Search space. The problem of finding the collision time for two planets is equivalent to finding the grid point (k, I) in the small blue band that is closest to the origin. The horizontal and vertical axes are given by and in Eq. (13), respectively. The slope is the ratio p/q = T₁/T₂ of orbital periods. The center of the intersection with the horizontal axis is , where is the difference between the time-oſ-passage of the collision point for the two planets. The width of the band depends on the minimal distance of the orbits and the planet radii. The solution can be rapidly found using the bases {b_2n,b_2n+1}