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Suppose I have a quad-edge data-structure, and I want to be able to perform the operations of DCEL (twin, next, prev) on the quad-edge data structure. Let the current edge I am operating on be e. I know that e.twin would simply be e.rot.rot. I am just unsure about next and prev.

(Quad-Edge visualization given below.)

Quad Edge Visualization

next should be equivalent to Lnext in the picture. Thus I tried to derive it and got e.rot.rot.rot.Onext.rot. prev should be equivalent to Lprev, which is the twin of Onext, i.e. e.Onext.rot.rot.

Is this correct? I am using quad-edges to prove properties on half-edges, so I want to make sure I got the basics correct before I proceed.

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It looks like you took these diagrams from this CMU publication. This publication also refers to the C++ header cell/edge.hh, describing basic and derived operations on edges.

So, according to this header, operations Rot, InvRot, Sym and Onext are considered basic ones, and all the other operations - derived ones. The DCEL operations next and prev are equivalent to quad-edge operations Lnext and Lprev, which are defined in this header as inline functions (I'm dropping the C++ syntax and keeping the meaning):

e.Lnext = e.InvRot.Onext.Rot
e.Lprev = e.Onext.Sym

So, your formulas look correct, with slightly overused operation Rot.

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