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Posted by Fabiouser in

12 Sep 2012 — 1:46pm

12 Sep 2012 — 1:46pm

Hello typophiles

Im learning the principles of designing a typeface via software assistant. So, when I need to control beziers to make a 'perfect curve' I fail again and again. I know that failing is good, but I think that the position of points and handles have some science on it.

Today I realize something that perhaps could help me, but I dont know if it make sense, have a look:

It seem to me that handles follow a certain logic that we can find in the structure of calligraphic variations…

This make sense to you?

Where do you [typographers] learn the 'art' of place perfect points and handles?

--------

Thanks

Fábio Santos

18 Sep 2012 — 6:22pm

Alt / Opt - Drag... Can something like this be done in Illustrator?

19 Sep 2012 — 2:12pm

Quick name the nine dimensions of space...

READY....

GOOOOO!!!!

19 Sep 2012 — 8:58pm

Up, in, and across, and around and through each of them. Or you can go with what that cartoon about 10 dimensional spacetime said.

20 Sep 2012 — 11:41am

I thought it went:

width

height

depth

time

something

something

something

something

something

20 Sep 2012 — 11:49am

No, no. It's

width

height

depth

time

something

something

something

something

darkside

20 Sep 2012 — 12:08pm

LOL!

20 Sep 2012 — 12:09pm

that cartoon I mentioned: http://www.tenthdimension.com/medialinks.php

20 Sep 2012 — 3:21pm

The four somethings in Hudson's list are almost certainly fandom, magic, strangeness and frink.

22 Sep 2012 — 4:32am

Pathetic

23 Sep 2012 — 4:04am

I don't get the feeling that's on the list.

12 Oct 2012 — 10:03am

Ahem, ahem... Going back to the actual topic:

That's because quadratic Bézier curves can only represent segments of a parabola. And only very few curves are parabola, i.e. this is a clear restriction. On the other hand, this is why people may prefer it; parabolæ are per se smooth, "good looking" curves. As there is only one control point, it's seams easy to control the tension/smoothness of the curve, because point with greatest tension (the vertex of the corresponding parabola) seems to follow the control point.

To understand how this point of maximal tension relates to the position of the control point I have written a simple applet which can be found here (requires

Wolfram CDF Player, it's worth it). In addition, I plotted the focus and directrix of the parabola. To explain these terms, loosely speaking, the focus plays the same role in a parabola as the center does in a circle. That means, the further this point is from the curve, the "opener" it will look. The directrix is the line where the parabola "sits". More than that is not necessary to understand how a quadratic Bézier curve works. Play around and see what you find out.*At this point, my last question becomes interesting. "Well, what if you could change the flattness of that quardatic Bézier without changing it's direction?". That's just what so-called

rationalBezier curves provide. The result is indeed a much more dynamic curve, which can not only represent parts of a parabola, but also of ellipses and hyperbolas. All three are per definition smooth looking curves; they do not contain any inflection points. In other words, none of them is distortable into a bumpy curve. Ellipses, parabolas and hyperbolas together make up the class of theconic sections, the class of utopic curves [angelic voices]. The applet from the link above also implements a rational quadratic Bézier, just change the parameter ω with the slider.So I asked myself why, for the sake of design, has this not been done before. One answer is that it's unpractical to have an extra parameter ω, which should first easily modifiable. The second, conclusive answer is earned by asking another question. What can cubic Béziers do in comparison to quadratic Béziers? I'll spoil it. Every quadratic Bezier curve can be exactly reproduced by a cubic bezier curve. You just have to place the control points of the cubic curve ⅔ along the "handles" of the quadratic. Not only that, but also every rational quadratic curve as described above can be drawn by a cubic Bézier, by changing the ratio in which the handle points "cut" the control polygon of the quadratic Bézier. For example, to draw an elliptic curve you would place the handle points less than ⅔ the distance to the (imaginary) intersection of the two handle lines. To draw an hyperbolic curve you would choose a distance greater than ⅔ (but please still less than the intersection). I have written another applet to illustrate this idea as well.

I hope that settles the conflicting speculations of Hrant and David somehow, which I wished to interrupt since the beginning but hadn't had time :-)

Summary and some more ideas:

________

* Note: the bending energy is a measure of tension.

** Actually, if you go further and create a loop, the inflection points disappear, but recurving serpentines are not what we're looking for.

12 Oct 2012 — 10:16am

It's an interesting -and to me valid- point that the simpler the curve the easier it is to make-look-nice; and I think most people feel that the simplest curve of all, the straight line :-) is the nicest looking one!

However, and without trying to agree with David :-) I would say that:

- Nice-to-look at does not equate to easy-to-make.

- It takes more effort to make a cubic act like a quadratic, so why bother?

- What about ensuring smoothness across segments? This is especially true of quadratics, where avoiding inflection points simply means those points could/would be between segments.

Nice of you to make those applets - I'll try to install the player to see.

hhp

14 Oct 2012 — 6:31pm

good notes, Lex, and sexy looking a for your avatar.

I thought that converting from quadratic to regular bezier could be done flawlessly, although this doesn't always seem to be the case in fontlab.

Techniques used to build something don't have to be the techniques with which it is displayed to the masses, by which I mean building in quadratic and publishing in bezier shouldn't even be something we have to think about. Instead we should just be able to choose whatever type of curves we want, from bezier to quadratic to spiro, or even raster based "paint" and not to worry about the translation in to whatever format.

***

An interesting blend of regular bezier and quadratic is to edit a bezier curve, not by moving the control points, but by grabbing the curve itself and moving it around.

***

I don't think that straight lines will ever surpass curves in terms of beauty, but I have been exploring of late how

smoothstraight lines can look connected together perhaps more obtusely than what one immediately thinks of.18 Oct 2012 — 5:55am

"...most people feel that the simplest curve of all, the straight line [...] is the nicest looking..."

uhhu, and the nicest day's the darkest night, and the wettest ocean the driest desert and the highest mountain the deepest valley... in Hrantasia ;)

"You just have to place the control points of the cubic curve ⅔ along the "handles" of the quadratic."

This has been clear since the first quadratic drawing tool. 1000's of fonts have been drawn using these tools.

Am I reading the dates of these posts wrong?

18 Oct 2012 — 7:07am

You're right - I was being too Modernist. In fact we Hrantistanis try to avoid those Hrantasian flatlanders.

hhp