it depends on the ratio of the speeds of the speen and the speeeeeeeeen (from your diagram). you could return to the original position after one rotation, by picking an angle that exactly paints over itself forever. the 4-for-full-coverage is a minimum that only works for some angles/ratios
December 20, 2024 at 9:25 PM
it depends on the ratio of the speeds of the speen and the speeeeeeeeen (from your diagram). you could return to the original position after one rotation, by picking an angle that exactly paints over itself forever. the 4-for-full-coverage is a minimum that only works for some angles/ratios
the rotational motion of the arc is a diagonal motion of the line segment, whose angle is determined by the ratio of the speens. each pass of the line covers 2r/2πr = 1/π of the vertical dimension, so you'll need to do "π rounded up" = 4 rotations to cover it all, more if it overlaps too much (2/2)
December 20, 2024 at 6:53 AM
the rotational motion of the arc is a diagonal motion of the line segment, whose angle is determined by the ratio of the speens. each pass of the line covers 2r/2πr = 1/π of the vertical dimension, so you'll need to do "π rounded up" = 4 rotations to cover it all, more if it overlaps too much (2/2)
visualize unwrapping the donut surface into a flat rectangle, where if you go off one side, you come back on the opposite side, like asteroids (1978). the vertical dimension is the circle circumference 2πr, the horizontal goes around the donut 2πR, and the arc is a vertical line of length 2r. (1/2)
December 20, 2024 at 6:52 AM
visualize unwrapping the donut surface into a flat rectangle, where if you go off one side, you come back on the opposite side, like asteroids (1978). the vertical dimension is the circle circumference 2πr, the horizontal goes around the donut 2πR, and the arc is a vertical line of length 2r. (1/2)