info/BezierInfo-2
1
0
Fork 0
mirror of https://github.com/Pomax/BezierInfo-2.git synced 2025-08-17 14:10:56 +02:00
Code Issues Releases Wiki Activity

Fix typos (#282)

Browse Source
This commit is contained in:
Steve Nicholson
2020-11-18 07:50:57 -08:00
committed by GitHub
parent 77284e1051
commit 43973e9bb3
1 changed files with 2 additions and 2 deletions
Download Patch File Download Diff File Expand all files Collapse all files

4
docs/chapters/tracing/content.en-GB.md
View File

@@ -4,9 +4,9 @@ Say you want to draw a curve with a dashed line, rather than a solid line, or yo
Now you have a problem.
The reason you have a problem is that Bézier curves are parametric functions with non-linear behaviour, whereas moving a train along a track is about as close to a practical example of linear behaviour as you can get. The problem we're faced with is that we can't just pick `t` values at some fixed interval and expect the Bézier functions to generate points that are spaced a fixed distance apart. In fact, let's look at the relation between "distance long a curve" and "`t` value", by plotting them against one another.
The reason you have a problem is that Bézier curves are parametric functions with non-linear behaviour, whereas moving a train along a track is about as close to a practical example of linear behaviour as you can get. The problem we're faced with is that we can't just pick `t` values at some fixed interval and expect the Bézier functions to generate points that are spaced a fixed distance apart. In fact, let's look at the relation between "distance along a curve" and "`t` value", by plotting them against one another.
The following graphic shows a particularly illustrative curve, and it's distance-for-t plot. For linear traversal, this line needs to be straight, running from (0,0) to (length,1). That is, it's safe to say, not what we'll see: we'll see something very wobbly, instead. To make matters even worse, the distance-for-t function is also of a much higher order than our curve is: while the curve we're using for this exercise is a cubic curve, which can switch concave/convex form twice at best, the distance function is our old friend the arc length function, which can have more inflection points.
The following graphic shows a particularly illustrative curve, and its distance-for-t plot. For linear traversal, this line needs to be straight, running from (0,0) to (length,1). That is, it's safe to say, not what we'll see: we'll see something very wobbly, instead. To make matters even worse, the distance-for-t function is also of a much higher order than our curve is: while the curve we're using for this exercise is a cubic curve, which can switch concave/convex form twice at best, the distance function is our old friend the arc length function, which can have more inflection points.
<graphics-element title="The t-for-distance function" width="550" src="./distance-function.js"></graphics-element>