diff --git a/components/sections/derivatives/content.en-GB.md b/components/sections/derivatives/content.en-GB.md index 5f7b2caf..81db71dc 100644 --- a/components/sections/derivatives/content.en-GB.md +++ b/components/sections/derivatives/content.en-GB.md @@ -1,6 +1,8 @@ # Derivatives -There's a number of useful things that you can do with Bézier curves based on their derivative, and one of the more amusing observations about Bézier curves is that their derivatives are, in fact, also Bézier curves. In fact, the differentiation of a Bézier curve is relatively straight forward, although we do need a bit of math. First, let's look at the derivative rule for Bézier curves, which is: +There's a number of useful things that you can do with Bézier curves based on their derivative, and one of the more amusing observations about Bézier curves is that their derivatives are, in fact, also Bézier curves. In fact, the differentiation of a Bézier curve is relatively straightforward, although we do need a bit of math. + +First, let's look at the derivative rule for Bézier curves, which is: \[ Bézier'(n,t) = n \cdot \sum_{i=0}^{n-1} (b_{i+1}-b_i) \cdot Bézier(n-1,t)_i @@ -12,7 +14,7 @@ which we can also write (observing that b in this formula is the same as Bézier'(n,t) = \sum_{i=0}^{n-1} Bézier(n-1,t)_i \cdot n \cdot (w_{i+1}-w_i) \] -Or, in plain text: the derivative of an nth degree Bézier curve is an (n-1)th degree Bézier curve, with one fewer term, and new weights w'0...w'n-1 derived from the original weights as n(wi+1 - wi), so for a 3rd degree curve, with four weights, the derivative has three new weights w'0 = 3(w1-w0), w'1 = 3(w2-w1) and w'2 = 3(w3-w2). +Or, in plain text: the derivative of an nth degree Bézier curve is an (n-1)th degree Bézier curve, with one fewer term, and new weights w'0...w'n-1 derived from the original weights as n(wi+1 - wi). So for a 3rd degree curve, with four weights, the derivative has three new weights: w'0 = 3(w1-w0), w'1 = 3(w2-w1) and w'2 = 3(w3-w2).