<p>Quickly Googling produces:</p>

<ol>
<li><a href="https://stackoverflow.com/questions/4148831/how-to-offset-a-cubic-bezier-curve">This related question</a> on StackOverflow</li>
<li>"Computing offsets of NURBS curves and surfaces" (paper)</li>
<li>Discussions of 2.</li>
</ol>

<p>To summarize them:</p>

<ol>
<li>Suggests that this is impossible to do exactly.</li>
<li>Gives an algorithm to compute an offset curve approximately (though to within a tiny epsilon).  If you can't access the paper, then:</li>
<li>Look at any of these.  <a href="https://www.google.com/url?sa=t&amp;rct=j&amp;q=&amp;esrc=s&amp;source=web&amp;cd=7&amp;cad=rja&amp;uact=8&amp;ved=0CDwQFjAGahUKEwiihLzeg47IAhVHN4gKHfsmCaM&amp;url=http%3A%2F%2Fwww.math.zju.edu.cn%2Fcagd%2FSeminar%2F2005_AutumnWinter%2F2005_Master_LH.ppt&amp;usg=AFQjCNGlb556TMftSL8d4xU7kT1x7Q9G7Q&amp;sig2=9rYaCjpqD-VMKu94ke5O7w" rel="nofollow noreferrer">This presentation</a> for example seems to cover it.</li>
</ol>

CC BY-SA 3.0
2015-09-23T23:40:56.507Z
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.
joojaa
CC BY-SA 3.0
2015-09-23T23:55:40.947Z
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.
joojaa
CC BY-SA 3.0
2015-09-24T00:04:09.457Z
So if you say impossible to do exactly in the general case then its ok.
joojaa
CC BY-SA 3.0
2015-09-24T02:16:05.580Z
Yes; I meant impossible in the general case (at least for Bézier). If you actually read the answers I was linking to when I said this, you'll find it says exactly that.
geometrian
CC BY-SA 3.0
2015-09-24T06:58:26.377Z
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.
joojaa
CC BY-SA 3.0
2015-09-23T20:49:14.113Z
2015-09-23T20:49:14.113Z
2017-05-23T12:39:19.850Z
geometrian
<p>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?</p>

CC BY-SA 3.0
2015-08-25T14:44:27.810Z
What do you mean exactly by offset curves?
Alan Wolfe
CC BY-SA 3.0
2015-08-25T14:47:56.580Z
An "offset" curve has constant distance to the given curve, aka [parallel curve](http://mathworld.wolfram.com/ParallelCurves.html).
Ecir Hana
CC BY-SA 3.0
2015-08-28T04:07:40.140Z
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.
Alan Wolfe
CC BY-SA 3.0
2015-09-16T06:19:04.597Z
Where did you see this claim? It is rather surprising to me (but I don't immediately see that it's impossible).
Etienne
<nurbs>
NURBS curve offset
Ecir Hana
# NURBS curve offset

(Asked by: username_0 on 2015-08-25)

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?

**Tags:** nurbs

> **Comments:**
> **username_1**: What do you mean exactly by offset curves?  
> **username_0**: An "offset" curve has constant distance to the given curve, aka [parallel curve](http://mathworld.wolfram.com/ParallelCurves.html).  
> **username_1**: 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.  
> **username_2**: Where did you see this claim? It is rather surprising to me (but I don't immediately see that it's impossible).  

---

## 1 answers

### Answer by username_3 on 2015-09-23. Score: 1

Quickly Googling produces:

1. [This related question](https://stackoverflow.com/questions/4148831/how-to-offset-a-cubic-bezier-curve) 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](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=7&cad=rja&uact=8&ved=0CDwQFjAGahUKEwiihLzeg47IAhVHN4gKHfsmCaM&url=http%3A%2F%2Fwww.math.zju.edu.cn%2Fcagd%2FSeminar%2F2005_AutumnWinter%2F2005_Master_LH.ppt&usg=AFQjCNGlb556TMftSL8d4xU7kT1x7Q9G7Q&sig2=9rYaCjpqD-VMKu94ke5O7w) for example seems to cover it.

> **Comments:**
> **username_4**: 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.  
> **username_4**: 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.  
> **username_4**: So if you say impossible to do exactly in the general case then its ok.  
> **username_3**: Yes; I meant impossible in the general case (at least for Bézier). If you actually read the answers I was linking to when I said this, you'll find it says exactly that.  
> **username_4**: 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.  

---

