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For $\frac {\pi} {2}$ I have found maximum 4 point on every $C$ which has no intersection under the following rotation ,also for $\frac {\pi} {3}\le\alpha$ < $\frac {\pi} {2}$ maximum 3 points and for $\alpha$ < $\frac {\pi} {3}$ there must be maximum 2 points satisfying the condition, I'm investigating for $\frac {\pi} {2}$ < $\alpha$... https://math.stackexchange.com/questions/2773404/prove-that-there-are-maximum-4-points-on-every-closed-jordan-curve-c-which-avo