Legato & points

geraintf's picture

one of the most noteworthy font releases of this year has got to be legato by evert bloemsma. being a curious sort, i had a peek at the outlines. i was surprised that rather a lot of points are being used to describe the curves.

the green i is the legato outlines, the orange i is what i would have *expected* to see (ie minimum number of points)

i wonder if additional points are being added at the 'font development/output stage' (ie. after the glyph has been drawn). if that's the case, why exactly? is it something to do with format conversion?

typophile input welcomed.

cheers

geraint

andreas's picture

These rounding errors are typical for Ikarus or Fontmaster Systems. The outline will be converted from a 15.000*15.000 em grid to a Type 1 grid of 1.000*1.000 em.

But ask him yourself.
His website is http://www.evertbloemsma.nl

geraintf's picture

thanks for your help andreas.

when you say 'rounding errors'... how does a rounding error manifest itself: are points being added during conversion, or is the *position* of existing points being slightly altered?

geraintf's picture

[edit]

geraintf's picture

also, there are multiple points on the few straight lines in the font:

r

is this also some sort of conversion artifact?

i may contact EB, but i ask here because i have seen this kind of thing in other fonts before.

andreas's picture

Try it yourself with FontMaster Light.

http://www.fontmaster.nl/

Use the DataMaster and convert a clean outline Type 1 font into the IK or BE Format. But use 15.000 em as destination grid. And then back to Type 1 and compare the outlines. In most cases you will have more unnecessary points and perhaps a slide change in the design.

geraintf's picture

ah ha. thanks for the explanation, andreas.

one last queston; if i was making a font, how could i avoid this kind of thing, and keep my 'clean outlines' all the way through to the output file?

hrant's picture

Legato is a landmark design, quite possibly the most significant font yet made (I'm not kidding).

But those outlines are pretty horrid.

hhp

eomine's picture

What software did you use to produce your samples?

See the image below. These are screen grabs from Freehand and CorelDraw.
The font is Legato Semibold (PS T1), which was available as a free download from FF some time ago. The top sample (blue) shows FH's conversion of Legato to outlines, and the bottom one (gray) is CorelDraw's conversion. FH introduced a lot of points to the outlines.

legato

hrant's picture

> FH introduced a lot of points to the outlines.

Strange. Why the hell would it do that?
Maybe it's related to the converter in Flash?
FH is not quadratic-based, is it?

hhp

John Hudson's picture

The FH outline conversion looks as if it has been converted via quadratics. The pattern of extra nodes is characteristic: more nodes on long, shallow curves, fewer on more rounded curves. Someone who has the font can open it in FontLab and test this hypothesis by converting all outlines to quadratics, then back to beziers, and comparing the result to the FH conversion.

geraintf's picture

eduardo, i used illustrator CS. here's a sample for comparison with your examples:

legato

geraintf's picture

a different set of points again! similar to the FH conversion, but yet more points (see the o for instance)

eomine's picture

Interesting.
I think Legato's outlines are fine after all. But I didn't know that different softwares could produce such different conversions to outlines.

ebloem's picture

Dear all,

ofcourse these are the real outlines of Legato:



Evert Bloemsma
www.evertbloemsma.nl

geraintf's picture

wow. thanks for this exposition, evert. the univers example you use is telling: i see the slightly 'cupped' terminals of this sample replicated in the legato outlines, for example.

so, if i understand you correctly, these design features restore tension in the black/white balance: a more subtle thing than distressing/replicating letterpress artifacts. but what would legato look like if printed letterpress via polymer plates?

the font can't have been easy to hint either:-) but the 'tension-giving' properties of the curves is only one of legato's innovations. i think you've done an amazing job.

hrant's picture

Evert, great to see you here.
Quite shocking that such "big" programs are ruining your outlines.

--

The issue of rigidity, especially in offset, is quite interesting to me. I agree that gentle curves have a lot of potential, and I think this is two-pronged: they fight optical distortion (like a "perfect" rectangle looks fatter in the middle); and they add a certain feeling, even (especially?) when they're so subtle as to escape conscious perception.

The problem I have with curves -and the reason I see a positive role for rigidity- is that they convey a style that might not be suitable to the given typography. One area is news. You don't want a feminine, ambiguous, suggestive voice giving you updates about Ukraine for example - you want some neutrality so you can make up your own mind, without a designer "helping" you think, or at least not helping you feel.

It's no secret that I admire your work. But it's also quite feminine; and it's interesting to note that Avance -perhaps because it has serifs- seems sort of androgenous, which is why I think it was an incredibly good choice for the book "John Kelly"!

--

As for the perceived advantages of letterpress, I actually think they're real, but have struggled to figure out why they exist. It can't be because of depth/tactility. And even digital fonts (like Evert's) can be "soft". The only thing I can think of is that the boundary between black and white is more gradual. Solution? I can only think of two:
1) Photopolymer letterpress.
2) Microscopically de-focus the laser in an imagesetter! :-) Note that this is not the same thing as putting down gray dots, or using lower resolution; it would simply make the notan ever so slightly softer.

hhp

ebloem's picture

Hello Geraint;
indeed, if Legato could be printed in letterpress with polymer it could have different shapes, straight lines might also be possible. But that

ebloem's picture

I am not sure about some translations

history english

A typeface like "Template Gothic" is also mentioned to represent a movement or a foundry as a whole; in this case Emigre.

The chart should be compact, and the "classifications" are rather unconventional. It is targetted at (future-) graphic designers; not type designers.

I had a discussion with Martin Majoor who mentioned the fact that many typeface families have been published with a serifed and a sanserifed part, like Thesis. The two typefaces mentioned as "hybrid" however, are there because they embodie both sanserifed and serifed principals in one shape, for example FF Sari; it has the contrast of a serifed face, but lacks the actual serifs.

Evert Bloemsma

dezcom's picture

I don't understand the concept of masculine/feminine here? Curves are not defining to femininity. Curves are just elements of form. There are few typefaces which limit themselves by gender. Legato certainly does not. Legato is a strong yet neutral type which has strong character yet does not interfere with the meaning of the text it is set in. I am just an observer who pretends to know nothing about what was intended in the design of the face. My reaction is purely from what I see as an individual. I have not read about Legato or ever spoken with Mr. Bloemsma so what I say is just one man's interpretation. What I see is a face which gets its life by tension. The counters seem to be ever-so-subtly moved off axis from their surrounding form, just enough to cause a tension but not so much to make it look like an error. The tension locks the two forms (form and counter) into life. This is like the Keats poem, "Ode on a Grecian Urn" where the two lovers are locked for eternity just short of kissing--"bold lover, never, never, shalt thou kiss".
It may be that Gill attemted this in his Gill Sans but his lacks the tension present in Legato. It appears to me that Gill was just trying to overcome weight and contrast problems when moving to bolder weights--where Evert is successfully using the tension for its own sake.

ChrisL

hrant's picture

Evert, indeed I think Lexicon is not a good news face. And I'm not alone!

> leaving away the strokes

That's all fine and dandy, but what you've done with Legato is much deeper, in my mind coming the closest so far to the true essence of typography.

To me the only real problem with Legato is that it's too constructed, too Modernist. I think one has to relinquish some control to make something that truly works with the human reality. This is what the Post-Modernists had a feeling for, but too many of them were more interested in simply discrediting Modernism, like misbehaving children.

BTW, I like Avance! In fact it has one strong functional advantage over Legato: serifs.

> we certainly do not observe just with our
> eyes but also with other senses, with our
> body as a whole.

That sounds very spiritual (and I'm not being sarcastic) but I don't see how that's possible. But anyway discussing metaphysics has always seemed pretty hopeless to me. :-)

> I always prefer to see type on rough paper which is not too white.

This might be related to my idea about limiting contrast. I think not-too-white paper does indeed help offset, but not letterpress.

BTW, that chart is pretty nice!

--

> Curves are not defining to femininity.

I guess all I can do is disagree.

Nothing is just a shape - everything has meaning, and not just through social conditioning either. Red alludes to blood. An inverted triangle grabs attention because of the nature of gravity. We are completely surrounded by snippets of physical realities such as these, and good design requires an appreciation of them.

--

Legato is a landmark design because it finally gives the importance of the white more than lip service - it creates notan, as opposed to merely a sequence of pretty blacks. People like G Noordzij go on and on about creating the white as well as the black, but they essentially ignore that chirography is inescapably anti-notan.

The difference between designing Black versus designing notan is the difference between Art and Craft, between wanting to make versus wanting to serve. Smeijers says that Design starts with the desire to create. No.

hhp

eomine's picture

Slightly off-topic:

> Nothing is just a shape - everything has meaning

This is interesting. I think this also relates to the blackletter revival discussion. Blackletter carries a lot of meanings, but most of them have been 'socially conditioned'. Sometimes we just have to separate 'shape' from 'meaning'.

hrant's picture

Totally. And this was an important part of the point I tried to make during the blackletter panel discussion at ATypI-Leipzig. And an encouraging thing happened during the subsequent Q&A: Gabriel Martinez Meave saying that in Mexico they have no idea about all this Nazi association business, that some of them in fact turn to blackletter for its wondrous shapes. To me, the deeply human Yin/Yang of the Fraktur "o" goes well beyond the crimes of a group of people - and is something totally missing in whiteletter. Whiteletter is "The Wedding Planner" to Blackletter's "The Celebration".

hhp

pablohoney77's picture

is "the celebration" as good as "monsoon wedding"?

hrant's picture

Monsoon Wedding looks cool - I'll put it on my "list".
Anything Indian has almost automatic appeal for me though. :-)

The Celebration is quite dark, the ending shockingly human.

hhp

ebloem's picture

>> we certainly do not observe just with our
>> eyes but also with other senses, with our
>> body as a whole.

>That sounds very spiritual (and I'm not being sarcastic) but I
>don't see how that's possible. But anyway discussing
>metaphysics has always seemed pretty hopeless to me. :-)

It is simple; the smell of paper and ink, the touch of paper, strengh and materials of the cover, the sound of turning the pages, the way a book is bound(?), and last but not least, the way it is printed; it is a physical encounter that surrounds the visual concentration of reading.

Evert

hrant's picture

Oh, I agree with all that, but it's notable that none of that is at the level of the typeface, and all of those things can be applied in offset printing as well (as far as I know).

It's the pushing ink onto the paper aspect of letterpress that needs to be considered here I think.

hhp

ebloem's picture

look at the picture... do you think it is offset?

hrant's picture

No, even in a photograph I can tell it's letterpress.
(Which of course means the impression is way too much! :-)

But the point is that there's no reason you couldn't use that paper, that binding, etc. in a book printed offset. And if we're talking about the type, I don't see how any of the non-visual things -at least of the ones you mentioned, and even the ones I can think of- can affect reading.

As I said above, I do think there's a functional difference between letterpress and offset even at the level of the type. But it's not tactility, or smell, or anything else related to our non-visual senses; it's the nature of the border between black and white, the contrast of the notan. And I think it might actually be possible to acquire this positive attribute in offset - by slightly defocusing the imagesetter laser.

BTW, come to think of it, there might in fact have been a short period during the phototype era just before the digital revolution when things were just as "fuzzy" as they needed to be to match the feel of letterpress "rendering" - except of course the actual fonts they were using were caricatures. I wonder, do typographers who were active at that time feel that what they had was "warmer" than the subsequent digital stuff, like how an LP is seen by some as somehow better than a CD?

hhp

aquatoad's picture

Evert, your image of letterpress univers reminds me on Nick Shinn's brown gothic. There's cupped ends and waisted stems everywhere. In your opinion does this face have the tension you are talking about?
http://www.shinntype.com/Assets/Brown_GIFS/bro_r_s.html

Also, Evert, you mention letterpress induced tension adding to the beauty, and the overall environs created by a letterpress project whet the appetite for reading. I'm inclined to agree with you.

Hrant, you seem to want to argue that letterpressing enhances legibility, which I haven't heard Evert say explicitly. Evert, do you think this letterpress-like tension in Legato's outlines contributes to its legibility? If so, how?

R

hrant's picture

Yes, I think letterpress (when not over-impressed) increases readability, and I think it's because it promotes a notan more naturally in tune with human perception.

But I don't think Legato's main readability advantage has anything to do with that; to me it's due to the font's breaking of the Black, to its tying of the White - a better (if still somewhat simplistic, too Modernist) balance of notan. By reducing the relevance of strokes (chirography) the Black is made to accomodate the White -hence notan- much better. The letters are less themselves are more parts of boumas.

hhp

aquatoad's picture

Notan: I don't follow. How does something that is not overly constructed, but with reduced reference to a traditional writing tool, make the black accomodate the white?

What I do understand is Evert's explanation of importance of stress angle creating more cohesive word shapes (dare I say, boumas), as opposed to the choppy rythm of vertical stress.

R

ebloem's picture

Hrant: at the level of type (literally!) there is an important difference between letterpress and offset which is that in letterpress letters are not

hrant's picture

There seems to be a bit of confusion in this thread - maybe because we're all [positively] excited about the topic so we're reading things we anticipate into what others are saying...

> How does something that is not overly
> constructed, but with reduced reference
> to a traditional writing tool, make the
> black accomodate the white?

1) I think Legato is actually overly constructed!
2) I don't think its "curviness" has much to do with readability.
3) But indeed by demoting chirography it becomes possible to create better notan. Consider the core of the G Noordzij argument: the moving front. It creates the Black, and it favors the Black (even if you're keeping your eye on the White while doing it) because it ties the two edges (outlines) of the black body. By doing this it makes the White a byproduct, and this reduces bouma cohesion, hence readability. We are extremely lucky to be able to design individual outlines (as opposed to bodies) directly, since it's this border than defines notan. Sadly, Evert is the only person I know who has used this freedom to good effect so far. Hopefully though the floodgates will soon open, and chirography (at least in the realm of text) will be seen as the relic, the romantic inhibitor, that it is.

Also: I don't think Legato's particular "stress angle" is very relevant*, and neither is vertical versus horizontal pattern**. To me the key to Legato is that it "breaks" the Black and connects the White, balancing their relationship***, thus creating better boumas. When Evert makes a serif font on this principle then we'll really see a new chapter in readability.

* Even though some people, like Jan Middendorp, have gleefully exclaimed "Look, the pen, look!" But it's not there. Just like Gary Much once saw the pen in my Paphos "j", but I can assure you it's not there.

** Note: not "rhythm", because there is no flow.

*** You could say making it a healthy marriage, not an owner&slave relationship.

--

> .... visible as on the photo.

That's too much impression - bad for readability.
The degree of pressing that I think helps readability is subvisible.

Notan is indeed a Japanese term (there is no equivalent in Western lanaguages that I know of - and I think this is itself telling), but ironically the letter-making crafts in the Orient are even more Black-centric than in the West! On the other hand, cross-character relationships are a lot less important in Chinese for example than in Latin.

> In the same way offset will never be able
> to replace all the aspects of letterpress

I agree, but my personal concern is to replicate what is possible, and to me the readability advantage of letterpress is. Running your hands over the bumps (which should be barely tactile however, not impressed up the wazoo) is not replicable - although that doesn't make me very sad.

Also, don't forget photopolymer letterpress, which allows digital type to replicate the whole deal. Unless you think irregularity (I mean the true kind, different than intentional variance) is anything more than bad craft.

> you could also try the effect of random irregularity

No, that's not the type of thing I'm talking about. That is indeed fake.
Like Johnny von Neuman said, real irregularity can never be intentional anyway.

> the unsharpness of the phototype era
> is the dark age of typography to me.

Me too, but I'm speculating that that's what we're taught these days: the "outstanding" feature of phototype was blur, so that's what is remembered. But maybe there was a period towards the end that -were it not for the lousy fonts generally used- the effect was closer to letterpress than we're at now. I admit it's speculation though.

hhp

John Hudson's picture

Even though some people, like Jan Middendorp, have gleefully exclaimed "Look, the pen, look!" But it's not there. Just like Gary Much once saw the pen in my Paphos "j", but I can assure you it's not there.

I'm not sure that a letter needs to be made with a pen in order for the pen to 'be there'. During my short talk in Prague, I focused on role of ductus in glyph identity, as against a conception of letter identity as a kind of skeleton on which the meat of the individuated glyph is hung. Of course, ductus shifts and changes, across historical periods and styles, and in the hands of individual designers, but within a particular style of lettering there is only so far from the expected patterns resulting from ductus that one can go before the letters start to look freakish. So when one 'sees the pen' in Legato, one is acknowledging the ductus in the identity of the letters, which is not mechanically consistent angle and weight, but is significant enough to establish a common relationship between the letters. Yes, Evert has manipulated the ductus -- most type designers do, in fact: Gerrit Noorzij might be unique in claiming that his type designs are as he writes them -- but nothing in Legato looks freakish. The idea of ductus fascinates me, because it is something that originates in tools, but which has become an idea in our comprehension of letterforms, and now exists to a large degree independent of tools. We observe ductus -- we see the artifact of the tool -- where the tool has not been.

hrant's picture

Of course chirography is not dependend on the actual use of a physical tool. My observation was about some pro-chirography people seeing what they want to see.

There might be ductus is Legato, but I don't think so: I think it's merely a difference in width in different parts, and some people like to see ductus in that because it validates their ideology (and practice). And I know there's no ductus in my stuff, but that doesn't stop people from thinking they're seeing it.

> before the letters start to look freakish.

I don't think unacceptable deviation from a "norm" has much to do with ductus, at least not at the text level. It has more to do with salient features of the letters (where ductus is coincidental). Not to mention the great capacity of humans to adapt to new forms.

> nothing in Legato looks freakish.

Exactly! :-)
I don't think he's manipulated the ductus. And if he has, its wasn't necessary. What I think he's done (and more to the point what I think should be done) is to manipulate notan, directly, without any limiting influences from archaic tools. The moving front is anti-notan.

The need to demote chirography comes from the need to promote readability.
And also the need to get the 40-year-old man to move out of his mom's house.

hhp

John Hudson's picture

The moving front is anti-notan.

So you keep saying, but this implies that notan cannot be achieved except by treating the boundary between black and white as an independent path. In other words, drawing outlines is the only sound approach to type design. Now who is being ideological? The problem with this view is that the vast majority of notan in the world is not created by drawing outlines, but with brushes and pens, and the whole notion of notan develops in arts of painting and calligraphy in which it is understood that black and white can be balanced or, for that matter, deliberately imbalanced, by the use of such tools.

Most type designers are pragmatists: they'll use whatever methods, tools, and tricks they can. Watching Tim Holloway drawing Arabic letters was a thrill: the tools and tricks of the Walter Tracy's drawing office pulled out in the lounge of a London hotel: two pencils with shaved sides taped together to sketch initial forms, and then the outline drawn, following the sketched ductus sometimes and diverging where Tim felt it should, and then the thick black chisel marker to ink the form. Was it chirographic? Was it 'synthetic'? Or was it something in-between? To me, it was just type design, as practised by one person who I happen to think is one of the most brilliant designers I have met.

...manipulate notan, directly...

Presumably, this is what I do almost all the time, since I design from scratch with digital outlines, and manipulate them until I am satisfied with the shape, weight and consistency of the letter. So I think you must be talking about something other than method, which makes me suspect that your crusade against chirography is misdirected. It seems to me that what you really want to focus on is analysis of notan, not how notan is made. As Gary Munch said to you on the ATypI discussion list, an experienced calligrapher isn't significantly limited in the kind of forms he can make with the various tools at his disposal. His point regarding your Paphos j, wasn't that he thought you had made it with a pen, but that he could have made it with a pen. So I tend to consider the method of creating the form from a purely pragmatic perspective -- not least because I'm a jobbing craftsman with deadlines --: what is the most effective way to arrive at the form? The question of what that form should be, what makes it more or less readable: these are much more interesting questions, regardless of how one goes about making the form.

hrant's picture

Always the same circular shenanigans.

I'm not talking about enforcing design practice. I'm not interested in condemning how some (most) people make fonts (or shapes) and how some people choose fonts, or anything like that. What I'm trying to do is simply get people to admit that chirography is anti-readability. That's all. That's plenty, since it will simply cause some people to make better fonts. Others can keep making their type any damn way they please. I'm talking about ideological integrity, about what type is, not how people should act.

> His point regarding your Paphos j, wasn't that
> he thought you had made it with a pen, but that
> he could have made it with a pen.

Wrong. Where did you get that?
He said he was happy that I'd come to incorporate ductus into my work. Puhleez.
It's in the Typo-L archives.

And of course Gary can make any shape with a pen. Just like I can go to the grocery store crawling on my back, naked. The point is: why? The answer is: chirography is easy. It reduces the need to think, the need to change.

Damn script fonts and revivals of Swiss grot junk sell much better than Legato, and get more peer accolade. That's all the reason some people need. Not me.

hhp

kris's picture

Evert,

I must confess that Legato is the first typeface I have ever come across that makes sense. It seems to be free of stylistic conceits, and instead focused on the human aspect of type/reading. Tremendous

hrant's picture

But death is a part of Life.
And there's more to Life than humans.

hhp

as8's picture

"get people to admit that chirography is anti-readability."

LOL -- dat is the real Hrant !

"But death is a part of Life."

'Since we cannot comprehend the death,
there is no death at all,' and feel good to use straight lines !

AS

John Hudson's picture

What I'm trying to do is simply get people to admit that chirography is anti-readability.

But who reads chirography? We're not in a manuscript culture. We all read typography, and there are very, very few text faces that are chirographic in any way that suggests to me that the designers are not aware of the needs of readability. What they have recognised, though, is that the impact of type design on the readability of a text, presuming the design is within the range of typical text design, is minimal. I think Legato is a fine typeface, and I too appreciate the fact that it is a thoughtful typeface: but until you can actually demonstrate a statistically significant increase in reading speed and accuracy, all your talk about readability and anti-readability in type design doesn't convince me. Reader-ability so far outweighs readability as a factor in reading that insisting on addressing the readability in typeface design is like trying to say that all shoes should be designed primarily to make walking easier, as if walking were difficult.

But I'm wondering why you characterised my message as 'always the same circular shenanigans', when the principal point of it was to suggest that you might get more mileage from concentrating on analysis of notan, rather than tilting at the windmill of chirography. I was trying to be helpful. Consider: let us imagine that everyone has accepted that 'chirography is anti-readability'. So now what? What do we do differently? These seem to me interesting questions.

as8's picture

hrant's picture

> But who reads chirography? We're not in a manuscript culture.

1) Well, exactly. This makes the familiarity factor [increasingly] weak.
2) I'm talking about chirographic type, and in saying "I'm not sure that a letter needs to be made with a pen in order for the pen to 'be there'.", you are too.

> until you can actually demonstrate a statistically significant increase
> in reading speed and accuracy, all your talk about readability and
> anti-readability in type design doesn't convince me.

That bar is unreasonably high, and furthermore not in tune with how humans come to believe things, and as a result how they practice things. Empiricism should be an assistant, not a master. But anyway, I'm not interested in convincing individuals - just making my case.

> as if walking were difficult.

Walking is actually a great example:
- Better shoes help, a lot.
- You don't know how difficult it used to be until you get better shoes.

> I'm wondering why you characterised my message as 'always the same circular shenanigans'

Sorry, I didn't mean you, but the whole thing, including me. As much as I admittedly like the sound of my own voice, I get tired of repeating myself droningly, and having to counter intelligent people who refuse to listen with self-doubt. I should really follow Brian Jaramillo's advice and set up a Hrant FAQ. So I can just simply and quickly point to it as needed. Hey, what about www.hhpov.com ? :-) It's all just a point of view after all.

As for addressing notan versus chirography: both need elaboration, and they're strongly related. As I said, it is to promote better notan that chirography needs to be demoted. Few people (mostly outside the craft, like Kevin) think notan is limited to individual letters; overcoming that is the first step, but then you have to see the damage done by the moving front (to reading).

> What do we do differently?

All kinds of stuff. For example, Underware would use their huge talent and skill to make really really great fonts, instead of wasting time shackled to a relic, making just very good fonts. Once you admit something, things inevitably change. I have no problem with a designer saying that he makes chirographic fonts because his clients want it, or his designer friends think it's cool. It might make me sad, but not angry. Denial makes me angry.

It would be great though if this wasn't a hypothetical question - let's work on that.

hhp

geraintf's picture

going back to the original topic:-)

looking at the true outlines evert posted, i notice the waisted strokes (L, for instance) have additional ('mid') points at the inner extrema of the tapers.

i realise adding points to extreme is a good idea, but is this advisable even if the curves are gentle (as in this case) and the curve is concave (ditto)?

cheers.

evert, i am enjoying your contribution: please don't be put off by my techicalities or hrant's POV;)

hrant's picture

I think you need those points for two reasons:
- Hinting.
- Something called Flex that straightens gentle curves to reduce problems during lo-fi rendering?

hhp

Mark Simonson's picture

That's right. Curves and lines don't actually exist as such in a font, but points do. Hints need something to "latch on to" so to speak, and if you don't put those extrema points in, you can't hint those parts of the outline.

Mark Simonson's picture

Incidentally, an extrema point is the point at which a curve is tangent to either the horizontal or vertical axis. This is the reason diagonal lines or curved lines which are diagonally oriented can't be hinted--no place to put an extrema point because the line or curve is never tangent to either the vertical or horizontal axis.

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