Two randomly chosen rotations, R and S, depicted as two randomly chosen points in the solid ball model of RP^3.  (Rotation R is a rotation with axis parallel to {-1,-2,5} and with amount of rotation Pi/3; rotation S is a rotation with axis parallel to {2,3,5}, with amount of rotation Pi/2.  For ease of visualization, R and S are plotted along with vector from the origin in RP^3, and color coded in same way as trajectories in RP^3.  Viewpoint is same as in Figures 2 and 3.)

The same two randomly chosen rotations, R and S, and also their product (depicted as three points in the solid ball model of RP^3).  The product was determined from the previous figure, as the intersection of the trajectory  alpha  and the plane bisecting segment m-n.

The plane perpendicular to the line segment whose endpoints are the topmost point (or "pinhole" of the "basketball") and a fixed other point, q

Trajectory lying in the plane from Figure 2.  The orangeish ball is the "basketball", while the trajectory really lies in the solid ball model of RP^3.  The point on the trajectory farthest from the origin in RP^3 has coordinates

{1.202235459768693, 0, 2.902453152139431}

























































{1.20223546, 0, 2.902453152}

This point represents a rotation (by pi about its axis), with  matrix given by

MatrixForm[{{-2^(-1/2), 0, 2^(-1/2)}, {0, -1, 0}, 

   {2^(-1/2), 0, 2^(-1/2)}}]

























































     1                       1

-(-------)                -------

  Sqrt[2]    0            Sqrt[2]







0            -1           0



   1                         1

-------                   -------

Sqrt[2]      0            Sqrt[2]

The point q' depicted on the orangeish "basketball" is S(R(pinhole)).  For ease in direct comparison with Figures 2 and 3, we use the same viewpoint here as in those figures.  The trajectory, technically located in the solid ball model of RP^3 and denoted alpha  in the text, is shown in its plane.  The points in  alpha   represent all rotations that move the "pinhole" to q', ie to the same point that SR does.  Hence the trajectory must contain the point representing the rotation SR.

The point representing the composite rotation SR is found from the intersection of the trajectory  alpha  with a certain plane.  First, two points are depicted. n  is one of the points where R's axis of rotation intersects the sphere, and plotted along with part of R's axis; its coordinates are {-.5735,-1.1471,2.8678}. Next, m=S(R(n)), with coordinates {2.8393,-.6258,1.1899}.  The plane  depicted is not alpha's plane from the previous figure, but rather is the plane bisecting the segment n-m.  It intersects the trajectory  alpha  at the point in RP^3 representing the rotation SR.  (This point has coordinates {.9858,.0295,2.0144}).

Trajectories in solid ball model of RP^3, with dots located where trajectories strike the plane y=0.  Colors indicate distance from the origin (orange-red indicating distance is near 0, purple indicating distance is about pi).  The trajectories are in fact "left cosets", as will be discussed in the final section.

Top view of trajectories (or "left cosets"), with dots where cosets strike the y=0 plane

A single right coset (in black), along with several of the left cosets it intersects.  The trajectories all lie on a torus, which is an element of SO(2)\SO(3)/SO(2).  The torus  can be obtained by rotating any one trajectory (even the right coset) about the z-axis.

A single right coset (in black), along with several of the left cosets it intersects (top view)

Each trajectory lies in a plane.  If the plane of the screen/printout is taken as this plane, then the polar equation of this curve is given as r below.  There, phi denotes the angle from the z-axis to the beginning point on the trajectory, ie the point where the trajectory reaches the boundary of the ball.  Despite appearances, the trajectories are not arcs of circles, except for the trajectory that is the equator.  (This is readily confirmed by computing the curvature of the trajectories.)

r[theta,phi]

Some other trajectories (or "left cosets")

Top view of the second set of trajectories