Convert LifeOnTheEdge to LifeOnTheSlope
JavaScript (ES6),  135 120  119 bytes
Saved 16 bytes thanks to @Shaggy!
Expects either an array of strings or a matrix of characters. Returns a matrix of characters.

a=&gt;[...a,...a[0]].map((_,y,b)=&gt;b.map((_,x)=&gt;'\\/ '[(c=(a[~(x-=y+a.length)&gt;&gt;1]||0)[x+y-~y&gt;&gt;1])=='_|'[x&amp;=1]|c=='L'?x:2]))

Try it online!