the great majority of people seem to prefer cubic béziers

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 [[http://demonstrations.wolfram.com/FocusAndDirectrixInAQuadraticBezierCurve/|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 rational Bezier 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 the conic 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 [[http://demonstrations.wolfram.com/RationalCubicBezierCurves/|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:

Quadratics Bézier curves can only represent segments of a parabola.

Every quadratic Bézier curve can be reconstructed from a cubic Bézier curve by placing the control points ⅔ of the distance to the (imaginary) intersection of both handle-lines.

If you place the control points at a smaller radio, you get an elliptic curve.

If you place them at a greater ratio, you get a hyperbolic curve.

To know if both control points sit at the same ratio, just check if the imaginary line that passes through the control points is parallel to the imaginary line that passes through the anchor points.

If your "control lines" intersect at at least one your handles, your curve has an inflection point, which you want to avoid.** In every point where the direction (concavity) of the curve changes, there should be a new anchor point.

Separating the control points (maintaining the length of the handles) means making the curve more elliptic. Bringing them closer makes it more hyperbolic.

Not because there's a specific constellation of the Bézier points for obtaining specially "simple" curves does it necessarily mean one should develop such a drawing habit. But if we know beforehand we're going after an outline of that nature, we have such tools at our disposition.

________
* 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.

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.

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 smooth straight lines can look connected together perhaps more obtusely than what one immediately thinks of.

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

Quick name the nine dimensions of space...

READY....

GOOOOO!!!!

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

I thought it went:

width

height

depth

time

something

something

something

something

something

No, no. It's

width

height

depth

time

something

something

something

something

darkside

LOL!

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

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

Pathetic

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

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 [[http://demonstrations.wolfram.com/FocusAndDirectrixInAQuadraticBezierCurve/|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 [[http://demonstrations.wolfram.com/RationalCubicBezierCurves/|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.

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

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."...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?

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

hhp