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I find that a very interesting remark.
It is true, provided that the on-curve node is exactly halfway between two off-curve nodes.
My way of looking at the design of a contour is mathematical. I think in terms of the contour being made up of sections joined together. Each section is one of two types.
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Two on-curve points with no other points between them.
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Two on-curve points with one, and only one, off-curve point between them.
Any other sequence of on-curve and off-curve points is resolvable into two or more sections of the kinds mentioned above. This is by placing an on-curve point midway between each sequential pair of off-curve points.
For me, having the on-curve points included is important, because they show the start and finish of each section of the contour. Sometimes I only add on-curve points after I have placed off-curve points and observed the shape of the curve.
When a section of a contour has two on-curve points with an off-curve point between them, I like the idea that the section of contour starts at one on-curve point and at that point starts to move towards the off-curve point, then gradually changes direction so that when it reaches the second on-curve point, the direction of travel is as if it came from the off-curve point.
I was under the impression that having two off-curve points in sequence was almost an error situation, with the meaning of inserting an on-curve point midway between them is carried out by a font rendering system so as to put matters right prior to rendering.
However, following Bhikkhu Pesala’s post and thinking about it, I suppose that adding an on-curve point between two off-curve points is not necessary and not doing so could make the font file smaller in terms of bytes.
I suppose that my thinking about font contours from a mathematical standpoint perhaps makes me regard the inclusion of an on-curve point between two off-curve points as desirable. However, maybe that is just a personal aesthetic choice because of my interest in mathematics.
William Overington
11 February 2008
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An S curve would be two, or more, sections. For example, two sections each of type 2.
Delete the on-curve point that is in the middle and then add an on-curve point midway between the two off-curve points.
Yes. An on-curve point midway between two off-curve points makes no difference to the shape of the curve. I just like to have the point explicitly present in the contour.
William Overington
12 February 2008
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Indeed.
My first complete alphabet font was Quest text. The latest version is available from the following link.
http://www.users.globalnet.co.uk/~ngo/QUESTTXT.TTF
The lowercase e includes many features of the design of the font. I used a single off-curve point for the curved corners, with on-curve points 128 font units from the off-curve point.
A later font is Eutopian Architecture.
http://www.users.globalnet.co.uk/~ngo/EUTOARCH.TTF
Consider please the lowercase o.
Here I use two off-curve points in order to produce the curve of the quadrant.
I chose the positions of the off-curve points by calculation.
Consider please the upper left quadrant of the outer contour.
The desire is to make a curve approximating to a quadrant of a circle to go from (206, 640) to (896, 1330), where the centre of the circle is at (896, 640).
The radius of the circle can be calculated as 690 font units (=896 - 206).
Using the square root of 2 along the way, yet using only whole number final results, the on-curve point can be calculated to be at (408, 1128).
The tangent line through that point is at 45 degrees to the horizontal and the two off-curve points are where the tangent line meets the vertical and horizontal bounding lines of the circle. The quadrant of the circle is approximated by two sections of contour. The whole circle is approximated by eight sections of contour.
William Overington
12 February 2008
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Thanks Bhikkhu Pesala, the above image posted Tue Feb 12, 2008 3:34 am should be made part of the manual!
I hope Erwin is able to implement a button to convert a glyph into a background image so we can be sure to maintain the original shape while removing all those unnecessary points.
As you mentioned, removing them makes future adjustments far easier and simpler.
Removing them also makes the font file much smaller.
As fast as computers are, removing them also eases the job of the rasterizer for both monitor and printer.
As Einstein once remarked,
“Make things as simple as possible, but not simpler.”
I feel that knowledge is important and enabling.
It is interesting that the circle which can be inserted from the Drawing toolbar does indeed place the off-curve points as I outlined when I mentioned the off-curve points in the lowercase o of my Eutopian Architecture font. The 83 font unit rise in a 200 font unit high quadrant is indeed related to the square root of 2.
The Samples toolbar provides five glyphs which each contain one or more circles. Some of those circles are produced using eight off-curve points to each circle and some are produced using twelve off-curve points to each circle. One glyph contains examples of both types.
William Overington
12 February 2008
Thanks Dave. I think you’re probably right. The manual could use some practical tips like this. For now, I have added it to my Smooth Curves tutorial.
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Well, I did use mathematics to calculate the places to put the off-curve nodes when I was trying to produce an approximation to a circle when designing the lowercase o in the Eutopian Architecture font. The calculation was essentially done by hand, though using Microsoft Calculator in View Scientific mode to carry out the parts of the calculation.
Some readers might like a copy of part of my transcript file from when I produced the disc glyph in the Stardisc font. I wanted a disc which had a radius of 1024 font units, centred on (1024, 1024).
For the disc, the offset of the off-curve point is 1024 * (root2 - 1)
root2 is 1.4142135623730950488016887242097
root2 - 1 is 0.4142135623730950488016887242097
1024 * (root2 - 1) is 424.15468787004932997292925359073, so use 424
1024 - 424 is 600
1024 + 424 is 1448
The font is available as follows.
http://www.users.globalnet.co.uk/~ngo/STARDISC.TTF
The Stardisc font has its own thread in the Gallery Forum of this webspace.
http://forum.high-logic.com:9080/t/stardisc/1610/1
In that thread I wrote as follows.
I calculated the points for the star, using the idea that the star is inscribed within a construction circle such that the top point of the star is at 2048 font units vertically and that the two lowest points of the star are at 0 font units vertically.
I appreciate that font designing is a mixture of the mathematical placement of points and the artistic placement of points and that various people have various views about using mathematics when designing fonts. I am simply mentioning that I have found mathematics to be useful in designing individual glyphs.
William Overington
13 February 2008