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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!

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1 Answer 1

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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.

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  • $\begingroup$ This sounds promising, however I must admit i'm not entirely sure how to find N from the roll and pitch angles? $\endgroup$
    – Arran Hobson Sayers
    Commented May 25, 2017 at 3:09
  • $\begingroup$ I appreciate your help so far and for clarifying. So, I have defined phi (roll) as rotation around x, theta (pitch) as rotation about y and gamma (yaw) as rotation about z - i hope this is what you meant as defining them. I've been looking at this and this seems to help my question, however im unsure as how to know what order to apply the rotations and how the order affects the answer. Additionally i'm unsure as to what to actually multiply the rotation matrices by to get N $\endgroup$
    – Arran Hobson Sayers
    Commented May 25, 2017 at 3:41
  • $\begingroup$ Also some further info about my coordinate system I may have missed. I am using a left hand coordinate system with x pointing left, y forwards and z upwards. $\endgroup$
    – Arran Hobson Sayers
    Commented May 25, 2017 at 3:53
  • $\begingroup$ 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. $\endgroup$
    – bubba
    Commented May 25, 2017 at 4:40

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