am required to know the azimuth and inclination to find the point on the sphere. I am not sure how to calculate these from the angle of the tangent plane to the x and y axes. From further research I believe rotation matrices may be the answer, but I am unsure how to apply them to this problem. Any help is much appreciated.

Many Thanks!

Use the roll and pitch angles to calculate the unit-length normal vector $\mathbf{N}$ of the plane. Then the point of tangency is $R\,\mathbf{N}$.
The roll and pitch angles are examples of "Euler angles", which you can look up here and in many other places. Euler angles are a mess because there are many different ways to define them. There are choices about which axes you rotate around and in which order. I can't really tell you how to calculate $\mathbf{N}$ from your two angles without some clear definition of what they mean.
• You are rotating some initial plane to get the final one that's tangent to the sphere. If the you apply the rotations to the normal of the initial plane, you'll get the normal $\mathbf{N}$ of the desired plane. – bubba May 25 '17 at 4:40