We knew Epicycloid as a kind of trace curve of a specific point attached to a circle and rolls on another circle. But there's a limit for the ratio R/r of the radius of two circle, R and r, which is that R/r must be rational, so that the curve will loop through its origins.(Without losing generality, we may assume the circle with radius r rolls on the circle with radius R, abbr. as R-circle and r-circle for now)
What if we choose R/r to be irrational real numbers and than draw the curve?
If we choose concentric circles with radius R and R+2r centered at the R-circle. My question is, is it sufficient to imply that such curve coveres the whole region between the two circle of concentric circles?
Ok Now I've sort of figured it out with @IvanNeretin. And because there's no response for now I'll wrote one.
In conclusion, the trajectory will be dense but not complete in such region.
First we can show that the turning point of each arc forms a dense but not complete set on inner edge of the concentric circles. Because we can degenerate its density into the non-existence of minimum absolute value within the set
{Rx+ry|x,y are integer}\{0}, the existence is equivalent with rationality ofR/r, so ifR/rirrational, it gives us the density as a result. It is, however, not complete because the set of such turning point is countable, but it must be uncountable if it was to cover the whole circle. This leads to contradiction and shows us that the set must be incomplete.Because the origins of each arc is incomplete, this immediately gives us that the total trajectory cannot be complete. However, because the density of the origin of the arc, for each two arc, we can surely find another arc in between, so that if we connect each two points on that pair of arc respectively we can certainly find a cross point of the in-between arc and the connection, which is also a part of the trajectory. Thus the trajectory must be dense.