"This site (obviously) works best with JS enabled
But it's not required.
If you're reading this text block, then you have scripts disabled: thankfully, that's perfectly fine, and this site is not going to punish you for making smart choices around privacy and security in your browser. All the content will show just fine, you can still read the text, navigate to sections, and see the graphics that are used to illustrate the concepts that individual sections talk about."
That is how I'd like to the rest of the internet to work as well.
Agreed, but it is worth noting that the inline apps on the Bézier page won’t work with JS disabled. And they are very helpful and well done. The quote is too nice and doesn’t explicitly admit that the experience is degraded & incomplete without the interactive examples; this one is worth an exception.
I'm under the impression, maybe wrongly so, that every other week we saw a primer on some basic CG stuff: Bézier curves, Fourier transforms, Dithering, Tonemapping, ..etc, of themes being fetched from a pool of maybe 10 items that cycle every once in a while but get upvoted because CG stuffs are inherently cool (and they're often well written like this one).
I think I'm gonna make `primersprimer.graphics` to list them or something.
The cynic would say primers on basic stuff are easy to create even if you're not really any sort of expert in the area, so people create a lot of them and lots of people who also aren't expert think that's great.
This, however, is pretty damn nifty, esp relative to most 'let me explain to you with prettier pictures what google (now chatgpt) just explained to me' fluff.
In its defense, this is also happens to be a really good link, and every time it gets posted, new people encounter it for the first time. (And I'm always happy to see it once more).
I learned a lot about Bezier curves when I was playing a lot of No Limits, a roller coaster designer/simulator. It used beziers for defining the track, and it was always difficult to get a smooth track because you needed the radius of the curve to be the same on both sides of every control point. Otherwise, there'd be a noticeable "bump" as the change in change of direction (not a typo) was sudden.
Also, the math of quadratic beziers made it so even if you did get the handles of each control point the right length to avoid a bump, you could get a "pumping" effect as the radius of curvature changed from wide to narrow and back to wide, though you could avoid this by avoiding trying to create circles in your track design.
No Limits 2 eventually came out which uses a different type of curve which was far easier to make smooth. It also included a way to define the track by graphing out the ride's G-forces and banking changes, and it would plot the control points for you. It was an absolute game changer for making smooth track.
Often in mathematics (and by extension most science I guess), when something is named after someone, it usually means there's a huge amount going on behind the scenes if you start looking.
A very simple explanation of Bézier curves is this:
- You have one polynomial describing the x-coordinate and one describing the y-coordinate, and both polynomials have the same degree (two for quadratic, three for cubic Bézier curves)
- The two polynomials share the same parameter t, which runs from 0 to 1.
I haven't thought about Bézier Curves since my undergrad a long time ago. I distinctly remember wondering at the time why so many lecturers added extra hurdles (i.e. the need to understand the intricacies of Bézier Curves) in their assignments rather than letting students focus on the computer science/programming concepts they were meant to be learning.
What concepts were you “meant” to be learning? Doesn’t the content of the lectures give you a clue as to the educators’ intent? Are you aware that Bézier curves cover many fundamental CS topics, such as recursion, function evaluation, function approximation, numerical analysis, linear algebra, root finding, and more? Do you know they’re used heavily in a lot of software you use? (Browsers, font rendering, maps, nonlinear numerical libraries, UI frameworks, games, etc.)
Same here. Bézout also was another mysterious killer.
Concepts coming from french mathematicians were made more obscure just to raise the bar. The irony is, in french Universities.
I recall a student who had enough failing the computer based assessments. He kindly asked the lead lecturer to show us all that he, at least, could land a perfect score. He made the mistake to try, got 8 points out of 20.
He admitted it wasn't easy when not prepared, and moved on with the next mined field.
Related, I think Freya Holmér's "The Beauty of Bézier Curves" is in the running for one of the best educational videos on YouTube.
https://youtu.be/aVwxzDHniEw?si=K7QYf4luKhgv2mgd
All her videos are great.
"This site (obviously) works best with JS enabled But it's not required.
If you're reading this text block, then you have scripts disabled: thankfully, that's perfectly fine, and this site is not going to punish you for making smart choices around privacy and security in your browser. All the content will show just fine, you can still read the text, navigate to sections, and see the graphics that are used to illustrate the concepts that individual sections talk about."
That is how I'd like to the rest of the internet to work as well.
Agreed, but it is worth noting that the inline apps on the Bézier page won’t work with JS disabled. And they are very helpful and well done. The quote is too nice and doesn’t explicitly admit that the experience is degraded & incomplete without the interactive examples; this one is worth an exception.
I'm under the impression, maybe wrongly so, that every other week we saw a primer on some basic CG stuff: Bézier curves, Fourier transforms, Dithering, Tonemapping, ..etc, of themes being fetched from a pool of maybe 10 items that cycle every once in a while but get upvoted because CG stuffs are inherently cool (and they're often well written like this one).
I think I'm gonna make `primersprimer.graphics` to list them or something.
The cynic would say primers on basic stuff are easy to create even if you're not really any sort of expert in the area, so people create a lot of them and lots of people who also aren't expert think that's great.
This, however, is pretty damn nifty, esp relative to most 'let me explain to you with prettier pictures what google (now chatgpt) just explained to me' fluff.
I think it's just that the internet is particularly well suited for explanations on the intersection between mathematics and graphics
Maybe your impression also has something to do with this being the 20th repost of the same link.
In its defense, this is also happens to be a really good link, and every time it gets posted, new people encounter it for the first time. (And I'm always happy to see it once more).
I learned a lot about Bezier curves when I was playing a lot of No Limits, a roller coaster designer/simulator. It used beziers for defining the track, and it was always difficult to get a smooth track because you needed the radius of the curve to be the same on both sides of every control point. Otherwise, there'd be a noticeable "bump" as the change in change of direction (not a typo) was sudden.
Also, the math of quadratic beziers made it so even if you did get the handles of each control point the right length to avoid a bump, you could get a "pumping" effect as the radius of curvature changed from wide to narrow and back to wide, though you could avoid this by avoiding trying to create circles in your track design.
No Limits 2 eventually came out which uses a different type of curve which was far easier to make smooth. It also included a way to define the track by graphing out the ride's G-forces and banking changes, and it would plot the control points for you. It was an absolute game changer for making smooth track.
I've had this bookmarked for a while. Fantastic resource. It's been on HN dozens of times since 2017.
https://news.ycombinator.com/item?id=14191577
Past comments: https://news.ycombinator.com/item?id=8804691, https://news.ycombinator.com/item?id=30100427
I used Bézier Curves to draw SVG curves for my weather dashboard EPD https://github.com/mt-empty/pi-inky-weather-epd
have been a graphic designer for the last 15 years, never know how complex a simple bezier curve is. solid resource.
Often in mathematics (and by extension most science I guess), when something is named after someone, it usually means there's a huge amount going on behind the scenes if you start looking.
And slow motion was never the same hereafter
A very simple explanation of Bézier curves is this:
- You have one polynomial describing the x-coordinate and one describing the y-coordinate, and both polynomials have the same degree (two for quadratic, three for cubic Bézier curves)
- The two polynomials share the same parameter t, which runs from 0 to 1.
That's all.
I haven't thought about Bézier Curves since my undergrad a long time ago. I distinctly remember wondering at the time why so many lecturers added extra hurdles (i.e. the need to understand the intricacies of Bézier Curves) in their assignments rather than letting students focus on the computer science/programming concepts they were meant to be learning.
What concepts were you “meant” to be learning? Doesn’t the content of the lectures give you a clue as to the educators’ intent? Are you aware that Bézier curves cover many fundamental CS topics, such as recursion, function evaluation, function approximation, numerical analysis, linear algebra, root finding, and more? Do you know they’re used heavily in a lot of software you use? (Browsers, font rendering, maps, nonlinear numerical libraries, UI frameworks, games, etc.)
Because if you have the time and opportunity to study something in depth, then it should be taken imo.
If I just want to get a working product I only need the basic algorithm, but understanding "all" of it is never wrong
Same here. Bézout also was another mysterious killer.
Concepts coming from french mathematicians were made more obscure just to raise the bar. The irony is, in french Universities.
I recall a student who had enough failing the computer based assessments. He kindly asked the lead lecturer to show us all that he, at least, could land a perfect score. He made the mistake to try, got 8 points out of 20.
He admitted it wasn't easy when not prepared, and moved on with the next mined field.