
Bézier moi - one-more-minute
https://cormullion.github.io/blog/2018/06/21/bezier.html
======
Jun8
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-
kissin...](https://motherboard.vice.com/en_us/article/ezvq9n/the-kissing-
circles-the-time-nature-published-a-poem-as-a-scientific-paper)) 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-
mat...](https://math.stackexchange.com/questions/1102872/unusual-mathematical-
terms):

~~~
sn41
> 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)

------
throwawayepgpp
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.

~~~
tmalsburg2
Can we have the title changed please. HN shouldn't be a place for crude sexual
jokes.

~~~
coldtea
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.

------
jacobolus
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.

~~~
ginko
Or you could describe the quarter circle exactly using a rational spline.

~~~
jacobolus
[https://beta.observablehq.com/@jrus/svg-elliptical-
arcs](https://beta.observablehq.com/@jrus/svg-elliptical-arcs)

------
Svip
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
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
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.

~~~
jacquesm
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.

~~~
Svip
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.

~~~
jacquesm
Do you have a semi-automatic or a clutch? What kind of engine? Pics :) ?

~~~
Svip
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](https://i.imgur.com/KJXYtOX.jpg) And here
next to a Aston Martin Vanquish:
[https://i.imgur.com/pPMDoEg.jpg](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.

~~~
jacquesm
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.

~~~
Svip
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.

~~~
jacquesm
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!

~~~
arnsholt
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.

~~~
jacquesm
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.

~~~
arnsholt
Yeah, that sounds more or less like what the XK veterans told me: You can
certainly try, but you probably don't want to.

~~~
jacquesm
Super cool car you have. My former accountant had one too, he always took it
on trips to France for holidays every year. A bit finicky to keep running well
but with a good mechanic that should be doable as long as you don't start
using it as a daily driver.

~~~
arnsholt
> My former accountant had one too, he always took it on trips to France for
> holidays every year.

Oh, that's funny. The owners of the hostel we stayed at in the Netherlands
used to do that, too. Apparently it was one of the V12 Jags from the 70s or
80s (Daimler Sovereign or XJ I think), and for good measure they had an after
market trailer hitch attached, so they could bring their caravan along for the
trip! I don't dare thinking about how much money that must've cost in fuel
alone...

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

~~~
phkahler
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.

