info/BezierInfo-2
1
0
Fork 0
mirror of https://github.com/Pomax/BezierInfo-2.git synced 2025-08-28 18:49:57 +02:00
Code Issues Releases Wiki Activity

Fix minor typo (#203)

Browse Source
This commit is contained in:
Yuri Pourre
2019-06-03 16:32:22 -03:00
committed by Pomax
parent 8d6c7083bc
commit 1506c97645
1 changed files with 1 additions and 1 deletions
Download Patch File Download Diff File Expand all files Collapse all files

2
components/sections/curvefitting/content.en-GB.md
View File

@@ -4,7 +4,7 @@ Given the previous section, one question you might have is "what if I don't want
And really this is just a variation on the question "how do I get the curve through these X points?", so let's look at that. Specifically, let's look at the answer: "curve fitting". This is in fact a rather rich field in geometry, applying to anything from data modelling to path abstraction to "drawing", so there's a fair number of ways to do curve fitting, but we'll look at one of the most common approaches: something called a [least squares](https://en.wikipedia.org/wiki/Least_squares) [polynomial regression](https://en.wikipedia.org/wiki/Polynomial_regression). In this approach, we look at the number of points we have in our data set, roughly determine what would be an appropriate order for a curve that would fit these points, and then tackle the question "given that we want an `nth` order curve, what are the coordinates we can find such that our curve is "off" by the least amount?".
Now, there are many ways to determine how "off" points are from the curve, which is where that "least squares" term comes in. The most common tool in the toolbox is to minimise the _squared distance_ between each point we have, and the corrsponding point on the curve we end up "inventing". A curve with a snug fit will have zero distance between those two, and a bad fit will have non-zero distances between every such pair. It's a workable metric. You might wonder why we'd need to square, rather than just ensure that distance is a positive value (so that the total error is easy to compute by just summing distances) and the answer really is "because it tends to be a little better". There's lots of literature on the web if you want to deep dive the specific merits of least squared error metrics versus least absolute error metrics, but those are <em>well</em> beyond the scope of this material.
Now, there are many ways to determine how "off" points are from the curve, which is where that "least squares" term comes in. The most common tool in the toolbox is to minimise the _squared distance_ between each point we have, and the corresponding point on the curve we end up "inventing". A curve with a snug fit will have zero distance between those two, and a bad fit will have non-zero distances between every such pair. It's a workable metric. You might wonder why we'd need to square, rather than just ensure that distance is a positive value (so that the total error is easy to compute by just summing distances) and the answer really is "because it tends to be a little better". There's lots of literature on the web if you want to deep dive the specific merits of least squared error metrics versus least absolute error metrics, but those are <em>well</em> beyond the scope of this material.
So let's look at what we end up with in terms of curve fitting if we start with the idea of performing least squares Bézier fitting. We're going to follow a procedure similar to the one described by Jim Herold over on his ["Least Squares Bézier Fit"](https://web.archive.org/web/20180403213813/http://jimherold.com/2012/04/20/least-squares-bezier-fit/) article, and end with some nice interactive graphics for doing some curve fitting.