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Here is my construction,

If e1 or e2 are positioned a non-zero angle with respect to f, a single intersection point is directly considered. If e1 is unfixed, the sweeping of e1 through the intersection point is allowed.

My question is as I am trying to use only one edge (e1 and e2 are edges), in the above are should be is or not

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The word ‘or’ is used here as a disjunctive conjunction.
That means that ‘or’ does not conjoin as ‘and’ does.

There are two kinds of logical disjunction:

  • Inclusive means “and/or” - at least one of them is true, or maybe both.
  • Exclusive (“xor”) means exactly one must be true, but they cannot both be.
    In our case ‘or’ is exclusive; we should choose from one of the alternatives.

The American Heritage Dictionary describes the usage as follows:

When all the elements in a series connected by ‘or’ are singular, the verb they govern is singular: Tom or Jack is coming. Beer, ale, or wine is included in the charge. When all the elements are plural, the verb is plural. When the elements do not agree in number, some grammarians have suggested that the verb should agree in number with the nearest element: Tom or his sisters are coming. The girls or their brother is coming. Cold symptoms or headache is the usual first sign. Other grammarians, however, have argued that such constructions are inherently illogical and that the only solution is to revise the sentence to avoid the problem of agreement: Either Tom is coming or his sisters are. The usual first sign may be either cold symptoms or a headache.

I cannot contradict any of them even if the second group seems to be more logical.

If a greater distinction is needed, another phrasing is available: entertaining or traveling, or both.

In your question if you insist on using plural form of the verb your statement should be phrased this way:

If e1 or e2 or both are positioned...

Otherwise you should use:

If e1 or e2 is positioned...

That is, you must choose one of the alternatives.

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I could not think of the pattern -> if + singular noun + or + singular noun + are + verb. Can anyone come up with? And if this is the case, it's is there.

Also, StoneyB makes it clear - If e1 is positioned.... or e2 is positioned = If e1 or e2 is positioned...

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