How to use Bézier curves in graphics

Tuesday, August 16, 2022 • 4:40 EDT

Graphists, animators, game programmers, font designers, and other graphics professionals and enthusiasts are often working with Bézier curves. Most people are first confronted with Bézier curves through an UI that may look like this: In this case the curve is composed of 4 user controllable points, meaning it's a Cubic Bézier. I'm voluntarily omitting a lot of technical details here, such as the root finding algorithm and floating point inaccuracies challenges: the point is to illustrate how the 1D deconstruction is essential in understanding and manipulating Bézier curves. The presence of this function in the 1st degree is not just a coincidence: the function is actually the corner stone of all the Bézier curves. A fundamental problem of text rendering is figuring out whether a given pixel lands inside or outside the character shape (which is composed of a chain of Bézier curves). Indeed, we can build up the Bézier formulas using exclusively nested : This way of formulating the curves is basically De Casteljau's algorithm. But can you tell for sure what their respective curves actually look like precisely? Well, with an horizontal ray, we would have to know when the y-coordinate of the curve is the same as the y-coordinate of , so we first have to solve , or , where is the component of the given Bézier curve . ( Continue reading.

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