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I want to calculate the footprint of some signal emitted by an airborne sensor. In order to already take the yaw angle into account (defined as rotation around the Z-axis), I approximated the footprint as the triangle S-P1-P2 in XY plane, where S is the sensor and P1 and P2 are defined by some external constraints and define the maximum extent of the footprint. Now I need to add the influence of a pitch angle, i.e. a rotation around the y-axis of the sensor coordinate system. How must I manipulate/transform the XY-coordinates of the original triangle in order to account for the distortion induced by this pitch angle?

Or should I rather calculate everything in 3D space, given that I probably can define S, P1 and P2 in XYZ space as well?

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Recently solved a similar problem this way. Start with a right handed coordinate system x,y,z where x is to the right side of the page, y is out of the page and z is down. (NED in a navigation system where north is x). Start with a unit vector along x - [1,0,0]. Then apply two Euler rotations. First pitch the unit vector (+ rotation about y), then yaw the unit vector (+ rotation about z). The yaw angle is half your sensors beam width. The resulting unit vector now represents the original beam at a pitch angle. The unit vector x and y components give you the projection back to the xy plane. So zero the z element in the unit vector and you can then compute the projected beam angle in the xy plane as an acrtangent.

You can check this by first doing only the yaw with no pitch. You will get a a projected angle that matches half the beam width with no change. At a positive pitch angle the resulting projection will be a bigger angle than the original half beam width. At a pitch of 90 degrees the resulting projection is 90 degrees regardless of the yaw angle.

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