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When designing a geometric sans, such as Futura, do you adjust the thickness of a circular arc for a better visual match to a straight stroke? I'm hoping for a formula like “a circular arc of inner radius r and outer radius R will seem as thick as a straight stroke of thickness f(r,R)”; maybe it's simply R–r, maybe it's not, for all I know.

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    When designing typefaces...trust your eyes...not your math. – DA01 Sep 15 '15 at 1:16
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    If I don't trust my eyes, it would be nice to have at least a qualitative rule of thumb. – Anton Sherwood Apr 18 '16 at 16:45
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    Ya gotta trust your eyes. Or--at least--other people's eyes. Type design is very much an art. – DA01 Apr 18 '16 at 17:50
  • I'm rather amazed that, in the simplest possible case, no one is willing to go out on a limb to say whether it's less or more. – Anton Sherwood May 10 '18 at 4:48
  • You've been waiting 3 years for an answer? Regardless, there isn't one. There's no formula for type design. It's an art. – DA01 May 10 '18 at 17:37
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The optical adjustment is there to compensate for an illusion that is caused by human perception. Mostly this illusion happens in the visual processing pathway somewhere between your eye and your brain.

Now its perfectly understandable that you get the answer use your eyes because thats the only simple answer. Your brain sees the illusion so you should be able to adjust for it.

But what about a more mathematical approach, is it possible? Yes, however, human visual system is very complex. So to generate a formula you need to device a way to measure what you, or a generic sample as proxy to human kind sees. Since tools for directly measuring a effect are really hard to come by, if they even exist yet. It means you end up doing a lot of statistical analysis on some indirect data.

Since not many are super interested in spending a Ph.D. worth of work on measuring this for one special case. They use eyes instead. Also dont worry about it so much, its just a font.

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    Actually it seems that a scientific article based on such a statistical analysis was published recently (2019). – Wolff Dec 11 '19 at 15:05
  • @Wolff yes, good find. Basically they just confirm what the question supposes, "such an effect exists". They do not as of yet know the magnitude well enough and/or how different variables affect things. But certainly the work of doing this in some future has begun. Lot more legwork needed. PS I have wondered about lack of such measures myself. Ive tried to do measures with some university students but my setups have bwwn somewhat inconsistent. – joojaa Dec 11 '19 at 17:16
  • They conclude an average ratio (see my answer), but you are right. It's "just" statistics. But isn't it the same with a lot of knowledge related to vision? Color profiles for example are not calculated from mathematical/physical formulas, but rather found by measuring the total effect of all the unknown factors. – Wolff Dec 11 '19 at 17:30
  • @Wolff but color has been extensively studied for 70 years because it has been relevant to several industrial actors. Also the retina has been possible to model. And the statistical samples are a bit larger than 42. Whereas scientific study of typgraphy not so much because graphic design is an "art". And there is the idea that you can not science art. But there are a lot of things this paper does not answerlike why a capital O should be higher than a capital T. Also it does not answer how this affects ovals. We can assume its same but cognitive psycology is not always entirely straightforward – joojaa Dec 11 '19 at 17:51
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+50

I'm unsure if you are talking about that curved lines appear thicker than straight lines or that horizontal lines appear thicker than vertical lines. This is only about the latter.

I found a scientific article called A Thickness Illusion: Horizontal Is Perceived as Thicker than Vertical by de Waard, Van der Burg, and Olivers. It was january 2019 in the journal Vision.

Its conclusion is based on statistical analysis with 28 participants who were presented to horizontal and vertical lines of different thickness and was instructed to tell if the lines looked similar in thickness or not.

I won't attempt to explain the study in any detail, but I dug out that sentence you are looking for:

[...] the vertical line must be 5.4% thicker than the horizontal line for them to be perceived as equally thick.

27 out of 28 participants perceived horizontal lines as thicker than verticals, lending evidence to previous observations by type designers. The sample only included participants with normal, corrected vision from both sexes and ranging from 17 to 57 years old. However, other factors may come into play that could influence these results.

I'm unsure if this ratio also applies to geometric shapes as a whole, but I don't think so. Below we first have a circle with uniform stroke width, then a circle where the stroke is 5% thicker on the vertical part and last a circle with uniform stroke width scaled 5% wider as a whole. Judge for yourself.

As mentioned in the article:

Interestingly, the effect size in the present study (5% in Experiment 1 and 3% in Experiment 2) is much smaller than the thickness difference between vertical and horizontal lines in some of the best-known geometric typefaces, such as Futura (13%) and Avenir (20%).

Typefaces compensates for the thickness illusion, but doesn't necessarily follow statistics. Aesthetics and culture also plays a big role. Don't forget that most western typefaces are influenced by writing with a broadpen where horizontal lines naturally become thinner than vertical lines.

I can recommend these related articles from Hoefler&Co.:

  • This is a good find it does hovever not tell how much the round edges need to overlap straight lines for example. But yes atleast there is somebody studying this. – joojaa Dec 11 '19 at 17:37

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