French vs. Bezier Curves

[Joe Eagar]

Hi all.  I was catching up on WoodTalk episodes recently and heard Shannon dissing his French curve template as being primitive in today's world of Google SketchUp.  Since this is an area I actively do research in, I just had to to comment.  Bezier curves (in fact, all polynomial curve splines) are not superior to French curve templates.  They have mathematical flaws that make them a bit less helpful than you might think.  

 

I don't want to bore people with the math, but basically you cannot create as nice of curves with Bezier splines as if you constructed them with a French curve template, at least not in the same amount of time, since you are basically "fighting the math."  Now, there is software available that uses French curve splines (see this youtube video for an Inkscape plugin), and I encourage anyone doing curves on the computer (which includes me, my research is motivated by my own needs), to use it.

 

But let's face it: computers are not really superior to something as simple as a French curve template.  If you are using such a template to lay out your work, do not switch to computer-generated Bezier splines.  Your work will suffer if you do.

[Tpt life]

I can appreciate the desire to spare us the math. Please help me understand the take away though. I see an opinion without any understanding of why it matters.

[Joe Eagar]

I can appreciate the desire to spare us the math. Please help me understand the take away though. I see an opinion without any understanding of why it matters.

 

Mathematically, a "fair" curve is one where the curvature function only increases or decreases (and, preferably, is linear if graphed on a logarithmic plot).

 

The curvature function basically tells you, if I fitted a circle to that curve at this point, how big would it be? (It's actually one over that value).  A French curve (it's also called a clothoid, Euler spiral, and Cornu spiral) has a linearly (if graphed on a linear plot) increasing curvature function.  Polynomials, however, tend to have oscillating plots.

 

Here's an example image:

 

 

Notice how the bottom image has a nice, smooth shape (it's actually a bit better than what you would get from a simple French curve, but not by much).  Think of each part of the curve as if it were an arc of a circle.  For the top curve(starting from the left), the radius of that circle starts small, flattens out in the middle, gets small again, and then (though it's hard to see) flattens out again.

 

The bottom curve, on the other hand, has a smooth transition of curvature (the circle radius gets gradually smaller).

[Tpt life]

I do not tend to work with programs that use only Bezier as a function to work a curve. I work in programs that use parabolic and hyperbolic functions to blend curves. This has been the case since TI started building Jr. High math calculators that would perform this function in the late eighties. I am not sure why you latch on to a circle only reference. The French curve is appropriately a parabolic curve is it not? I appreciate the argument greatly and certainly mean no disrespect but also wonder if some amount of misrepresentation has been offered to you or if instead my own understanding is flawed.

Edit: A brief math review confirms for me that Eular is indeed a parabolic function.

[wtnhighlander]

Also wondering just how much this level of detail really matters in the making of furniture. I see nothing displeasing about either of the illustrated curves.

Sounds like another "Golden Ratio vs. Whole Number Ratio" discussion.

[Tpt life]

I had to Google around awhile. OP, are you simply saying do not trust the Bezier plugin for the best curves? I just had a hard time understanding because I do not use a Bezier plugin and so did not realize it existed. I am unsure why someone would create a plugin limited in this way. Sorry to bother.

[Joe Eagar]

I do not tend to work with programs that use only Bezier as a function to work a curve. I work in programs that use parabolic and hyperbolic functions to blend curves. This has been the case since TI started building Jr. High math calculators that would perform this function in the late eighties. I am not sure why you latch on to a circle only reference. The French curve is appropriately a parabolic curve is it not? I appreciate the argument greatly and certainly mean no disrespect but also wonder if some amount of misrepresentation has been offered to you or if instead my own understanding is flawed.

Edit: A brief math review confirms for me that Eular is indeed a parabolic function.

 

The curvature at a point on a plane curve is the reciprocal of the osculating circle at that point.  That's why I used the circle analogy.

 

Industrial designers have done a lot of survey research on what curves look best to people.  The result: a "fair" curve is one whose curvature function only increases or decreases (monotonic). More recent research is more specific: an ascetic curve is one whose curvature plot looks linear on a logarithmic scale.  This all based on consumer surveys, by the way (manufacturers have had a strong financial interest in getting this research right).

 

@Wtnhighlander, it does matter in furniture.  If you had two woodworkers build a sculpted chair, but one could only use real-world drafting tools like splines and French curve templates, and the other had to use traditional CAD software, the former's result would "look" better than the latter's (unless the latter was using software that itself was based on French curves).  This shows up a lot in manufacturing design, as well as computer graphics more generally (e.g. 3D animated films).

[drzaius]

Excellent thread with interesting comments by all.  I'd never even considered this before.  To my eye, the "good" curve is much nicer than the the "bad".

[Tpt life]

I think if I printed a bad, I would sketch toward a good before cutting. I also think usage dictates. A fair curve on a drawer pull leaves less of a mechanical hold. This is more about the visual effect.

[wtnhighlander]

I must be in the minority of folks, because although the illustrated curves are certainly different, I see absolutely nothing to make one "better" that the other. I'm betting context plays a strong part in that choice, and that simple comparison has no context.

[Eric.]

I absolutely see a "good" and "bad" curve.  The bad one is graceless and gives nausea to my eyeballs.  Fortunately I'm in no danger of ending up with a "bad" curve, since before this thread I'd never heard of "Bezier," nor do I use a computer to generate my shapes in the shop.  French curves, drawing bows, pencils.  Thank you very much.

[Vyrolan]

I must be in the minority of folks, because although the illustrated curves are certainly different, I see absolutely nothing to make one "better" that the other. I'm betting context plays a strong part in that choice, and that simple comparison has no context.

Yea I think it is one of those things where when you see it, it "looks right" or "looks wrong". Often the observer can't even explain why they think either way. Think about all the times Marc has said he's gotten Nicole's opinion just to get another check that it "looks right".

It also just may not be something you value visually. Just like we always say non-woodworkers don't care about fancier joinery...you may just not value the "quality" of the curves. /shrug

[wtnhighlander]

It also just may not be something you value visually. Just like we always say non-woodworkers don't care about fancier joinery...you may just not value the "quality" of the curves. /shrug

That's where context makes the difference. Of the two curves illustrated one may look better (to me, at least) on a certain piece of work, where the other curve may look better on something else.

Just like the whole "Golden Ratio" thing. Why should I accept that it is the "correct" ratio to use, if it lools off to me, or to my client?

[Joe Eagar]

That's where context makes the difference. Of the two curves illustrated one may look better (to me, at least) on a certain piece of work, where the other curve may look better on something else.

Just like the whole "Golden Ratio" thing. Why should I accept that it is the "correct" ratio to use, if it lools off to me, or to my client?

 

Note that unlike the Golden Ratio, this is based on a great deal of consumer research.  However, if you and your client like a given design that's fine; this sort of research is based on what looks good to most people, not what looks good to everyone.

 

Also, context doesn't matter.  There is no contextual use that will make the "bad" curve look good, though as C Shaffer pointed out, the process of physically making a curve for a given context may turn it into a "good" one.

[trz]

There cannot be a right or wrong between A or B, it all depends on what the application calls for, whether it be a continuous curve or a curve- flat- curve.

[Vyrolan]

...whether it be a continuous curve or a curve- flat- curve.

 

Well, curve+flat+curve is really two curves, and I think the argument would be that both of those should each be "good" curves.  The "research" no doubt indicates that there can be no application that "calls for" a "bad" curve.  That's an excessively-broad statement though as in most real-world situations the number of variables is truly enormous.

[mikem]

I think it is really hard to justify which curve is better till you apply it in actual use.  I can see where both would work well, just depending on the application it is being used for. 

 

 

Where is this "research" published?

[Vyrolan]

Where is this "research" published?

 

I don't know...hence my air-quoting it as "research".   I was more referring to the very formal mathematical approach that the Joe Eager was taking on the topic.

[Joe Eagar]

Vyrolan:

 

http://www.hindawi.com/journals/jam/2013/732457/abs/ 

 

I know I look absurdly young (especially in the last profile picture) but I'm 27, and yes, I do active research in this area (albeit unpublished, it's part of my job).

 

There cannot be a right or wrong between A or B, it all depends on what the application calls for, whether it be a continuous curve or a curve- flat- curve.

 

What do you mean, curve-flat-curve?  Do you mean transitions from straight lines to curves?  You do realize that that is one of the definitions of a French curve, that it transitions from a straight line to an arc (it's used heavily in road and rail layout for this reason; such transitions prevent sudden changes in angular velocity).

[mikem]

Age has nothing to do with this, but rather backing up what you are stating.  I get the math behind it, I don't think there are many here would disagree that you get two different types of curves produced.  My point is what published "market research" states one is preferable over the other.  As I stated earlier, it really depends on the application on which type of curve is used.  

[Janello]

If I were building a wooden Pinocchio, I'd use the bad curve for his shoes. I mean, what's "good" about Pinocchio anyway really, he has a damn wooden nose shaped like a damn carrot!

[Tpt life]

Trying to figure out how early curve makers managed a billion computations math flow without computers. Is the ancient "French" curve thoroughly consistent? Don't most of us tweak when we find things unbalanced? How many design, print, paste and cut with zero alteration? Is this that myriad of factors? It has been fun scratching my head but I am not sure a huge shift is called for.

[h3nry]

Is the ancient "French" curve thoroughly consistent?

 

The ancient french curve is not that ancient and not that French - having been invented in about 1904 by a German - Ludwig Burmester.

 

I think early curve makers used battens and splines.

 

Burmester did however lay down the mathematics and produced 28 different templates that would enable a draftsman to draw essentially any fair curve. Only three of his curves are in common use today, which are apparently good for drawing elliptical, parabolic and hyperbolic sections respectively.

 

That is about all I could discover from some quick googling - even wikipedia was surprisingly unhelpful.

 

I would love to learn how to use these templates properly for furniture and woodwork design as I am just starting to look at curves more complex than circular arcs in my projects.

[Tpt life]

Yeah, I was finding admitted conflicts in this reporting as I did my own study. The best I could glean was that Burmester standardized the drafting aid.

[dwacker]

Personally I find drawing on a computer a tedious labor, pretty much the same for hand drawing. I'd rather not care and just make a quick sketch of what is in my knot and just go to it in the shop. Im pretty sure this is why the lord made poplar.