Bézier moi(cormullion.github.io)
cormullion.github.io
Bézier moi
https://cormullion.github.io/blog/2018/06/21/bezier.html
6 comments
> probably the only risque mathematical term
Not a term, but there's the "Hairy Ball Theorem":
https://en.wikipedia.org/wiki/Hairy_ball_theorem
Not a term, but there's the "Hairy Ball Theorem":
https://en.wikipedia.org/wiki/Hairy_ball_theorem
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?
Now, who has an article about 3D wireframes?
Osculating polynomial is another one. It's basically a generalization of Taylor polynomials and Lagrange polynomials.
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.
It's also a renaissance French chanson for 4 voices.
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://www.youtube.com/watch?v=4abvrxNj89s
And then there is this movie which I'll never watch.
https://en.wikipedia.org/wiki/Baise-moi
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 :-)
Not french, but smiled anyway reading the title.
Ehh... nice curves!
Can we have the title changed please. HN shouldn't be a place for crude sexual jokes.
HN shouldn't be the place to change original articles.
And it shouldn't be the place to police writings with prudish 19th century ideas of "sexuality" as sin.
And it shouldn't be the place to police writings with prudish 19th century ideas of "sexuality" as sin.
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Woa, I've never received so many downvotes on HN. This speaks volumes.
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.
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.
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.
Or you could describe the quarter circle exactly using a rational spline.
The best could also be... a true quarter circle :)
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/
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/
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.
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.
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.
Yes, rust is a real problem with those cars but you can protect against it. There are also many body panels (doors, fenders) made out of fiberglass with internal re-inforcement, purists obviously hate these but they deal with the rusty panels problem once and for all. The roof is already fibreglass so at least that won't ever rust even in the originals.
I've never seen a DS Berline made in 75, the one I drove a lot was made Nov '74 and it was the newest that I've seen, my other one was from '68 (officially an ID 19B). There are however a lot of Breaks from '75.
As for the rust problem, the 'good' DSs were all striped down to bare metal, sandblasted and welded up then coated in tectyl or some other rust protector inside and out, an original, unrestored DS in good condition is something I haven't seen in many years. Maybe there are a couple in storage somewhere. Here in NL we still have several DS specialists, most notably Auto Renaissance in Amsterdam. There is also a company that converts Breaks into Decapotables, but that's not really my cup of tea. Though in convertible form it still is the most beautiful car there is to me.
Much good luck with your goddess, if you ever need parts that you can't get let me know and I'll see how warm my contacts in that scene are.
Edit: Amazing, I just checked, the license of my other DS/ID (not the one I took on long trips abroad) is still alive a good 20 years after I last saw the car, but it appears to be taken of the road when I sold it in 1998. DE-70-72 sits in a storage space somewhere. Let's hope one day someone puts it back on the road, I'd be happy to buy it today if I knew who the owner was.
I've never seen a DS Berline made in 75, the one I drove a lot was made Nov '74 and it was the newest that I've seen, my other one was from '68 (officially an ID 19B). There are however a lot of Breaks from '75.
As for the rust problem, the 'good' DSs were all striped down to bare metal, sandblasted and welded up then coated in tectyl or some other rust protector inside and out, an original, unrestored DS in good condition is something I haven't seen in many years. Maybe there are a couple in storage somewhere. Here in NL we still have several DS specialists, most notably Auto Renaissance in Amsterdam. There is also a company that converts Breaks into Decapotables, but that's not really my cup of tea. Though in convertible form it still is the most beautiful car there is to me.
Much good luck with your goddess, if you ever need parts that you can't get let me know and I'll see how warm my contacts in that scene are.
Edit: Amazing, I just checked, the license of my other DS/ID (not the one I took on long trips abroad) is still alive a good 20 years after I last saw the car, but it appears to be taken of the road when I sold it in 1998. DE-70-72 sits in a storage space somewhere. Let's hope one day someone puts it back on the road, I'd be happy to buy it today if I knew who the owner was.
I suppose saying that the DS was built into '75 is more interesting than useful. You'll be hard pressed to find a D Berline model from '75. According to the Original Citroën DS book from 1996, only 847 DS's were built in 1975 (compared to 40'039 in '74).
I am also fortunate to know a great DS specialist here in Denmark, who mostly imports his spare parts from the Netherlands and Germany, since that's where the good parts come from these days (Chinese parts have only gotten better in looking good, not actually working). In appreciation (and for helping working on my cars), I build him a website.
That being said, I do like actually working on the cars myself, but there are limits to what I can do when I live in a flat in the city. But don't worry, my cars are both parked in garages, as I want these to last a few more decades. The previous owner of my DS actually sandblasted it.
I am also fortunate to know a great DS specialist here in Denmark, who mostly imports his spare parts from the Netherlands and Germany, since that's where the good parts come from these days (Chinese parts have only gotten better in looking good, not actually working). In appreciation (and for helping working on my cars), I build him a website.
That being said, I do like actually working on the cars myself, but there are limits to what I can do when I live in a flat in the city. But don't worry, my cars are both parked in garages, as I want these to last a few more decades. The previous owner of my DS actually sandblasted it.
Do you have a semi-automatic or a clutch? What kind of engine? Pics :) ?
It's a DS 23 with a 5-speed manual.
Here is a picture of it at a car park: https://i.imgur.com/KJXYtOX.jpg And here next to a Aston Martin Vanquish: https://i.imgur.com/pPMDoEg.jpg
As you can see, it has been decorated as a Pallas, but it probably was not an original Pallas.
Here is a picture of it at a car park: https://i.imgur.com/KJXYtOX.jpg And here next to a Aston Martin Vanquish: https://i.imgur.com/pPMDoEg.jpg
As you can see, it has been decorated as a Pallas, but it probably was not an original Pallas.
Thank you for the posting of those images, beautiful car and very well maintained. Most of the better DS's are no 'pallassified', mostly because the parts are laying around anyway and if you have a DS you might as well have the nicest version. A 23i Pallas is as nice as they come. I wonder if it originally had the half-automatic gearbox. My first had that, it was a really nice system once you got the hang of it, but the 19B had a regular five speed manual as well and that was pretty much trouble free.
The biggest advantage of the half-automatic was that people that would ask to drive it were easily put off by the exotic gearbox :) You had to start it by pressing the gear shift lever to the left after making contact to avoid starting it in gear, clever interlock.
The biggest advantage of the half-automatic was that people that would ask to drive it were easily put off by the exotic gearbox :) You had to start it by pressing the gear shift lever to the left after making contact to avoid starting it in gear, clever interlock.
My mechanic actually has almost exactly the same model (same colour too!) from '74, but his has the Citromatic gearbox. Mine has always had the 5-speed manual. If not they would have needed to replace the engine as well.
The engine that went with the 5-speed manual was different than the other engines, because the transmission was bigger. It was therefore not possible to get an engine with the manual starter at the front of the car, if it was 5-speed.
The engine that went with the 5-speed manual was different than the other engines, because the transmission was bigger. It was therefore not possible to get an engine with the manual starter at the front of the car, if it was 5-speed.
Hehe, I wonder how many people realize what the hole in front of their DS grille is for :) The wheelnut wrench doubles as emergency starter. The best bit it still to show people how to change a tire on a DS. Magic!
Out of curiosity, is it possible to manually start it?
The reason I ask is that I've inherited a 1950 Jaguar, which also has a hole for crank starting the engine, but it's only these very early models that have the hole in the radiator for the crank. Apparently, the force required to start the XK engine manually is too much to be feasible for mere humans to achieve, and I've been told the best way to start it if the starter fails is to find a hill you can push the car down and putting it in gear.
The reason I ask is that I've inherited a 1950 Jaguar, which also has a hole for crank starting the engine, but it's only these very early models that have the hole in the radiator for the crank. Apparently, the force required to start the XK engine manually is too much to be feasible for mere humans to achieve, and I've been told the best way to start it if the starter fails is to find a hill you can push the car down and putting it in gear.
If you feel like breaking your wrist, then yes, you can try.
The problem is that if it misfires while you crank it (which is very easy because you are revolving the engine very slowly) then it will kick back against the direction of turn. Alternatively, the hole and the freeing mechanism will be so rusty that once it does catch it will continue to turn with the crank it. 2 chances for $1. It is supposed to free up once it starts running by pushing the crank forward against a slant cut into the receiver on the engine but that works as often as it doesn't.
I've done it, once. Even if you know what you are doing it is still dangerous and I would definitely not recommend it. Young, stupid, and so on. The good part is that the sleeve is long enough that it likely will not throw the crank out the way some other cars will do, it goes in quite far.
As for pushstarting it, that will work with the manual gearboxes, but it does not work with the semi-automatics.
The problem is that if it misfires while you crank it (which is very easy because you are revolving the engine very slowly) then it will kick back against the direction of turn. Alternatively, the hole and the freeing mechanism will be so rusty that once it does catch it will continue to turn with the crank it. 2 chances for $1. It is supposed to free up once it starts running by pushing the crank forward against a slant cut into the receiver on the engine but that works as often as it doesn't.
I've done it, once. Even if you know what you are doing it is still dangerous and I would definitely not recommend it. Young, stupid, and so on. The good part is that the sleeve is long enough that it likely will not throw the crank out the way some other cars will do, it goes in quite far.
As for pushstarting it, that will work with the manual gearboxes, but it does not work with the semi-automatics.
NURBS for the win... You know all the patents on NURBS have run out by now...
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.
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.
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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-kissin...) that summarizes Descartes's Theorem (https://en.wikipedia.org/wiki/Descartes%27_theorem):
EDIT: Turns out there are many other such math terms, e.g. see https://math.stackexchange.com/questions/1102872/unusual-mat...: