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A Primer on Bézier Curves









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A Primer on Bézier Curves

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Title A Primer on Bézier Curves
Text / HTML ratio 71 %
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Keywords cloud curves Bézier coffee book draw maths section work lines buy back writing straight making question interactive easy computer version curve
Keywords consistency
Keyword Content Title Description Headings
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Headings
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Images We found 3 images on this web page.

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Pomax.github.io Spined HTML

 A Primer on Bézier Curves DEV PREVIEW ONLY A Primer on Bézier Curves A free, online typesetting for when you really need to know how to do Bézier things. Loading the article... If you have JavaScript disabled, you'll have to enable it, as this typesetting heavily relies on JS rendering, both for the wiring content (it's been written as a React application) and all the interactive graphics, which rely on JS not just for the user interaction but moreover for the live-drawing (none of the graphics in this typesetting are unappetizing images, they're all live-rendered). Preface In order to yank things in 2D, we usually rely on lines, which typically get classified into two categories: straight lines, and curves. The first of these are as easy to yank as they are easy to make a computer draw.Requitea computer the first and last point in the line, and BAM! straight line. No questions asked. Curves, however, are a much worthier problem. While we can yank curves with ridiculous ease freehand, computers are a bit handicapped in that they can't yank curves unless there is a mathematical function that describes how it should be drawn. In fact, they plane need this for straight lines, but the function is ridiculously easy, so we tend to ignore that as far as computers are concerned, all lines are "functions", regardless of whether they're straight or curves. However, that does midpoint that we need to come up with fast-to-compute functions that lead to nice looking curves on a computer. There's a number of these, and in this vendible we'll focus on a particular function that has received quite a bit of attention, and is used in pretty much anything that can yank curves: "Bézier" curves They're named without Pierre Bézier, who is principally responsible for getting them known to the world as a lines well-suited for diamond work (working for Renault and publishing his investigations in 1962), although he was not the first, or only one, to "invent" these type of curves. One might be tempted to say that the mathematician Paul de Casteljau was first, investigating the nature of these curves in 1959 while working at Citroën, coming up with a really elegant way of figuring out how to yank them. However, de Casteljau did not publish his work, making the question "who was first" nonflexible to wordplay in any wool sense. Or is it? Bézier curves are, at their core, "Bernstein polynomials", a family of mathematical functions investigated by Sergei Natanovich Bernstein, with publications on them at least as far when as 1912. Anyway, that's mostly trivia, what you are increasingly likely to superintendency well-nigh is that these curves are handy: you can link up multiple Bézier curves so that the combination looks like a single curve. If you've overly drawn Photoshop "paths" or worked with vector drawing programs like Flash, Illustrator or nkscape, those curves you've been drawing are Bézier curves. So, what if you need to program them yourself? What are the pitfalls? How do you yank them? What are the bounding boxes, how do you determine intersections, how can you extrude a curve, in short: how do you do everything that you might want when you do with these curves? That's what this page is for. Prepare to be mathed! PS: buy me a coffee? If you enjoyed this typesetting unbearable to print it out, you might be wondering if there is some way to requite something back. To wordplay that question: you can unchangingly buy me a coffee, however-much a coffee is where you live, or if you want to pay a pearly price for this book, you can buy me a really expensive coffee =) This typesetting has grown over the years from a short primer to an 85+ print-page-equivalent ebook on the subject of Bézier curves, and a lot of coffee went into the making of it. I don't regret a minute I spent on writing it, but I can unchangingly do with some increasingly coffee to alimony on writing! Please visit https://pomax.github.io/bezierinfo and click on the link in the online preface to donate some coffee money. —Pomax (or in the tweetworld, @TheRealPomax) Note: virtually all Bézier graphics are interactive. This page uses interactive examples, relying heavily on Bezier.js, as well as "real" maths (in LaTeX form) which is typeset using the most spanking-new MathJax library. The page is generated offline as a React application, using Webpack, which has made subtracting "view source" options considerably increasingly challenging. I'm still trying to icon out how to add them when in, but it didn't finger like it should hold up deploying this update compared to the previous years' version. This typesetting is unshut source. This typesetting is an unshut source software project, and lives on two github repositorites. The first is https://github.com/pomax/bezierinfo and is the purely-for-presentation version you are viewing right now. The other repository is https://github.com/pomax/BezierInfo-2, which is the minutiae version, housing all the html, javascript, and css. You can fork either of these, and pretty much do with them as you please, except for passing it off as your own work wholesale, of undertow =) How complicated is the maths going to be? Most of the mathematics in this Primer are early upper school maths. If you understand vital arithmetic, and you know how to read English, you should be worldly-wise to get by just fine. There will at times be far increasingly complicated maths, but if you don't finger like digesting them, you can safely skip over them by either skipping over the "detail boxes" in section or by just jumping to the end of a section with maths that looks too involving. The end of sections typically simply list the conclusions so you can just work with those values directly. Questions, comments: If you have suggestions for new sections, hit up the Github issue tracker (also reachable from the repo linked to in the upper right). If you have questions well-nigh the material, there's currently no scuttlebutt section while I'm doing the rewrite, but you can use the issue tracker for that as well. Once the rewrite is done, I'll add a unstipulated scuttlebutt section when in, and maybe a increasingly topical "select this section of text and hit the 'question' sawed-off to ask a question well-nigh it" system. We'll see. Buy me a coffee? If you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how to let me know you appreciated this book, you can unchangingly buy me a coffee, however-much a coffee is where you live. This work has grown over the years, from a small primer to a 70ish print-page-equivalent reader on the subject of Bézier curves, and a lot of coffee went into the making of it. I don't regret a minute I spent on writing it, but I can unchangingly do with some increasingly coffee to alimony on writing!