7 ms·

Bézier moi

Jun8 8y ago
Fantastic article and well written, too. For those of you, like me, who don't know French and miss the pun of the title: https://en.wikipedia.org/wiki/Baise-moi https://en.wikipedia.org/wiki/Baise-moi. It also mentions the concept of the osculating circle, probably the only risque mathematical term (anyone know of any other ones?), which led me of the following poem (from this post https://motherboard.vice.com/en_us/article/ezvq9n/the-kissing-circles-the-time-nature-published-a-poem-as-a-scientific-paper https://motherboard.vice.com/en_us/article/ezvq9n/the-kissin...) that summarizes Descartes's Theorem (https://en.wikipedia.org/wiki/Descartes%27_theorem https://en.wikipedia.org/wiki/Descartes%27_theorem): "For pairs of lips to kiss maybe Involves no trigonometry. This not so when four circles kiss Each one the other three. To bring this off the four must be As three in one or one in three. If one in three, beyond a doubt Each gets three kisses from without. If three in one, then is that one Thrice kissed internally. " EDIT: Turns out there are many other such math terms, e.g. see https://math.stackexchange.com/questions/1102872/unusual-mathematical-terms https://math.stackexchange.com/questions/1102872/unusual-mat...:
- sn41 8y ago
  > probably the only risque mathematical term Not a term, but there's the "Hairy Ball Theorem": https://en.wikipedia.org/wiki/Hairy_ball_theorem https://en.wikipedia.org/wiki/Hairy_ball_theorem
- kccqzy 8y ago
  Osculating polynomial is another one. It's basically a generalization of Taylor polynomials and Lagrange polynomials.
- nkg 8y ago
  I have been using bezier curves for years in Illustrator or Inkscape without any knowledge of how it worked under the hood. I guess that is how you know what a great UI is. Now, who has an article about 3D wireframes?
jacobolus 8y ago
Deciding which is the “best” approximation to a quarter circle depends on your choice of error metric. Just choosing the one with the correct endpoints and endpoint tangents which intersects the circle halfway along is not the best you can do by most choices of error metric. A more traditional choice might be the cubic curve where the maximum distance between a curve point and the circle is minimized. Or it might instead be desirable to preserve the arclength of the circle and minimize curvature deviation. Or it might be desirable to preserve the circle’s area. Or ... Depending on the application different choices might be better. For some purposes it is not strictly essential that the endpoints of the curve directly fall on the circle.
- raphlinus 8y ago
  Or in the case of my recent Euler explorer, minimizes the L2 norm of tangent error. That said, most of the error norms will give similar results, as long as you're actually minimizing something. You're absolutely right about this though, the main reason people pick that one is that the math comes out easy, it's just a bit of trig.
- ttoinou 8y ago
  The best could also be... a true quarter circle :)
- ginko 8y ago
  Or you could describe the quarter circle exactly using a rational spline.
  - jacobolus 8y ago
    https://beta.observablehq.com/@jrus/svg-elliptical-arcs https://beta.observablehq.com/@jrus/svg-elliptical-arcs
throwawayepgpp 8y ago
Off-topic, but the title (Bézier moi) sounds like "Baisez-moi", which in French pretty much means "Fuck me". Seeing how most people involved in the article are French I assume the title was chosen on purpose, in case any reader was wondering about it.
- catoc 8y ago
  Not french, but smiled anyway reading the title. Ehh... nice curves!
- PavlovsCat 8y ago
  It's also a renaissance French chanson for 4 voices. https://www.youtube.com/watch?v=4abvrxNj89s https://www.youtube.com/watch?v=4abvrxNj89s And then there is this movie which I'll never watch. https://en.wikipedia.org/wiki/Baise-moi https://en.wikipedia.org/wiki/Baise-moi
  - catoc 8y ago
    Incidently that movie is brilliant. Graphic but briljant. And has nothing whatsoever to do with bezier curves; thus completely off topic, so I will leave it at that. (but briljant :-)
- tmalsburg2 8y ago
  Can we have the title changed please. HN shouldn't be a place for crude sexual jokes.
  - enz 8y ago
    This is the original title of the linked article.
    - tmalsburg2 8y ago
      So ...?
  - coldtea 8y ago
Svip 8y ago
The inclusion of a Citroën DS is nice (as an owner of '74 model, I am a little biased), but it's weird that he doesn't mention the Citroën CX, the fruits of de Casteljau's labours. It's call the CX because it refers to the coefficient of drag (Cd in English, but CX in French). I have the definitive book on the CX, de originale Citroën CX (it's in Dutch)[0], and they spend a lot of time discussing the mathematics of the design. It's only unfortunate that I don't own a CX (although I used to have a '85 TRS 22, but it was in too bad shape to keep). [0] See here for an English article discussing the book: https://citroenvie.com/new-citroen-cx-book-available/ https://citroenvie.com/new-citroen-cx-book-available/
- jacquesm 8y ago
  I've had both. The DS was pretty much indestructible, drove it all throughout Europe including the former Eastblock when it was still called that. The occasional repair of the exhaust system with tin cans and steel wire were the only repairs it needed. The CX was a different matter. I had two of these, the first dropped its engine on A4 somewhere between Amsterdam and The Hague, the second one blew a hole in a piston. Both within a few weeks of buying them. So maybe it is not so bad that you don't have a CX, and consider me jealous with your DS, the 74 is the last year they were made iirc, and if you maintain it well will likely outlast you. As comfy a ride as you'll likely ever have, even today very little comes close.
  - Svip 8y ago
    The DS was built all the way into 1975, although considering '74 the last is not far off. As I understand CX reliability, it's generally advised to avoid mid-80s carburettor and diesel models. If you want a CX, they say Prestige, Turbo I or II or GTI. All other models are basically not worth the trouble. But the TRS I had was also badly maintained by the previous owner, which is why I eventually abandoned it and got a '98 Xantia Activa V6 instead. That being said, I still want a CX, just a good one. And they do exist (good ones). As for the DS being indestructible, the engine definitely was, but the body work would often erode in northern climates like mine (Denmark). And particularly at 40+ years of age, there are signs of its age. Fortunately, mine runs fine, so I hope to keep it at least for another 20 years.
PaulHoule 8y ago
NURBS for the win... You know all the patents on NURBS have run out by now...
- phkahler 8y ago
  The problem with NURBS is that you very quickly need trimmed NURBS to model anything of even modest complexity, and those are unpleasant for software to work with. I've been toying with the idea (not new) of triangular NURBS which could do away with the need for trimming entirely. Questions remain about which type of triangular surface patch will meet the requirements.
deleted 8y ago
[deleted]

Jun8 8y ago

Fantastic article and well written, too. For those of you, like me, who don't know French and miss the pun of the title: https://en.wikipedia.org/wiki/Baise-moi https://en.wikipedia.org/wiki/Baise-moi. It also mentions the concept of the osculating circle, probably the only risque mathematical term (anyone know of any other ones?), which led me of the following poem (from this post https://motherboard.vice.com/en_us/article/ezvq9n/the-kissing-circles-the-time-nature-published-a-poem-as-a-scientific-paper https://motherboard.vice.com/en_us/article/ezvq9n/the-kissin...) that summarizes Descartes's Theorem (https://en.wikipedia.org/wiki/Descartes%27_theorem https://en.wikipedia.org/wiki/Descartes%27_theorem): "For pairs of lips to kiss maybe Involves no trigonometry. This not so when four circles kiss Each one the other three. To bring this off the four must be As three in one or one in three. If one in three, beyond a doubt Each gets three kisses from without. If three in one, then is that one Thrice kissed internally. " EDIT: Turns out there are many other such math terms, e.g. see https://math.stackexchange.com/questions/1102872/unusual-mathematical-terms https://math.stackexchange.com/questions/1102872/unusual-mat...:

sn41 8y ago
> probably the only risque mathematical term Not a term, but there's the "Hairy Ball Theorem": https://en.wikipedia.org/wiki/Hairy_ball_theorem https://en.wikipedia.org/wiki/Hairy_ball_theorem
kccqzy 8y ago
Osculating polynomial is another one. It's basically a generalization of Taylor polynomials and Lagrange polynomials.
nkg 8y ago
I have been using bezier curves for years in Illustrator or Inkscape without any knowledge of how it worked under the hood. I guess that is how you know what a great UI is. Now, who has an article about 3D wireframes?

jacobolus 8y ago

Deciding which is the “best” approximation to a quarter circle depends on your choice of error metric. Just choosing the one with the correct endpoints and endpoint tangents which intersects the circle halfway along is not the best you can do by most choices of error metric. A more traditional choice might be the cubic curve where the maximum distance between a curve point and the circle is minimized. Or it might instead be desirable to preserve the arclength of the circle and minimize curvature deviation. Or it might be desirable to preserve the circle’s area. Or ... Depending on the application different choices might be better. For some purposes it is not strictly essential that the endpoints of the curve directly fall on the circle.

raphlinus 8y ago
Or in the case of my recent Euler explorer, minimizes the L2 norm of tangent error. That said, most of the error norms will give similar results, as long as you're actually minimizing something. You're absolutely right about this though, the main reason people pick that one is that the math comes out easy, it's just a bit of trig.
ttoinou 8y ago
The best could also be... a true quarter circle :)
ginko 8y ago
Or you could describe the quarter circle exactly using a rational spline.
- jacobolus 8y ago
  https://beta.observablehq.com/@jrus/svg-elliptical-arcs https://beta.observablehq.com/@jrus/svg-elliptical-arcs

throwawayepgpp 8y ago

Off-topic, but the title (Bézier moi) sounds like "Baisez-moi", which in French pretty much means "Fuck me". Seeing how most people involved in the article are French I assume the title was chosen on purpose, in case any reader was wondering about it.

catoc 8y ago
Not french, but smiled anyway reading the title. Ehh... nice curves!
PavlovsCat 8y ago
It's also a renaissance French chanson for 4 voices. https://www.youtube.com/watch?v=4abvrxNj89s https://www.youtube.com/watch?v=4abvrxNj89s And then there is this movie which I'll never watch. https://en.wikipedia.org/wiki/Baise-moi https://en.wikipedia.org/wiki/Baise-moi
- catoc 8y ago
  Incidently that movie is brilliant. Graphic but briljant. And has nothing whatsoever to do with bezier curves; thus completely off topic, so I will leave it at that. (but briljant :-)
tmalsburg2 8y ago
Can we have the title changed please. HN shouldn't be a place for crude sexual jokes.
- enz 8y ago
  This is the original title of the linked article.
  - tmalsburg2 8y ago
    So ...?
- coldtea 8y ago

Svip 8y ago

The inclusion of a Citroën DS is nice (as an owner of '74 model, I am a little biased), but it's weird that he doesn't mention the Citroën CX, the fruits of de Casteljau's labours. It's call the CX because it refers to the coefficient of drag (Cd in English, but CX in French). I have the definitive book on the CX, de originale Citroën CX (it's in Dutch)[0], and they spend a lot of time discussing the mathematics of the design. It's only unfortunate that I don't own a CX (although I used to have a '85 TRS 22, but it was in too bad shape to keep). [0] See here for an English article discussing the book: https://citroenvie.com/new-citroen-cx-book-available/ https://citroenvie.com/new-citroen-cx-book-available/

jacquesm 8y ago
I've had both. The DS was pretty much indestructible, drove it all throughout Europe including the former Eastblock when it was still called that. The occasional repair of the exhaust system with tin cans and steel wire were the only repairs it needed. The CX was a different matter. I had two of these, the first dropped its engine on A4 somewhere between Amsterdam and The Hague, the second one blew a hole in a piston. Both within a few weeks of buying them. So maybe it is not so bad that you don't have a CX, and consider me jealous with your DS, the 74 is the last year they were made iirc, and if you maintain it well will likely outlast you. As comfy a ride as you'll likely ever have, even today very little comes close.
- Svip 8y ago
  The DS was built all the way into 1975, although considering '74 the last is not far off. As I understand CX reliability, it's generally advised to avoid mid-80s carburettor and diesel models. If you want a CX, they say Prestige, Turbo I or II or GTI. All other models are basically not worth the trouble. But the TRS I had was also badly maintained by the previous owner, which is why I eventually abandoned it and got a '98 Xantia Activa V6 instead. That being said, I still want a CX, just a good one. And they do exist (good ones). As for the DS being indestructible, the engine definitely was, but the body work would often erode in northern climates like mine (Denmark). And particularly at 40+ years of age, there are signs of its age. Fortunately, mine runs fine, so I hope to keep it at least for another 20 years.

PaulHoule 8y ago

NURBS for the win... You know all the patents on NURBS have run out by now...

phkahler 8y ago
The problem with NURBS is that you very quickly need trimmed NURBS to model anything of even modest complexity, and those are unpleasant for software to work with. I've been toying with the idea (not new) of triangular NURBS which could do away with the need for trimming entirely. Questions remain about which type of triangular surface patch will meet the requirements.

deleted 8y ago

[deleted]