What do the following $\varepsilon$-balls in $S^{1}$ look like? And induced topology

Question

Define a metric on $S^{1}:=[0,1)/_{0\sim1}$ by $$d_{1}(x,y)=\min\{|x-y\mod1|,1-|x-y\mod1|\}.$$ Given an $x\in S^{1}$ and $\varepsilon>0$, what does the '$\varepsilon$-ball around $x$' (or interval) in $S^{1}$ look like in this metric?

Also, is $d(x,y):=|x-y|$ a metric on $S^{1}$ and does it induce the same topology as $d_{1}$?