Re: A single font can display ANY alphabet, pictograph, or rune

Hi, all, especially Gary Shannon.

I have been thinking of something like this for quite some time now;
you've made more progress than I, especially with your alphabetic-
codings of the various strokes.

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[WHAT I DID:  (skip to "curves" if you want totally new ideas)]

I based my ideas on 1) the LED outputs of elevators in Vancouver
during the World's Fair there and 2) the "imaginary lines" frame
provided for teaching Chinese script.

I vacillated at first between what would show up on your
illustrations as a two-cell-by-two-cell system and one that would
show up as a three-cell-by-three-cell system.

I also vacillated at first over whether or not to include curved
strokes, and how many kinds of curves to include, if so.

I eventually decided on a two-cell-by-two-cell system with all
straight strokes.

The reason was that I get 28 different strokes this way; and 2^28 =
256 * 1,048,576 is way, way more glyphs than I would need in any one
language.

I began by naming the points within the character.  I used the number-
key pad on my computer's keyboard.

7 8 9
4 5 6
1 2 3

The strokes I named by just concatenating their endpoints.

Six Horizontal: 12, 23, 45, 56, 78, 89
7-8-9
4-5-6
1-2-3
(your system does not require 23, 56, and 89 to be listed separately)

Six Vertical: 14, 25, 36, 47, 58, 69
7 8 9
| | |
4 5 6
| | |
1 2 3
(your system does not require 25, 36, 58, and 69 to be listed
separately)

Eight short 45-degree diagonals:
15, 26, 48, 59
7 8 9
./ /
4 5 6
./ /
1 2 3
(your system does not require 26 and 59 to be listed separately)

and 24, 35, 57, 68
7 8 9
.\ \
4 5 6
.\ \
1 2 3
(your system does not require 35 and 68 to be listed separately).

And eight longer obliques, not 45-degrees from horizontal or vertical:
16, 49;
18, 29 (your system does not require 29 to be listed separately);
27, 38 (your system does not require 38 to be listed separately);
34, 67.

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[CHINESE IMAGINARY LINES]

[NOTE: the frame on which a Chinese character is drawn, consists of
the six horizontal (12, 23, 45, 56, 78, 89)
and six vertical (14, 25, 36, 47, 58, 69)
segments,
the four diagonal segments 15, 35, 57, and 59,
the four lines 19, 28, 37, and 46,
and the eight triangles 125, 145, 235, 356, 457, 569, 578, and 589.]

7-8-9
|\|/|
4-5-6
|/|\|
1-2-3

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[CURVES]

Including curved strokes is something I did indeed give a lot of
thought to.

I came up with several different degrees of complexity.
1) arcs of circles
2) arcs of conics
3) cubic splines
4) traces of rational functions
4a) linear-fractional
4b) quadratic-fractional
4c) cubic-fractional
4d) quartic-fractional

Reasons for and against each idea:

1) Limit the kinds of curves to arcs of circles.
While the eye can detect the differnece between a curve and its
approximation built out of straight-line segments, the eye cannot
detect the difference between a curve and its approximation built out
of straight-line-segments together with circular-arcs.
An arc of a circle can be specified by its endpoints plus any other
point on it; or, its endpoints plus the center of the circle of which
it is an arc (provided all arcs are less than semi-circles).

2) Include arcs of any conic section; ellipses, parabolae, and
hyperbolae, as well as circles.
The aviation industry's cybernetically-controlled machining for
producing airplane-parts, requires the ability to cut any arc of any
conic; but any more-complicated curve can be, and is, approximated by
stringing together arcs of conics (and straight lines).
A conic is fully specified by any five points on it; an arc of a
conic could be specified by picking its endpoints as two of those
five points.

3) Include cubic splines.
The yacht-designing and yacht-building industry approximates all
curves by "cubic splines".
A particular piece of the approximation can be specified by
specifying its endpoints and its tangents at those endpoints, and
requiring it to have the smallest possible curvature consistent with
those requirements.
This has the advantage that, given any sequence of points that must
appear on the curve, and directions of tangents that the curve must
have at those points, one can approximate the curve by a smallest-
curvature approximation that in fact goes through exactly those
points with exactly those tangents.

4) Traces of "rational functions".

In projective geometry and algebraic geometry, a curve in n-space
(n=2 for our purposes) is the trace of a pair of "rational
functions", where by "rational function" is meant a fraction whose
numerator and denominator are both polynomials.

A curve in the plane would be, for instance, the set of all points
(x,y) where (x=P(t)/R(t), y=Q(t)/R(t)), allowing t to vary throughout
the set of all real numbers.  The functions P, Q, and R, are
polynomials in t.

An arc of this curve can be selected by choosing start- and end-
values for t; for example, let t go from 0 to 1.

These curves have the advantage that any natural smooth motion of the
hand can actually be closely approximated by one of these.  Some hand-
motions would actually fall into category 4c below, but not 4b;
others would fall in 4b, but not 4a; etc.

4a) If P, Q, and R are all at-most first-degree in t, we get one set
of curves, including all straight lines, and (if I remember aright)
some conics.
(x=(At+B)/(Et+F), y=(Ct+D)/(Et+F)), for t running from (say) g to
(say) h, inclusive.
By quantizing and restricting the allowed values of A, B, C, D, E, F,
g, and h; we can select some finite set of curves.

4b) If P, Q, and R are all 2nd-degree or lower in t, we get all the
kinds of curves from 4a), plus all conics, plus perhaps some others.
(x=(Att+Bt+C)/(Gtt+Ht+I), y=(Dtt+Et+F)/(Gtt+Ht+I)), j <= t <= k.

4c) Let P, Q, and R be up to 3rd-degree in t; we get all of 4b), plus
all cubics (plus, afaik, a few others.)

4d) In reading about curves in algebraic geometry, I occasionally
have come across quartic curves; especially in 3-space as the
intersections of two quadric surfaces.  I have not come across higher-
degree curves much; little enough has been written about higher-
degree curves that encyclopedias merely refer to them, rather than
discuss them.  As for surfaces and 3-folds etc., they are generally
of degree 3 or less in these articles.
So, I figured, if there was any need to get more precise than cubic
splines, it would surely stop at quartics.
That would be if P and Q and R were all fourth-degree or smaller in t.

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[MY PERSONAL RECOMMENDATION]

I recommend including circular arcs, but not any more-complicated
elementary curve.

In my experience the eye can detect that a "curve" has been
approximated by several straight-line segments, even if they are
somewhat small; but cannot locally detect the difference between,
say, an arc of a circle vs. an arc of an ellipse vs. an arc of a
parabola vs. an arc of a hyperbola, even if they are somewhat large.

The eye has an even harder time telling an arc of a conic from an arc
of a cubic from an arc of a quartic.

Stringing together line-segments and circle-arcs to approximate a
curve will, IMO, probably look like a smooth curve, at the size of a
glyph.

If you are more interested in approximating what the hand does than
what the eye sees, you might want to go with 4c) instead.

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[NUMBER OF CELLS PER GLYPH, RISERS, AND DESCENDERS]

To learn cursive in American schools, students are given "imaginary-
line" paper.

On this paper, the "non-imaginary" lines are bold.  The space between
the "non-imaginary" lines is cut in half by a paler solid line; each
half is again halved by a yet paler dotted line.

Well-written risers never rise above the bold line above the "base"
line "on" which the student is writing; well-written descenders never
go below the pale solid half-way line below the base line.

Most

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Sorry, have to finish later.

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I like, and am inspired by, your work.

Thanks.

Tom H.C. in MI

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