# NURBS curve offset

One of the benefits of NURBS curve over, say Bezier curve, is the ability to create offset curves exactly. How to proceed with such computation? Do I just translate the control points?

• What do you mean exactly by offset curves? Aug 25, 2015 at 14:44
• An "offset" curve has constant distance to the given curve, aka parallel curve. Aug 25, 2015 at 14:47
• I wonder what property of NURBS make them able to do this. I would think it would be that they are rational, but then rational bezier curves would also have this property. Aug 28, 2015 at 4:07
• Where did you see this claim? It is rather surprising to me (but I don't immediately see that it's impossible). Sep 16, 2015 at 6:19

Quickly Googling produces:

1. This related question on StackOverflow
2. "Computing offsets of NURBS curves and surfaces" (paper)
3. Discussions of 2.

To summarize them:

1. Suggests that this is impossible to do exactly.
2. Gives an algorithm to compute an offset curve approximately (though to within a tiny epsilon). If you can't access the paper, then:
3. Look at any of these. This presentation for example seems to cover it.
• Bit of nitpicking its not impossible to do exactly. Its just not possible to do exactly for all possible cases. There are lots of cases where this can in fact be done exactly its just a bit hard to generalize this. Sep 23, 2015 at 23:40
• @joojaa IMO possible cases are exceptions rather than the rule. Can you link to a reference ? Sep 23, 2015 at 23:50
• Yes but thats enough to refute impossible. Not in front of a desktop right now bit consider this: Since rational splines can do exact circular arcs that means you can do exact offsets of said arc. Since linear splines are possible and their offsets contains lines and circular arcs that too is possible. A big set of realworld uses there. Sep 23, 2015 at 23:55
• So if you say impossible to do exactly in the general case then its ok. Sep 24, 2015 at 0:04
• Point being, what you say should be self contained. You should update your post so that others who dont have the time dont make wrong conclusions. Remember discussions under posts are not permanent. What you dont say on the otherhand you can leave to the link. Sep 24, 2015 at 6:58