I have a point in a 3D coordinate system 1 (CS1). There can be two situations: the point is constant or the point is moving along a straight line from one known position to another at constant speed.
The CS1 is rotating in another (static) 3D coordinate system (CS2). The rotations of CS1 are known, i.e. the starting and ending angles are known, and the angular speeds are constant, so we can get a precise rotation matrix at any moment of time.
I need to find the length of the point's trajectory in the CS2.
In the simplest case, when the point isn't moving in CS1 and CS1 is rotating around one axis of CS2, the trajectory is a simple arc. In more complex cases, my current solution is to find a few points along the way (having point's position in CS1 and rotation angles of CS1 in CS2) and interpolate them with cubic spline, then get the length of the spline.
Is there a more precise and/or straightforward way to find the trajectory of the point in CS2? Thanks.
Hint:
WLOG one of the points describes a unit circle in the plane $XY$ and the second point a circle in a parallel plane, with some phase difference but at the same unit angular speed.
$$P(t)=(\cos t,\sin t,0),$$ $$Q(t)=(r\cos(t-t_0),r\sin(t-t_0),z_0).$$
Now a point that travels between them at constant speed has the coordinates $$R(t)=(1-v(t-t_1))P(t)+v(t-t_1)Q(t)$$ and $$\dot R(t)=-vP(t)+vQ(t)+(1-v(t-t_1))\dot P(t)+v(t-t_1)\dot Q(t).$$
After simplification, the element of arc seems to be just the square root of a quadratic trinomial, and this has an analytical antiderivative. But you will definitely need a CAS.