How to make an object to take smooth turns at edges between points?

I want an object to move between points, say from point A to B then B to C then C to D and so on. But I want the object to make smooth turns at the edges instead of an abrupt 90 degrees turn. Like the object takes a curved path at the edge.

You can use THREE paths for this like the fiddle below, or a direct implementation for a bezier curve if you know how you will compute the tangents
https://jsfiddle.net/wilt/mhxtdfp8/40/

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It seems the fiddle is broken. Here is an updated version: https://jsfiddle.net/wb6j52q7/

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