= Central eclipse contact-radius relations
{title2=$R_L\pm R_S=a\sin(\pi T_\pm/P)$}
For a circular edge-on <eclipsing binary> with central passage, define $T_+=t_4-t_1$ between external contacts and $T_-=t_3-t_2$ between internal contacts, with $R_L\ge R_S$. The projected separation from conjunction is $a|\sin[2\pi(t-t_c)/P]|$. At contact it equals $R_L+R_S$ or $R_L-R_S$, proving the title formula. Taking half the sum and difference recovers the radii. When durations are small compared with $P$, this becomes $R_{L,S}=v_{\rm rel}(T_+\pm T_-)/4$ with $v_{\rm rel}=2\pi a/P$. A nonzero impact parameter changes the contact chords.
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