On AI sketch vectorization: Understanding and drawing Bézier curves

MARCH 18, 2022

I'm reading through Virtual Sketching's paper.1 Specifically, I've read the part about the quadratic Bézier curves used by the model to further understand what the model is learning. I'm slowly preparing a lecture on machine learning-based sketch vectorization, and for that, I want to thoroughly understand how this model works.

Googling, I came across a site2 which talks in detail about Bézier curves in the context of CSS and web animation and referenced diagrams from Live 65's slides. A curve with two control points (or three control points that all lie in the same line) is a line. A Bézier curve with three control points is a quadratic curve. And a Bézier curve with four control points is a cubic curve.

A line and quadratic and cubic Bézier curves.

The paper uses curves with three control points, so it follows the following formula to render curves.

P(t) = (1−t)²P₁ + 2(1−t)tP₂ + t²P₃ with t ∈ [0, 1]

The cited page uses De Casteljau’s algorithm to build 3-point Bézier curves, and shows how curves with more control points can be built using interpolation.2

Take a look at yesterday's post for coding examples to draw quadratic Bézier curves.

  1. Mo, Haoran and Simo-Serra, Edgar and Gao, Chengying and Zou, Changqing and Wang, Ruomei. General Virtual Sketching Framework for Vector Line Art, 2021. 

  2. Bézier Curves. Animation.