Need to draw a polygon that parallels a randomized line

[Mikey]

Is the poly centered on the line?  How many verticies can there be? 
How many segments?  Is there some limit as to where it can wander?  Is
there some limit as to what area it can encompass?

When you say it "parallels" the line, do you mean that you only have
four sides on a rhombus?

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