A Primer on Bézier Curves (2013)
I enjoyed reading it, but I wouldn&#x27;t call it a &quot;primer&quot;, since I can hardly think of what else there is to know about Bezier curves! It&#x27;s a rather comprehensive exposition.<p>Here&#x27;s a primer on Bezier curves if you had to program them in 30 minutes. You won&#x27;t need more than this:<p>1. Bezier curves are just taking weighted averages, repeatedly. A weighted average of two points, A and B, is a linear combination <i>P(t) = (1-t)A + tB</i>. As <i>t</i> goes from 0 to 1 with constant speed, <i>P(t)</i> goes from A to B with constant speed.<p>A weighted average like this gives you <i>one</i> point out of <i>two</i>. Now, say you have <i>four</i> points, in order: A1, A2, A3, A4. For a number <i>t</i> between 0 and 1, take weighted averages of consecutive points: A1 and A2, A2 and A3, A3 and A4. That gives you a sequence <i>three</i> points. Repeat this procedure until you have only one point left. That&#x27;s your Bezier curve defined by A1..A4 at time t. This is called a 3rd degree curve, and generalizes to sequences of any length (you can figure out how!).<p>2. The above tells you how to draw this with a computer. Just pick, say, 100 values of <i>t</i> between 0 and 1, and connect the dots with straight lines. That is, you converted a curve into a poly-line, consisting of straight segments. This can be used to compute intersections (just see if any one segment intersects any other), compute tangents (extend any segment to get a tangent at that point), normals (in 2D: rotate a segment by 90 degrees clockwise).<p>3. Most importantly, <i>why</i> you would want to do something like this: Bezier curves are easy for machines to trace, and for humans to understand. Here are some properties of a (cubic) Bezier curve given by points A1, A2, A3, A4:<p>-it starts at A1, and ends at A4<p>-it is tangent to segments A1A2 and A3A4 at endpoints<p>-by moving A2 and A3, it is easy to make a C- and S- shaped curve.<p>Beyond that, things aren&#x27;t as intuitive. This is why <i>all</i> graphics software gives you 3rd degree Bezier curves as a tool: it&#x27;s flexible enough while easy enough.<p>In practice, Bezier curves came to replace the French Curve[1]. The irony is that there is little French about the French curve (nobody even knows why they&#x27;re called that!), but Bezier curves are French through and trough: the two key people behind them, Paul de Casteljau and Pierre Bezier, have spent their entire lives in France, and came up with the curves for manufacturing at Citroen and Renault. Arguably, Bezier curves are the true French curves.<p>4. Bonus. In Photoshop&#x2F;Illustrator&#x2F;etc: the UI for Bezier curves is this: click and drag to define A1 and A2 (A1 is is where you click, A2 is where you release). Consecutive click-drag-releases define three points: e.g. A3 on mouse down, A4 on mouse up, and A5 such that A4 is the midpoint of A3 and A5. The curve you get is a union of cubic Beziers defined by A1..A4, A4..A7, A7..A10, etc.<p>The curve is smooth because the UI forces the tangents to be aligned (unless you do a single click without dragging: this makes 2 tangents of length 0, i.e. a vertex).<p>When drawing curves using this UI, you really are drawing <i>tangents</i> to the curve you want to get: draw _— to get an S, roughly.<p>That&#x27;s it folks! For way, way more - read the article :) (Which is an interactive book, at this point).<p>[1]<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;French_curve" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;French_curve</a>