The condition implies the track pass all points on the ray of original point (O) toward destination (D)
let A is smaller that B.
First, we know A*B make a pattern that repeats. (1)
When B-A=4n-d (n=N;d=1-3)
let car starts at point O toward D. when total move A*B, it follows as certain path to reach a point (O-1) that car run toward different direction. This process repeats and follow the path with the similar pattern and run another A*B to O-2. We will find it looks that this pattern turn a certain angle d*90 degree. Repeats again and again. when run 4 A*B, we will find it get back to the starting point and start a new round. Thus this pairs of numbers gives a close feature and can not cover all points along the destination direction (ray O-D).
Because of 1),to cover all the points at the direction of ray 0-D, after run A*B, the car must turn back to the direction toward (O to D). then it can continue move on but start a different point ahead. This cycle repeats can cover all the points at the O-D direction. at least, this requires B-A=4n.
next step is to test if car reach a point on the ray O-D after A*B.
make xy system with O as (0,0). Car run along y up. Suppose after A*B, the car reach to a point O' which is not on y-axis. Think of the return trip (you need set machine in opposite way (turn set up 2 vs. original turn set up 1)) and think of car goes along y-axis (with set up 1). By study the patterns, we will find car can reach the O' from O by either setup, (symmetry). it is not possible unless O' is on y-axis.
Next step is to test if O' is ahead in the direction of ray O-D. Think of return way, we will find the track of O-O' must in a shape with characteristics of plane symmetry. if car turns at the first point P from O and the first point P' from O' (in a return way). The largest PP' could only less than 2A, since we can not have any line down or up greater than A. As OP is A and O'P' is A. So OP and OP' overlap. O' is ahead of O.
In this case, from O' starts a new cycle and keep moving to cover all the points on the ray O-D.
let A is smaller that B.
First, we know A*B make a pattern that repeats. (1)
When B-A=4n-d (n=N;d=1-3)
let car starts at point O toward D. when total move A*B, it follows as certain path to reach a point (O-1) that car run toward different direction. This process repeats and follow the path with the similar pattern and run another A*B to O-2. We will find it looks that this pattern turn a certain angle d*90 degree. Repeats again and again. when run 4 A*B, we will find it get back to the starting point and start a new round. Thus this pairs of numbers gives a close feature and can not cover all points along the destination direction (ray O-D).
Because of 1),to cover all the points at the direction of ray 0-D, after run A*B, the car must turn back to the direction toward (O to D). then it can continue move on but start a different point ahead. This cycle repeats can cover all the points at the O-D direction. at least, this requires B-A=4n.
next step is to test if car reach a point on the ray O-D after A*B.
make xy system with O as (0,0). Car run along y up. Suppose after A*B, the car reach to a point O' which is not on y-axis. Think of the return trip (you need set machine in opposite way (turn set up 2 vs. original turn set up 1)) and think of car goes along y-axis (with set up 1). By study the patterns, we will find car can reach the O' from O by either setup, (symmetry). it is not possible unless O' is on y-axis.
Next step is to test if O' is ahead in the direction of ray O-D. Think of return way, we will find the track of O-O' must in a shape with characteristics of plane symmetry. if car turns at the first point P from O and the first point P' from O' (in a return way). The largest PP' could only less than 2A, since we can not have any line down or up greater than A. As OP is A and O'P' is A. So OP and OP' overlap. O' is ahead of O.
In this case, from O' starts a new cycle and keep moving to cover all the points on the ray O-D.
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