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Would it be easier to use rational beziers (NURBS) rather than just beziers?

by maire 8y ago

jacobolus 8y ago
Easier for what? Rational curves give you another degree of freedom. CAD people like them because they can exactly represent conics. Neither polynomials nor NURBS are IMO especially intuitive to work with as a design tool. As an approximate output target I think cubic polynomials are okay: a few cubic segments strung together are plenty flexible for typical drawing/modeling purposes.
- maire 8y ago
  "Easier for what?" is a legitimate question. I also had the question after I submitted my question because the "for what" wasn't specified in the article. If the target application only takes beziers, then that is the only tool you have. My original question was that a rational bezier would be easier to use since a rational bezier would make an exact representation of a Euler Curve instead of an approximate representation. I agree with you that full NURBs are difficult. But if you ignore the "Non Uniform" (NU) part of NURBs and just use the "Rational BSpline" (RB) part of NURBs then they are as easy to use as just beziers. The rational term just means a projection from n+1 space to n space. It is what gives an exact representation of a conic. If you want to fully model the parametric equation then you would need the "non uniform" part. I don't know why you would want to do this since you can analyze the original curve directly.
  - jacobolus 8y ago
    You cannot represent a section of an Euler spiral exactly with a polynomial or rational function. You can make a fine approximation if you either (a) increase the degree of the polynomial or (b) increase the number of segments. There is no particular advantage of NURBS for this vs. polynomials as far as I can tell.
    - maire 8y ago
      You are right. I just came across this article which talks about using rational cubic trigonometric Bézier curves. https://link.springer.com/article/10.1186/s40064-016-2544-3 https://link.springer.com/article/10.1186/s40064-016-2544-3
      - jacobolus 8y ago
        I don’t really see the point of this paper. The reason to try to approximate Euler spirals by Bézier curves is that Bézier curves are ubiquitous in existing implementations. Trigonometric Bézier curves (https://ac.els-cdn.com/S0893965908001262/1-s2.0-S0893965908001262-main.pdf?_tid=8924d36c-e3b8-40a2-8e39-946f8ff09f00&acdnat=1544395363_14596f5d6adba170cd34f4745364919e https://ac.els-cdn.com/S0893965908001262/1-s2.0-S08939659080...) are not to my knowledge implemented anywhere.

Would it be easier to use rational beziers (NURBS) rather than just beziers?

by maire 8y ago

Would it be easier to use rational beziers (NURBS) rather than just beziers?

Easier for what? Rational curves give you another degree of freedom. CAD people like them because they can exactly represent conics. Neither polynomials nor NURBS are IMO especially intuitive to work with as a design tool. As an approximate output target I think cubic polynomials are okay: a few cubic segments strung together are plenty flexible for typical drawing/modeling purposes.

maire 8y ago
"Easier for what?" is a legitimate question. I also had the question after I submitted my question because the "for what" wasn't specified in the article. If the target application only takes beziers, then that is the only tool you have. My original question was that a rational bezier would be easier to use since a rational bezier would make an exact representation of a Euler Curve instead of an approximate representation. I agree with you that full NURBs are difficult. But if you ignore the "Non Uniform" (NU) part of NURBs and just use the "Rational BSpline" (RB) part of NURBs then they are as easy to use as just beziers. The rational term just means a projection from n+1 space to n space. It is what gives an exact representation of a conic. If you want to fully model the parametric equation then you would need the "non uniform" part. I don't know why you would want to do this since you can analyze the original curve directly.
- jacobolus 8y ago
  You cannot represent a section of an Euler spiral exactly with a polynomial or rational function. You can make a fine approximation if you either (a) increase the degree of the polynomial or (b) increase the number of segments. There is no particular advantage of NURBS for this vs. polynomials as far as I can tell.
  - maire 8y ago
    You are right. I just came across this article which talks about using rational cubic trigonometric Bézier curves. https://link.springer.com/article/10.1186/s40064-016-2544-3 https://link.springer.com/article/10.1186/s40064-016-2544-3
    - jacobolus 8y ago
      I don’t really see the point of this paper. The reason to try to approximate Euler spirals by Bézier curves is that Bézier curves are ubiquitous in existing implementations. Trigonometric Bézier curves (https://ac.els-cdn.com/S0893965908001262/1-s2.0-S0893965908001262-main.pdf?_tid=8924d36c-e3b8-40a2-8e39-946f8ff09f00&acdnat=1544395363_14596f5d6adba170cd34f4745364919e https://ac.els-cdn.com/S0893965908001262/1-s2.0-S08939659080...) are not to my knowledge implemented anywhere.